AI App UI Design

AI App UI Design — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Automatic summarization

  Automatic summarization is the process of shortening a set of data computationally, to create a subset (a summary) that represents the most important or relevant information within the original content. Artificial intelligence (AI) algorithms are commonly developed and employed to achieve this, specialized for different types of data. Text summarization is usually implemented by natural language processing methods, designed to locate the most informative sentences in a given document. On the other hand, visual content can be summarized using computer vision algorithms. Image summarization is the subject of ongoing research; existing approaches typically attempt to display the most representative images from a given image collection, or generate a video that only includes the most important content from the entire collection. Video summarization algorithms identify and extract from the original video content the most important frames (key-frames), and/or the most important video segments (key-shots), normally in a temporally ordered fashion. Video summaries simply retain a carefully selected subset of the original video frames and, therefore, are not identical to the output of video synopsis algorithms, where new video frames are being synthesized based on the original video content. == Commercial products == In 2022 Google Docs released an automatic summarization feature. == Approaches == There are two general approaches to automatic summarization: extraction and abstraction. === Extraction-based summarization === Here, content is extracted from the original data, but the extracted content is not modified in any way. Examples of extracted content include key-phrases that can be used to "tag" or index a text document, or key sentences (including headings) that collectively comprise an abstract, and representative images or video segments, as stated above. For text, extraction is analogous to the process of skimming, where the summary (if available), headings and subheadings, figures, the first and last paragraphs of a section, and optionally the first and last sentences in a paragraph are read before one chooses to read the entire document in detail. Other examples of extraction that include key sequences of text in terms of clinical relevance (including patient/problem, intervention, and outcome). === Abstractive-based summarization === Abstractive summarization methods generate new text that did not exist in the original text. This has been applied mainly for text. Abstractive methods build an internal semantic representation of the original content (often called a language model), and then use this representation to create a summary that is closer to what a human might express. Abstraction may transform the extracted content by paraphrasing sections of the source document, to condense a text more strongly than extraction. Such transformation, however, is computationally much more challenging than extraction, involving both natural language processing and often a deep understanding of the domain of the original text in cases where the original document relates to a special field of knowledge. "Paraphrasing" is even more difficult to apply to images and videos, which is why most summarization systems are extractive. === Aided summarization === Approaches aimed at higher summarization quality rely on combined software and human effort. In Machine Aided Human Summarization, extractive techniques highlight candidate passages for inclusion (to which the human adds or removes text). In Human Aided Machine Summarization, a human post-processes software output, in the same way that one edits the output of automatic translation by Google Translate. == Applications and systems for summarization == There are broadly two types of extractive summarization tasks depending on what the summarization program focuses on. The first is generic summarization, which focuses on obtaining a generic summary or abstract of the collection (whether documents, or sets of images, or videos, news stories etc.). The second is query relevant summarization, sometimes called query-based summarization, which summarizes objects specific to a query. Summarization systems are able to create both query relevant text summaries and generic machine-generated summaries depending on what the user needs. An example of a summarization problem is document summarization, which attempts to automatically produce an abstract from a given document. Sometimes one might be interested in generating a summary from a single source document, while others can use multiple source documents (for example, a cluster of articles on the same topic). This problem is called multi-document summarization. A related application is summarizing news articles. Imagine a system, which automatically pulls together news articles on a given topic (from the web), and concisely represents the latest news as a summary. Image collection summarization is another application example of automatic summarization. It consists in selecting a representative set of images from a larger set of images. A summary in this context is useful to show the most representative images of results in an image collection exploration system. Video summarization is a related domain, where the system automatically creates a trailer of a long video. This also has applications in consumer or personal videos, where one might want to skip the boring or repetitive actions. Similarly, in surveillance videos, one would want to extract important and suspicious activity, while ignoring all the boring and redundant frames captured. At a very high level, summarization algorithms try to find subsets of objects (like set of sentences, or a set of images), which cover information of the entire set. This is also called the core-set. These algorithms model notions like diversity, coverage, information and representativeness of the summary. Query based summarization techniques, additionally model for relevance of the summary with the query. Some techniques and algorithms which naturally model summarization problems are TextRank and PageRank, Submodular set function, Determinantal point process, maximal marginal relevance (MMR) etc. === Keyphrase extraction === The task is the following. You are given a piece of text, such as a journal article, and you must produce a list of keywords or key[phrase]s that capture the primary topics discussed in the text. In the case of research articles, many authors provide manually assigned keywords, but most text lacks pre-existing keyphrases. For example, news articles rarely have keyphrases attached, but it would be useful to be able to automatically do so for a number of applications discussed below. Consider the example text from a news article: "The Army Corps of Engineers, rushing to meet President Bush's promise to protect New Orleans by the start of the 2006 hurricane season, installed defective flood-control pumps last year despite warnings from its own expert that the equipment would fail during a storm, according to documents obtained by The Associated Press". A keyphrase extractor might select "Army Corps of Engineers", "President Bush", "New Orleans", and "defective flood-control pumps" as keyphrases. These are pulled directly from the text. In contrast, an abstractive keyphrase system would somehow internalize the content and generate keyphrases that do not appear in the text, but more closely resemble what a human might produce, such as "political negligence" or "inadequate protection from floods". Abstraction requires a deep understanding of the text, which makes it difficult for a computer system. Keyphrases have many applications. They can enable document browsing by providing a short summary, improve information retrieval (if documents have keyphrases assigned, a user could search by keyphrase to produce more reliable hits than a full-text search), and be employed in generating index entries for a large text corpus. Depending on the different literature and the definition of key terms, words or phrases, keyword extraction is a highly related theme. ==== Supervised learning approaches ==== Beginning with the work of Turney, many researchers have approached keyphrase extraction as a supervised machine learning problem. Given a document, we construct an example for each unigram, bigram, and trigram found in the text (though other text units are also possible, as discussed below). We then compute various features describing each example (e.g., does the phrase begin with an upper-case letter?). We assume there are known keyphrases available for a set of training documents. Using the known keyphrases, we can assign positive or negative labels to the examples. Then we learn a classifier that can discriminate between positive and negative examples as a function of the features. Some classifiers make a binary classification for a test example, while others assign a probability of being a keyphrase. For ins

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  Latent and observable variables

  In statistics, latent variables (from Latin: present participle of lateo 'lie hidden') are variables that can only be inferred indirectly through a mathematical model from other observable variables that can be directly observed or measured. Such latent variable models are used in many disciplines, including engineering, medicine, ecology, physics, machine learning/artificial intelligence, natural language processing, bioinformatics, chemometrics, demography, economics, management, political science, psychology and the social sciences. Latent variables may correspond to aspects of physical reality. These could in principle be measured, but may not be for practical reasons. Among the earliest expressions of this idea is Francis Bacon's polemic the Novum Organum, itself a challenge to the more traditional logic expressed in Aristotle's Organon: But the latent process of which we speak, is far from being obvious to men’s minds, beset as they now are. For we mean not the measures, symptoms, or degrees of any process which can be exhibited in the bodies themselves, but simply a continued process, which, for the most part, escapes the observation of the senses. In this situation, the term hidden variables is commonly used, reflecting the fact that the variables are meaningful, but not observable. Other latent variables correspond to abstract concepts, like categories, behavioral or mental states, or data structures. The terms hypothetical variables or hypothetical constructs may be used in these situations. The use of latent variables can serve to reduce the dimensionality of data. Many observable variables can be aggregated in a model to represent an underlying concept, making it easier to understand the data. In this sense, they serve a function similar to that of scientific theories. At the same time, latent variables link observable "sub-symbolic" data in the real world to symbolic data in the modeled world. == Examples == === Psychology === Latent variables, as created by factor analytic methods, generally represent "shared" variance, or the degree to which variables "move" together. Variables that have no correlation cannot result in a latent construct based on the common factor model. The "Big Five personality traits" have been inferred using factor analysis. extraversion spatial ability wisdom: “Two of the more predominant means of assessing wisdom include wisdom-related performance and latent variable measures.” Spearman's g, or the general intelligence factor in psychometrics === Economics === Examples of latent variables from the field of economics include quality of life, business confidence, morale, happiness and conservatism: these are all variables which cannot be measured directly. However, by linking these latent variables to other, observable variables, the values of the latent variables can be inferred from measurements of the observable variables. Quality of life is a latent variable which cannot be measured directly, so observable variables are used to infer quality of life. Observable variables to measure quality of life include wealth, employment, environment, physical and mental health, education, recreation and leisure time, and social belonging. === Medicine === Latent-variable methodology is used in many branches of medicine. A class of problems that naturally lend themselves to latent variables approaches are longitudinal studies where the time scale (e.g. age of participant or time since study baseline) is not synchronized with the trait being studied. For such studies, an unobserved time scale that is synchronized with the trait being studied can be modeled as a transformation of the observed time scale using latent variables. Examples of this include disease progression modeling and modeling of growth (see box). == Inferring latent variables == There exists a range of different model classes and methodology that make use of latent variables and allow inference in the presence of latent variables. Models include: linear mixed-effects models and nonlinear mixed-effects models Hidden Markov models Factor analysis Item response theory Analysis and inference methods include: Principal component analysis Instrumented principal component analysis Partial least squares regression Latent semantic analysis and probabilistic latent semantic analysis EM algorithms Metropolis–Hastings algorithm === Bayesian algorithms and methods === Bayesian statistics is often used for inferring latent variables. Latent Dirichlet allocation The Chinese restaurant process is often used to provide a prior distribution over assignments of objects to latent categories. The Indian buffet process is often used to provide a prior distribution over assignments of latent binary features to objects.

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  Information gain ratio

  In decision tree learning, information gain ratio is a ratio of information gain to the intrinsic information. It was proposed by Ross Quinlan, to reduce a bias towards multi-valued attributes by taking the number and size of branches into account when choosing an attribute. Information gain is also known as mutual information. == Information gain calculation == Information gain is the reduction in entropy produced from partitioning a set with attributes a {\displaystyle a} and finding the optimal candidate that produces the highest value: IG ( T , a ) = H ( T ) − H ( T | a ) , {\displaystyle {\text{IG}}(T,a)=\mathrm {H} {(T)}-\mathrm {H} {(T|a)},} where T {\displaystyle T} is a random variable and H ( T | a ) {\displaystyle \mathrm {H} {(T|a)}} is the entropy of T {\displaystyle T} given the value of attribute a {\displaystyle a} . The information gain is equal to the total entropy for an attribute if for each of the attribute values a unique classification can be made for the result attribute. In this case the relative entropies subtracted from the total entropy are 0. == Split information calculation == The split information value for a test is defined as follows: SplitInformation ( X ) = − ∑ i = 1 n N ( x i ) N ( x ) ∗ log ⁡ 2 N ( x i ) N ( x ) {\displaystyle {\text{SplitInformation}}(X)=-\sum _{i=1}^{n}{{\frac {\mathrm {N} (x_{i})}{\mathrm {N} (x)}}\log {_{2}}{\frac {\mathrm {N} (x_{i})}{\mathrm {N} (x)}}}} where X {\displaystyle X} is a discrete random variable with possible values x 1 , x 2 , . . . , x i {\displaystyle {x_{1},x_{2},...,x_{i}}} and N ( x i ) {\displaystyle N(x_{i})} being the number of times that x i {\displaystyle x_{i}} occurs divided by the total count of events N ( x ) {\displaystyle N(x)} where x {\displaystyle x} is the set of events. The split information value is a positive number that describes the potential worth of splitting a branch from a node. This in turn is the intrinsic value that the random variable possesses and will be used to remove the bias in the information gain ratio calculation. == Information gain ratio calculation == The information gain ratio is the ratio between the information gain and the split information value: IGR ( T , a ) = IG ( T , a ) / SplitInformation ( T ) {\displaystyle {\text{IGR}}(T,a)={\text{IG}}(T,a)/{\text{SplitInformation}}(T)} IGR ( T , a ) = − ∑ i = 1 n P ( T ) log ⁡ P ( T ) − ( − ∑ i = 1 n P ( T | a ) log ⁡ P ( T | a ) ) − ∑ i = 1 n N ( t i ) N ( t ) ∗ log ⁡ 2 N ( t i ) N ( t ) {\displaystyle {\text{IGR}}(T,a)={\frac {-\sum _{i=1}^{n}{\mathrm {P} (T)\log \mathrm {P} (T)}-(-\sum _{i=1}^{n}{\mathrm {P} (T|a)\log \mathrm {P} (T|a)})}{-\sum _{i=1}^{n}{{\frac {\mathrm {N} (t_{i})}{\mathrm {N} (t)}}\log {_{2}}{\frac {\mathrm {N} (t_{i})}{\mathrm {N} (t)}}}}}} == Example == Using weather data published by Fordham University, the table was created below: Using the table above, one can find the entropy, information gain, split information, and information gain ratio for each variable (outlook, temperature, humidity, and wind). These calculations are shown in the tables below: Using the above tables, one can deduce that Outlook has the highest information gain ratio. Next, one must find the statistics for the sub-groups of the Outlook variable (sunny, overcast, and rainy), for this example one will only build the sunny branch (as shown in the table below): One can find the following statistics for the other variables (temperature, humidity, and wind) to see which have the greatest effect on the sunny element of the outlook variable: Humidity was found to have the highest information gain ratio. One will repeat the same steps as before and find the statistics for the events of the Humidity variable (high and normal): Since the play values are either all "No" or "Yes", the information gain ratio value will be equal to 1. Also, now that one has reached the end of the variable chain with Wind being the last variable left, they can build an entire root to leaf node branch line of a decision tree. Once finished with reaching this leaf node, one would follow the same procedure for the rest of the elements that have yet to be split in the decision tree. This set of data was relatively small, however, if a larger set was used, the advantages of using the information gain ratio as the splitting factor of a decision tree can be seen more. == Advantages == Information gain ratio biases the decision tree against considering attributes with a large number of distinct values. For example, suppose that we are building a decision tree for some data describing a business's customers. Information gain ratio is used to decide which of the attributes are the most relevant. These will be tested near the root of the tree. One of the input attributes might be the customer's telephone number. This attribute has a high information gain, because it uniquely identifies each customer. Due to its high amount of distinct values, this will not be chosen to be tested near the root. == Disadvantages == Although information gain ratio solves the key problem of information gain, it creates another problem. If one is considering an amount of attributes that have a high number of distinct values, these will never be above one that has a lower number of distinct values. == Difference from information gain == Information gain's shortcoming is created by not providing a numerical difference between attributes with high distinct values from those that have less. Example: Suppose that we are building a decision tree for some data describing a business's customers. Information gain is often used to decide which of the attributes are the most relevant, so they can be tested near the root of the tree. One of the input attributes might be the customer's credit card number. This attribute has a high information gain, because it uniquely identifies each customer, but we do not want to include it in the decision tree: deciding how to treat a customer based on their credit card number is unlikely to generalize to customers we haven't seen before. Information gain ratio's strength is that it has a bias towards the attributes with the lower number of distinct values. Below is a table describing the differences of information gain and information gain ratio when put in certain scenarios.

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  Transkribus

  Transkribus is a platform for the text recognition, image analysis and structure recognition of historical documents. The platform was created in the context of the two EU projects "tranScriptorium" (2013–2015) and "READ" (Recognition and Enrichment of Archival Documents – 2016–2019). It was developed by the University of Innsbruck. Since July 1, 2019 the platform has been directed and further developed by the READ-COOP, a non-profit cooperative. The platform integrates tools developed by research groups throughout Europe, including the Pattern Recognition and Human Language Technology (PRHLT) group of the Technical University of Valencia and the Computational Intelligence Technology Lab (CITlab) group of University of Rostock. Comparable programs that offer similar functions are eScriptorium and OCR4All.

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  Underwater computer vision

  Underwater computer vision is a subfield of computer vision. In recent years, with the development of underwater vehicles ( ROV, AUV, gliders), the need to be able to record and process huge amounts of information has become increasingly important. Applications range from inspection of underwater structures for the offshore industry to the identification and counting of fishes for biological research. However, no matter how big the impact of this technology can be to industry and research, it still is in a very early stage of development compared to traditional computer vision. One reason for this is that, the moment the camera goes into the water, a whole new set of challenges appear. On one hand, cameras have to be made waterproof, marine corrosion deteriorates materials quickly and access and modifications to experimental setups are costly, both in time and resources. On the other hand, the physical properties of the water make light behave differently, changing the appearance of a same object with variations of depth, organic material, currents, temperature etc. == Applications == Seafloor survey Vehicle navigation and positioning Biological monitoring {possibly aquatic biomonitoring) Video mosaics as visual navigation maps Submarine pipeline inspection Wreckage visualization Maintenance of underwater structures Drowning detection systems == Medium differences == === Illumination === In air, light comes from the whole hemisphere on cloudy days, and is dominated by the sun. In water direct lighting comes from a cone about 96° wide above the scene. This phenomenon is called Snell's window. Artificial lighting can be used where natural light levels are insufficient and where the light path is too long to produce acceptable colour, as the loss of colour is a function of the total distance through water from the source to the camera lens port. === Light attenuation === Unlike air, water attenuates light exponentially. This results in hazy images with very low contrast. The main reasons for light attenuation are light absorption (where energy is removed from the light) and light scattering, by which the direction of light is changed. Light scattering can further be divided into forward scattering, which results in an increased blurriness and backward scattering that limits the contrast and is responsible for the characteristic veil of underwater images. Both scattering and attenuation are heavily influenced by the amount of organic matter dissolved or suspended in the water. Light attenuation in water is also a function of the wavelength. This means that different colours are attenuated at different rates, leading to colour degradation.with depth and distance. Red and orange light are attenuated faster, followed by yellows and greens. Blue is the least attenuated visible wavelength. === Artificial lighting === == Challenges == In high level computer vision, human structures are frequently used as image features for image matching in different applications. However, the sea bottom lacks such features, making it hard to find correspondences in two images. In order to be able to use a camera in the water, a watertight housing is required. However, refraction will happen at the water-glass and glass-air interface due to differences in density of the materials. This has the effect of introducing a non-linear image deformation. The motion of the vehicle presents another special challenge. Underwater vehicles are constantly moving due to currents and other phenomena. This introduces another uncertainty to algorithms, where small motions may appear in all directions. This can be specially important for video tracking. In order to reduce this problem image stabilization algorithms may be applied. == Relevant technology == === Image restoration === Image restoration< techniques are intended to model the degradation process and then invert it, obtaining the new image after solving. It is generally a complex approach that requires plenty of parameters that vary a lot between different water conditions. === Image enhancement === Image enhancement only tries to provide a visually more appealing image without taking the physical image formation process into account. These methods are usually simpler and less computational intensive. === Color correction === Various algorithms exist that perform automatic color correction. The UCM (Unsupervised Color Correction Method), for example, does this in the following steps: It firstly reduces the color cast by equalizing the color values. Then it enhances contrast by stretching the red histogram towards the maximum and finally saturation and intensity components are optimized. == Underwater stereo vision == It is usually assumed that stereo cameras have been calibrated previously, geometrically and radiometrically. This leads to the assumption that corresponding pixels should have the same color. However this can not be guaranteed in an underwater scene, because of dispersion and backscatter. However, it is possible to digitally model this phenomenon and create a virtual image with those effects removed == Other application fields == Imaging sonars have become more and more accessible and gained resolution, delivering better images. Sidescan sonars are used to produce complete maps of regions of the sea floor stitching together sequences of sonar images. However, sonar images often lack proper contrast and are degraded by artefacts and distortions due to noise, attitude changes of the AUV/ROV carrying the sonar or non uniform beam patterns. Another common problem with sonar computer vision is the comparatively low frame rate of sonar images.

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  Vapnik–Chervonenkis theory

  Vapnik–Chervonenkis theory (also known as VC theory) was developed during 1960–1990 by Vladimir Vapnik and Alexey Chervonenkis. The theory is a form of computational learning theory, which attempts to explain the learning process from a statistical point of view. == Introduction == VC theory covers at least four parts (as explained in The Nature of Statistical Learning Theory): Theory of consistency of learning processes What are (necessary and sufficient) conditions for consistency of a learning process based on the empirical risk minimization principle? Nonasymptotic theory of the rate of convergence of learning processes How fast is the rate of convergence of the learning process? Theory of controlling the generalization ability of learning processes How can one control the rate of convergence (the generalization ability) of the learning process? Theory of constructing learning machines How can one construct algorithms that can control the generalization ability? VC Theory is a major subbranch of statistical learning theory. One of its main applications in statistical learning theory is to provide generalization conditions for learning algorithms. From this point of view, VC theory is related to stability, which is an alternative approach for characterizing generalization. In addition, VC theory and VC dimension are instrumental in the theory of empirical processes, in the case of processes indexed by VC classes. Arguably these are the most important applications of the VC theory, and are employed in proving generalization. Several techniques will be introduced that are widely used in the empirical process and VC theory. The discussion is mainly based on the book Weak Convergence and Empirical Processes: With Applications to Statistics. == Overview of VC theory in empirical processes == === Background on empirical processes === Let ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} be a measurable space. For any measure Q {\displaystyle Q} on ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} , and any measurable functions f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } , define Q f = ∫ f d Q {\displaystyle Qf=\int fdQ} Measurability issues will be ignored here, for more technical detail see. Let F {\displaystyle {\mathcal {F}}} be a class of measurable functions f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } and define: ‖ Q ‖ F = sup { | Q f | : f ∈ F } . {\displaystyle \|Q\|_{\mathcal {F}}=\sup\{\vert Qf\vert \ :\ f\in {\mathcal {F}}\}.} Let X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} be independent, identically distributed random elements of ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} . Then define the empirical measure P n = n − 1 ∑ i = 1 n δ X i , {\displaystyle \mathbb {P} _{n}=n^{-1}\sum _{i=1}^{n}\delta _{X_{i}},} where δ here stands for the Dirac measure. The empirical measure induces a map F → R {\displaystyle {\mathcal {F}}\to \mathbf {R} } given by: f ↦ P n f = 1 n ( f ( X 1 ) + . . . + f ( X n ) ) {\displaystyle f\mapsto \mathbb {P} _{n}f={\frac {1}{n}}(f(X_{1})+...+f(X_{n}))} Now suppose P is the underlying true distribution of the data, which is unknown. Empirical Processes theory aims at identifying classes F {\displaystyle {\mathcal {F}}} for which statements such as the following hold: uniform law of large numbers: ‖ P n − P ‖ F → n 0 , {\displaystyle \|\mathbb {P} _{n}-P\|_{\mathcal {F}}{\underset {n}{\to }}0,} That is, as n → ∞ {\displaystyle n\to \infty } , | 1 n ( f ( X 1 ) + . . . + f ( X n ) ) − ∫ f d P | → 0 {\displaystyle \left|{\frac {1}{n}}(f(X_{1})+...+f(X_{n}))-\int fdP\right|\to 0} uniformly for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . uniform central limit theorem: G n = n ( P n − P ) ⇝ G , in ℓ ∞ ( F ) {\displaystyle \mathbb {G} _{n}={\sqrt {n}}(\mathbb {P} _{n}-P)\rightsquigarrow \mathbb {G} ,\quad {\text{in }}\ell ^{\infty }({\mathcal {F}})} In the former case F {\displaystyle {\mathcal {F}}} is called Glivenko–Cantelli class, and in the latter case (under the assumption ∀ x , sup f ∈ F | f ( x ) − P f | < ∞ {\displaystyle \forall x,\sup \nolimits _{f\in {\mathcal {F}}}\vert f(x)-Pf\vert <\infty } ) the class F {\displaystyle {\mathcal {F}}} is called Donsker or P-Donsker. A Donsker class is Glivenko–Cantelli in probability by an application of Slutsky's theorem. These statements are true for a single f {\displaystyle f} , by standard LLN, CLT arguments under regularity conditions, and the difficulty in the Empirical Processes comes in because joint statements are being made for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . Intuitively then, the set F {\displaystyle {\mathcal {F}}} cannot be too large, and as it turns out that the geometry of F {\displaystyle {\mathcal {F}}} plays a very important role. One way of measuring how big the function set F {\displaystyle {\mathcal {F}}} is to use the so-called covering numbers. The covering number N ( ε , F , ‖ ⋅ ‖ ) {\displaystyle N(\varepsilon ,{\mathcal {F}},\|\cdot \|)} is the minimal number of balls { g : ‖ g − f ‖ < ε } {\displaystyle \{g:\|g-f\|<\varepsilon \}} needed to cover the set F {\displaystyle {\mathcal {F}}} (here it is obviously assumed that there is an underlying norm on F {\displaystyle {\mathcal {F}}} ). The entropy is the logarithm of the covering number. Two sufficient conditions are provided below, under which it can be proved that the set F {\displaystyle {\mathcal {F}}} is Glivenko–Cantelli or Donsker. A class F {\displaystyle {\mathcal {F}}} is P-Glivenko–Cantelli if it is P-measurable with envelope F such that P ∗ F < ∞ {\displaystyle P^{\ast }F<\infty } and satisfies: ∀ ε > 0 sup Q N ( ε ‖ F ‖ Q , F , L 1 ( Q ) ) < ∞ . {\displaystyle \forall \varepsilon >0\quad \sup \nolimits _{Q}N(\varepsilon \|F\|_{Q},{\mathcal {F}},L_{1}(Q))<\infty .} The next condition is a version of Dudley's theorem. If F {\displaystyle {\mathcal {F}}} is a class of functions such that ∫ 0 ∞ sup Q log ⁡ N ( ε ‖ F ‖ Q , 2 , F , L 2 ( Q ) ) d ε < ∞ {\displaystyle \int _{0}^{\infty }\sup \nolimits _{Q}{\sqrt {\log N\left(\varepsilon \|F\|_{Q,2},{\mathcal {F}},L_{2}(Q)\right)}}d\varepsilon <\infty } then F {\displaystyle {\mathcal {F}}} is P-Donsker for every probability measure P such that P ∗ F 2 < ∞ {\displaystyle P^{\ast }F^{2}<\infty } . In the last integral, the notation means ‖ f ‖ Q , 2 = ( ∫ | f | 2 d Q ) 1 2 {\displaystyle \|f\|_{Q,2}=\left(\int |f|^{2}dQ\right)^{\frac {1}{2}}} . === Symmetrization === The majority of the arguments about how to bound the empirical process rely on symmetrization, maximal and concentration inequalities, and chaining. Symmetrization is usually the first step of the proofs, and since it is used in many machine learning proofs on bounding empirical loss functions (including the proof of the VC inequality which is discussed in the next section). It is presented here: Consider the empirical process: f ↦ ( P n − P ) f = 1 n ∑ i = 1 n ( f ( X i ) − P f ) {\displaystyle f\mapsto (\mathbb {P} _{n}-P)f={\dfrac {1}{n}}\sum _{i=1}^{n}(f(X_{i})-Pf)} Turns out that there is a connection between the empirical and the following symmetrized process: f ↦ P n 0 f = 1 n ∑ i = 1 n ε i f ( X i ) {\displaystyle f\mapsto \mathbb {P} _{n}^{0}f={\dfrac {1}{n}}\sum _{i=1}^{n}\varepsilon _{i}f(X_{i})} The symmetrized process is a Rademacher process, conditionally on the data X i {\displaystyle X_{i}} . Therefore, it is a sub-Gaussian process by Hoeffding's inequality. Lemma (Symmetrization). For every nondecreasing, convex Φ: R → R and class of measurable functions F {\displaystyle {\mathcal {F}}} , E Φ ( ‖ P n − P ‖ F ) ≤ E Φ ( 2 ‖ P n 0 ‖ F ) {\displaystyle \mathbb {E} \Phi (\|\mathbb {P} _{n}-P\|_{\mathcal {F}})\leq \mathbb {E} \Phi \left(2\left\|\mathbb {P} _{n}^{0}\right\|_{\mathcal {F}}\right)} The proof of the Symmetrization lemma relies on introducing independent copies of the original variables X i {\displaystyle X_{i}} (sometimes referred to as a ghost sample) and replacing the inner expectation of the LHS by these copies. After an application of Jensen's inequality different signs could be introduced (hence the name symmetrization) without changing the expectation. The proof can be found below because of its instructive nature. The same proof method can be used to prove the Glivenko–Cantelli theorem. A typical way of proving empirical CLTs, first uses symmetrization to pass the empirical process to P n 0 {\displaystyle \mathbb {P} _{n}^{0}} and then argue conditionally on the data, using the fact that Rademacher processes are simple processes with nice properties. === VC Connection === It turns out that there is a fascinating connection between certain combinatorial properties of the set F {\displaystyle {\mathcal {F}}} and the entropy numbers. Uniform covering numbers can be controlled by the notion of Vapnik–Chervonenkis classes of sets – or shortly VC sets. Consider a collection C {\displaystyle {\mathcal {C}}} of subsets of the sample space X {\displaystyle

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  Ho–Kashyap algorithm

  The Ho–Kashyap algorithm is an iterative method in machine learning for finding a linear decision boundary that separates two linearly separable classes. It was developed by Yu-Chi Ho and Rangasami L. Kashyap in 1965, and usually presented as a problem in linear programming. == Setup == Given a training set consisting of samples from two classes, the Ho–Kashyap algorithm seeks to find a weight vector w {\displaystyle \mathbf {w} } and a margin vector b {\displaystyle \mathbf {b} } such that: Y w = b {\displaystyle \mathbf {Yw} =\mathbf {b} } where Y {\displaystyle \mathbf {Y} } is the augmented data matrix with samples from both classes (with appropriate sign conventions, e.g., samples from class 2 are negated), w {\displaystyle \mathbf {w} } is the weight vector to be determined, and b {\displaystyle \mathbf {b} } is a positive margin vector. The algorithm minimizes the criterion function: J ( w , b ) = | | Y w − b | | 2 {\displaystyle J(\mathbf {w} ,\mathbf {b} )=||\mathbf {Yw} -\mathbf {b} ||^{2}} subject to the constraint that b > 0 {\displaystyle \mathbf {b} >\mathbf {0} } (element-wise). Given a problem of linearly separating two classes, we consider a dataset of elements { ( x i , y i ) } i ∈ 1 : N {\displaystyle \{(\mathbf {x_{i}} ,y_{i})\}_{i\in 1:N}} where y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} . Linearly separating them by a perceptron is equivalent to finding weight and bias w , b {\displaystyle \mathbf {w} ,b} for a perceptron, such that: [ y 1 x 1 1 ⋮ ⋮ y N x N 1 ] [ w b ] > 0 {\displaystyle {\begin{bmatrix}y_{1}\mathbf {x} _{1}&1\\\vdots &\vdots \\y_{N}\mathbf {x} _{N}&1\\\end{bmatrix}}{\begin{bmatrix}\mathbf {w} \\b\end{bmatrix}}>0} == Algorithm == The idea of the Ho–Kashyap algorithm is as follows: Given any b {\displaystyle \mathbf {b} } , the corresponding w {\displaystyle \mathbf {w} } is known: It is simply w = Y + b {\displaystyle \mathbf {w} =\mathbf {Y} ^{+}\mathbf {b} } , where Y + {\displaystyle \mathbf {Y} ^{+}} denotes the Moore–Penrose pseudoinverse of Y {\displaystyle \mathbf {Y} } . Therefore, it only remains to find b {\displaystyle \mathbf {b} } by gradient descent. However, the gradient descent may sometimes decrease some of the coordinates of b {\displaystyle \mathbf {b} } , which may cause some coordinates of b {\displaystyle \mathbf {b} } to become negative, which is undesirable. Therefore, whenever some coordinates of b {\displaystyle \mathbf {b} } would have decreased, those coordinates are unchanged instead. As for the coordinates of b {\displaystyle \mathbf {b} } that would increase, those would increase without issue. Formally, the algorithm is as follows: Initialization: Set b ( 0 ) {\displaystyle \mathbf {b} (0)} to an arbitrary positive vector, typically b ( 0 ) = 1 {\displaystyle \mathbf {b} (0)=\mathbf {1} } (a vector of ones). Set the iteration counter k = 0 {\displaystyle k=0} . Set w ( 0 ) = Y + b ( 0 ) {\displaystyle \mathbf {w} (0)=\mathbf {Y} ^{+}\mathbf {b} (0)} Loop until convergence, or until iteration counter exceeds some k m a x {\displaystyle k_{max}} . Error calculation: Compute the error vector: e ( k ) = Y w ( k ) − b ( k ) {\displaystyle \mathbf {e} (k)=\mathbf {Yw} (k)-\mathbf {b} (k)} . Margin update: Update the margin vector: b ( k + 1 ) = b ( k ) + 2 η k ( e ( k ) + | e ( k ) | ) {\displaystyle \mathbf {b} (k+1)=\mathbf {b} (k)+2\eta _{k}(\mathbf {e} (k)+|\mathbf {e} (k)|)} where η k {\displaystyle \eta _{k}} is a positive learning rate parameter, and | e ( k ) | {\displaystyle |\mathbf {e} (k)|} denotes the element-wise absolute value. Weight calculation: Compute the weight vector using the pseudoinverse: w ( k + 1 ) = Y + b ( k + 1 ) {\displaystyle \mathbf {w} (k+1)=\mathbf {Y} ^{+}\mathbf {b} (k+1)} . Convergence check: If | | e ( k ) | | ≤ θ {\displaystyle ||\mathbf {e} (k)||\leq \theta } for some predetermined threshold θ {\displaystyle \theta } (close to zero), then return b ( k + 1 ) , w ( k + 1 ) {\displaystyle \mathbf {b} (k+1),\mathbf {w} (k+1)} . if e ( k ) ≤ 0 {\displaystyle \mathbf {e} (k)\leq \mathbf {0} } (all components non-positive), return "Samples not separable.". Return "Algorithm failed to converge in time.". == Properties == If the training data is linearly separable, the algorithm converges to a solution (where e ( k ) = 0 {\displaystyle \mathbf {e} (k)=\mathbf {0} } ) in a finite number of iterations. If the data is not linearly separable, the algorithm may or may not ever reach the point where e ( k ) = 0 {\displaystyle \mathbf {e} (k)=\mathbf {0} } . However, if it does happen that e ( k ) ≤ 0 {\displaystyle \mathbf {e} (k)\leq \mathbf {0} } at some iteration, this proves non-separability. The convergence rate depends on the choice of the learning rate parameter ρ {\displaystyle \rho } and the degree of linear separability of the data. == Relationship to other algorithms == Perceptron algorithm: Both seek linear separators. The perceptron updates weights incrementally based on individual misclassified samples, while Ho–Kashyap is a batch method that processes all samples to compute the pseudoinverse and updates based on an overall error vector. Linear discriminant analysis (LDA): LDA assumes underlying Gaussian distributions with equal covariances for the classes and derives the decision boundary from these statistical assumptions. Ho–Kashyap makes no explicit distributional assumptions and instead tries to solve a system of linear inequalities directly. Support vector machines (SVM): For linearly separable data, SVMs aim to find the maximum-margin hyperplane. The Ho–Kashyap algorithm finds a separating hyperplane but not necessarily the one with the maximum margin. If the data is not separable, soft-margin SVMs allow for some misclassifications by optimizing a trade-off between margin size and misclassification penalty, while Ho–Kashyap provides a least-squares solution. == Variants == Modified Ho–Kashyap algorithm changes weight calculation step w ( k + 1 ) = Y + b ( k + 1 ) {\displaystyle \mathbf {w} (k+1)=\mathbf {Y} ^{+}\mathbf {b} (k+1)} to w ( k + 1 ) = w ( k ) + η k Y + | e ( k ) | {\displaystyle \mathbf {w} (k+1)=\mathbf {w} (k)+\eta _{k}\mathbf {Y} ^{+}|\mathbf {e} (k)|} . Kernel Ho–Kashyap algorithm: Applies kernel methods (the "kernel trick") to the Ho–Kashyap framework to enable non-linear classification by implicitly mapping data to a higher-dimensional feature space.

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  Dynamic time warping

  In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences, which may vary in speed. For instance, similarities in walking could be detected using DTW, even if one person was walking faster than the other, or if there were accelerations and decelerations during the course of an observation. DTW has been applied to temporal sequences of video, audio, and graphics data — indeed, any data that can be turned into a one-dimensional sequence can be analyzed with DTW. A well-known application has been automatic speech recognition, to cope with different speaking speeds. Other applications include speaker recognition and online signature recognition. It can also be used in partial shape matching applications. In general, DTW is a method that calculates an optimal match between two given sequences (e.g. time series) with certain restriction and rules: Every index from the first sequence must be matched with one or more indices from the other sequence, and vice versa The first index from the first sequence must be matched with the first index from the other sequence (but it does not have to be its only match) The last index from the first sequence must be matched with the last index from the other sequence (but it does not have to be its only match) The mapping of the indices from the first sequence to indices from the other sequence must be monotonically increasing, and vice versa, i.e. if j > i {\displaystyle j>i} are indices from the first sequence, then there must not be two indices l > k {\displaystyle l>k} in the other sequence, such that index i {\displaystyle i} is matched with index l {\displaystyle l} and index j {\displaystyle j} is matched with index k {\displaystyle k} , and vice versa We can plot each match between the sequences 1 : M {\displaystyle 1:M} and 1 : N {\displaystyle 1:N} as a path in a M × N {\displaystyle M\times N} matrix from ( 1 , 1 ) {\displaystyle (1,1)} to ( M , N ) {\displaystyle (M,N)} , such that each step is one of ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) {\displaystyle (0,1),(1,0),(1,1)} . In this formulation, we see that the number of possible matches is the Delannoy number. The optimal match is denoted by the match that satisfies all the restrictions and the rules and that has the minimal cost, where the cost is computed as the sum of absolute differences, for each matched pair of indices, between their values. The sequences are "warped" non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in time series classification. Although DTW measures a distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold. In addition to a similarity measure between the two sequences (a so called "warping path" is produced), by warping according to this path the two signals may be aligned in time. The signal with an original set of points X(original), Y(original) is transformed to X(warped), Y(warped). This finds applications in genetic sequence and audio synchronisation. In a related technique sequences of varying speed may be averaged using this technique see the average sequence section. This is conceptually very similar to the Needleman–Wunsch algorithm. == Implementation == This example illustrates the implementation of the dynamic time warping algorithm when the two sequences s and t are strings of discrete symbols. For two symbols x and y, d ( x , y ) {\displaystyle d(x,y)} is a distance between the symbols, e.g., d ( x , y ) = | x − y | {\displaystyle d(x,y)=|x-y|} . int DTWDistance(s: array [1..n], t: array [1..m]) { DTW := array [0..n, 0..m] for i := 0 to n for j := 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := 1 to m cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } where DTW[i, j] is the distance between s[1:i] and t[1:j] with the best alignment. We sometimes want to add a locality constraint. That is, we require that if s[i] is matched with t[j], then | i − j | {\displaystyle |i-j|} is no larger than w, a window parameter. We can easily modify the above algorithm to add a locality constraint (differences marked). However, the above given modification works only if | n − m | {\displaystyle |n-m|} is no larger than w, i.e. the end point is within the window length from diagonal. In order to make the algorithm work, the window parameter w must be adapted so that | n − m | ≤ w {\displaystyle |n-m|\leq w} (see the line marked with () in the code). int DTWDistance(s: array [1..n], t: array [1..m], w: int) { DTW := array [0..n, 0..m] w := max(w, abs(n-m)) // adapt window size () for i := 0 to n for j:= 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) DTW[i, j] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } == Warping properties == The DTW algorithm produces a discrete matching between existing elements of one series to another. In other words, it does not allow time-scaling of segments within the sequence. Other methods allow continuous warping. For example, Correlation Optimized Warping (COW) divides the sequence into uniform segments that are scaled in time using linear interpolation, to produce the best matching warping. The segment scaling causes potential creation of new elements, by time-scaling segments either down or up, and thus produces a more sensitive warping than DTW's discrete matching of raw elements. == Complexity == The time complexity of the DTW algorithm is O ( N M ) {\displaystyle O(NM)} , where N {\displaystyle N} and M {\displaystyle M} are the lengths of the two input sequences. The 50 years old quadratic time bound was broken in 2016: an algorithm due to Gold and Sharir enables computing DTW in O ( N 2 / log ⁡ log ⁡ N ) {\displaystyle O({N^{2}}/\log \log N)} time and space for two input sequences of length N {\displaystyle N} . This algorithm can also be adapted to sequences of different lengths. Despite this improvement, it was shown that a strongly subquadratic running time of the form O ( N 2 − ϵ ) {\displaystyle O(N^{2-\epsilon })} for some ϵ > 0 {\displaystyle \epsilon >0} cannot exist unless the Strong exponential time hypothesis fails. While the dynamic programming algorithm for DTW requires O ( N M ) {\displaystyle O(NM)} space in a naive implementation, the space consumption can be reduced to O ( min ( N , M ) ) {\displaystyle O(\min(N,M))} using Hirschberg's algorithm. == Fast computation == Fast techniques for computing DTW include PrunedDTW, SparseDTW, FastDTW, and the MultiscaleDTW. A common task, retrieval of similar time series, can be accelerated by using lower bounds such as LB_Keogh, LB_Improved, or LB_Petitjean. However, the Early Abandon and Pruned DTW algorithm reduces the degree of acceleration that lower bounding provides and sometimes renders it ineffective. In a survey, Wang et al. reported slightly better results with the LB_Improved lower bound than the LB_Keogh bound, and found that other techniques were inefficient. Subsequent to this survey, the LB_Enhanced bound was developed that is always tighter than LB_Keogh while also being more efficient to compute. LB_Petitjean is the tightest known lower bound that can be computed in linear time. == Average sequence == Averaging for dynamic time warping is the problem of finding an average sequence for a set of sequences. NLAAF is an exact method to average two sequences using DTW. For more than two sequences, the problem is related to that of multiple alignment and requires heuristics. DBA is currently a reference method to average a set of sequences consistently with DTW. COMASA efficiently randomizes the search for the average sequence, using DBA as a local optimization process. == Supervised learning == A nearest-neighbour classifier can achieve state-of-the-art performance when using dynamic time warping as a distance measure. == Amerced Dynamic Time Warping == Amerced Dynamic Time Warping (ADTW) is a variant of DTW designed to better control DTW's permissiveness in the alignments that it allows. The windows that classical DTW uses to constrain alignments introduce a step function. Any warping of the path is allowed within the window and none beyond it. In contrast, ADTW employs an additive penalty that is incurred each time that the path is warped. Any amount of warping is allowed, but each warping action incurs a direct penalty. ADTW significantly outperforms DTW with windowing when applied as a nearest neighbor classifier on a set of benchmark time series classification tasks. == Alternative approaches == In functional data analysis, time series are regarde

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  Conference app

  A conference app, also known as an event app or meeting app, is a mobile app developed to help attendees and meeting planners manage their conference experience. It typically includes conference proceedings and venue information, allowing users to create personalized schedules and engage with other users. A conference app can be a native app or web-based. In recent years, conference apps have gained in popularity as a sustainable solution for event management by reducing paper produced by printed materials. Advanced features often include real-time notifications for updates or changes, integration with virtual meeting platforms for hybrid or fully online events, and analytics tools for organizers to measure attendance and engagement. Additionally, some apps support sponsorship and exhibitor features, enabling businesses to showcase their products or services directly within the app.

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  Loss function

  In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s. In optimal control, the loss is the penalty for failing to achieve a desired value. In financial risk management, the function is mapped to a monetary loss. == Examples == === Regret === Leonard J. Savage argued that using non-Bayesian methods such as minimax, the loss function should be based on the idea of regret, i.e., the loss associated with a decision should be the difference between the consequences of the best decision that could have been made under circumstances will be known and the decision that was in fact taken before they were known. === Quadratic loss function === The use of a quadratic loss function is common, for example when using least squares techniques. It is often more mathematically tractable than other loss functions because of the properties of variances, as well as being symmetric: an error above the target causes the same loss as the same magnitude of error below the target. If the target is t {\displaystyle t} , then a quadratic loss function is λ ( x ) = C ( t − x ) 2 {\displaystyle \lambda (x)=C(t-x)^{2}\;} for some constant C {\displaystyle C} ; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. This is also known as the squared error loss (SEL). Many common statistics, including t-tests, regression models, design of experiments, and much else, use least squares methods applied using linear regression theory, which is based on the quadratic loss function. The quadratic loss function is also used in linear-quadratic optimal control problems. In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a quadratic form in the deviations of the variables of interest from their desired values; this approach is tractable because it results in linear first-order conditions. In the context of stochastic control, the expected value of the quadratic form is used. The quadratic loss assigns more importance to outliers than to the true data due to its square nature, so alternatives like the Huber, log-cosh and SMAE losses are used when the data has many large outliers. === 0-1 loss function === In statistics and decision theory, a frequently used loss function is the 0-1 loss function L ( y ^ , y ) = { 0 if y = y ^ 1 if y ≠ y ^ {\displaystyle L({\hat {y}},y)={\begin{cases}0&{\text{if }}y={\hat {y}}\\1&{\text{if }}y\neq {\hat {y}}\end{cases}}} In information theory, this loss function is known as Hamming distortion. == Constructing loss and objective functions == In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. The existing methods for constructing objective functions are collected in the proceedings of two dedicated conferences. In particular, Andranik Tangian showed that the most usable objective functions — quadratic and additive — are determined by a few indifference points. He used this property in the models for constructing these objective functions from either ordinal or cardinal data that were elicited through computer-assisted interviews with decision makers. Among other things, he constructed objective functions to optimally distribute budgets for 16 Westfalian universities and the European subsidies for equalizing unemployment rates among 271 German regions. == Expected loss == In some contexts, the value of the loss function itself is a random quantity because it depends on the outcome of a random variable X {\displaystyle X} . === Statistics === Both frequentist and Bayesian statistical theory involve making a decision based on the expected value of the loss function; however, this quantity is defined differently under the two paradigms. ==== Frequentist expected loss ==== We first define the expected loss in the frequentist context. It is obtained by taking the expected value with respect to the probability distribution, P θ {\displaystyle P_{\theta }} , of the observed data, X {\displaystyle X} . This is also referred to as the risk function of the decision rule δ {\displaystyle \delta } and the parameter θ {\displaystyle \theta } . Here the decision rule depends on the outcome of X {\displaystyle X} . The risk function is given by: R ( θ , δ ) = E θ ⁡ L ( θ , δ ( X ) ) = ∫ X L ( θ , δ ( x ) ) d P θ ( x ) . {\displaystyle R(\theta ,\delta )=\operatorname {E} _{\theta }L{\big (}\theta ,\delta (X){\big )}=\int _{X}L{\big (}\theta ,\delta (x){\big )}\,\mathrm {d} P_{\theta }(x).} Here, θ {\displaystyle \theta } is a fixed but possibly unknown state of nature, X {\displaystyle X} is a vector of observations stochastically drawn from a population, E θ {\displaystyle \operatorname {E} _{\theta }} is the expectation over all population values of X {\displaystyle X} , d P θ {\displaystyle \mathrm {d} P_{\theta }} is a probability measure over the event space of X {\displaystyle X} (parametrized by θ {\displaystyle \theta } ) and the integral is evaluated over the entire support of X {\displaystyle X} . ==== Bayes Risk ==== In a Bayesian approach, the expectation is calculated using the prior distribution π ∗ {\displaystyle \pi ^{}} of the parameter θ {\displaystyle \theta } : ρ ( π ∗ , a ) = ∫ Θ ∫ X L ( θ , a ( x ) ) d P ( x | θ ) d π ∗ ( θ ) = ∫ X ∫ Θ L ( θ , a ( x ) ) d π ∗ ( θ | x ) d M ( x ) {\displaystyle \rho (\pi ^{},a)=\int _{\Theta }\int _{\mathbf {X}}L(\theta ,a({\mathbf {x}}))\,\mathrm {d} P({\mathbf {x}}\vert \theta )\,\mathrm {d} \pi ^{}(\theta )=\int _{\mathbf {X}}\int _{\Theta }L(\theta ,a({\mathbf {x}}))\,\mathrm {d} \pi ^{}(\theta \vert {\mathbf {x}})\,\mathrm {d} M({\mathbf {x}})} where M ( x ) {\displaystyle M(\mathbf {x} )} is known as the predictive likelihood wherein θ {\displaystyle \theta } has been "integrated out," π ∗ ( θ | x ) {\displaystyle \pi ^{}(\theta |\mathbf {x} )} is the posterior distribution, and the order of integration has been changed. One then should choose the action a ∗ {\displaystyle a^{}} which minimises this expected loss, which is referred to as Bayes Risk. In the latter equation, the integrand inside d x {\displaystyle \mathrm {d} x} is known as the Posterior Risk, and minimising it with respect to decision a {\displaystyle a} also minimizes the overall Bayes Risk. This optimal decision, a ∗ {\displaystyle a^{}} is known as the Bayes (decision) Rule - it minimises the average loss over all possible states of nature θ {\displaystyle \theta } , over all possible (probability-weighted) data outcomes. One advantage of the Bayesian approach is to that one need only choose the optimal action under the actual observed data to obtain a uniformly optimal one, whereas choosing the actual frequentist optimal decision rule as a function of all possible observations, is a much more difficult problem. Of equal importance though, the Bayes Rule reflects consideration of loss outcomes under different states of nature, θ {\displaystyle \theta } . ==== Examples in statistics ==== For a scalar parameter θ {\displaystyle \theta } , a decision function whose output θ ^ {\displaystyle {\hat {\theta }}} is an estimate of θ {\displaystyle \theta } , and a quadratic loss function (squared error loss) L ( θ , θ ^ ) = ( θ − θ ^ ) 2 , {\displaystyle L(\theta ,{\hat {\theta }})=(\theta -{\hat {\theta }})^{2},} the risk function becomes the mean squared error of the estimate, R ( θ , θ ^ ) = E θ ⁡ [ ( θ − θ ^ ) 2 ] . {\displaystyle R(\theta ,{\hat {\thet

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  Bayesian hierarchical modeling

  Bayesian hierarchical modelling is a statistical model written in multiple levels (hierarchical form) that estimates the posterior distribution of model parameters using the Bayesian method. The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and account for all the uncertainty that is present. This integration enables calculation of updated posterior over the (hyper)parameters, effectively updating prior beliefs in light of the observed data. Frequentist statistics may yield conclusions seemingly incompatible with those offered by Bayesian statistics due to the Bayesian treatment of the parameters as random variables and its use of subjective information in establishing assumptions on these parameters. As the approaches answer different questions the formal results are not technically contradictory but the two approaches disagree over which answer is relevant to particular applications. Bayesians argue that relevant information regarding decision-making and updating beliefs cannot be ignored and that hierarchical modeling has the potential to overrule classical methods in applications where respondents give multiple observational data. Moreover, the model has proven to be robust, with the posterior distribution less sensitive to the more flexible hierarchical priors. Hierarchical modeling, as its name implies, retains nested data structure, and is used when information is available at several different levels of observational units. For example, in epidemiological modeling to describe infection trajectories for multiple countries, observational units are countries, and each country has its own time-based profile of daily infected cases. In decline curve analysis to describe oil or gas production decline curve for multiple wells, observational units are oil or gas wells in a reservoir region, and each well has each own time-based profile of oil or gas production rates (usually, barrels per month). Hierarchical modeling is used to devise computation based strategies for multiparameter problems. == Philosophy == Statistical methods and models commonly involve multiple parameters that can be regarded as related or connected in such a way that the problem implies a dependence of the joint probability model for these parameters. Individual degrees of belief, expressed in the form of probabilities, come with uncertainty. Amidst this is the change of the degrees of belief over time. As was stated by Professor José M. Bernardo and Professor Adrian F. Smith, "The actuality of the learning process consists in the evolution of individual and subjective beliefs about the reality." These subjective probabilities are more directly involved in the mind rather than the physical probabilities. Hence, it is with this need of updating beliefs that Bayesians have formulated an alternative statistical model which takes into account the prior occurrence of a particular event. == Bayes' theorem == The assumed occurrence of a real-world event will typically modify preferences between certain options. This is done by modifying the degrees of belief attached, by an individual, to the events defining the options. Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital j having survival probability θ j {\displaystyle \theta _{j}} , the survival probability will be updated with the occurrence of y, the event in which a controversial serum is created which, as believed by some, increases survival in cardiac patients. In order to make updated probability statements about θ j {\displaystyle \theta _{j}} , given the occurrence of event y, we must begin with a model providing a joint probability distribution for θ j {\displaystyle \theta _{j}} and y. This can be written as a product of the two distributions that are often referred to as the prior distribution P ( θ ) {\displaystyle P(\theta )} and the sampling distribution P ( y ∣ θ ) {\displaystyle P(y\mid \theta )} respectively: P ( θ , y ) = P ( θ ) P ( y ∣ θ ) {\displaystyle P(\theta ,y)=P(\theta )P(y\mid \theta )} Using the basic property of conditional probability, the posterior distribution will yield: P ( θ ∣ y ) = P ( θ , y ) P ( y ) = P ( y ∣ θ ) P ( θ ) P ( y ) {\displaystyle P(\theta \mid y)={\frac {P(\theta ,y)}{P(y)}}={\frac {P(y\mid \theta )P(\theta )}{P(y)}}} This equation, showing the relationship between the conditional probability and the individual events, is known as Bayes' theorem. This simple expression encapsulates the technical core of Bayesian inference which aims to deconstruct the probability, P ( θ ∣ y ) {\displaystyle P(\theta \mid y)} , relative to solvable subsets of its supportive evidence. == Exchangeability == The usual starting point of a statistical analysis is the assumption that the n values y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} are exchangeable. If no information – other than data y – is available to distinguish any of the θ j {\displaystyle \theta _{j}} 's from any others, and no ordering or grouping of the parameters can be made, one must assume symmetry of prior distribution parameters. This symmetry is represented probabilistically by exchangeability. Generally, it is useful and appropriate to model data from an exchangeable distribution as independently and identically distributed, given some unknown parameter vector θ {\displaystyle \theta } , with distribution P ( θ ) {\displaystyle P(\theta )} . === Finite exchangeability === For a fixed number n, the set y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable if the joint probability P ( y 1 , y 2 , … , y n ) {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})} is invariant under permutations of the indices. That is, for every permutation π {\displaystyle \pi } or ( π 1 , π 2 , … , π n ) {\displaystyle (\pi _{1},\pi _{2},\ldots ,\pi _{n})} of (1, 2, …, n), P ( y 1 , y 2 , … , y n ) = P ( y π 1 , y π 2 , … , y π n ) . {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})=P(y_{\pi _{1}},y_{\pi _{2}},\ldots ,y_{\pi _{n}}).} The following is an exchangeable, but not independent and identical (iid), example: Consider an urn with a red ball and a blue ball inside, with probability 1 2 {\displaystyle {\frac {1}{2}}} of drawing either. Balls are drawn without replacement, i.e. after one ball is drawn from the n {\displaystyle n} balls, there will be n − 1 {\displaystyle n-1} remaining balls left for the next draw. Let Y i = { 1 , if the i th ball is red , 0 , otherwise . {\displaystyle {\text{Let }}Y_{i}={\begin{cases}1,&{\text{if the }}i{\text{th ball is red}},\\0,&{\text{otherwise}}.\end{cases}}} The probability of selecting a red ball in the first draw and a blue ball in the second draw is equal to the probability of selecting a blue ball on the first draw and a red on the second, both of which are 1/2: P ( y 1 = 1 , y 2 = 0 ) = P ( y 1 = 0 , y 2 = 1 ) = 1 2 {\displaystyle P(y_{1}=1,y_{2}=0)=P(y_{1}=0,y_{2}=1)={\frac {1}{2}}} . This makes y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} exchangeable. But the probability of selecting a red ball on the second draw given that the red ball has already been selected in the first is 0. This is not equal to the probability that the red ball is selected in the second draw, which is 1/2: P ( y 2 = 1 ∣ y 1 = 1 ) = 0 ≠ P ( y 2 = 1 ) = 1 2 {\displaystyle P(y_{2}=1\mid y_{1}=1)=0\neq P(y_{2}=1)={\frac {1}{2}}} . Thus, y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} are not independent. If x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are independent and identically distributed, then they are exchangeable, but the converse is not necessarily true. === Infinite exchangeability === Infinite exchangeability is the property that every finite subset of an infinite sequence y 1 {\displaystyle y_{1}} , y 2 , … {\displaystyle y_{2},\ldots } is exchangeable. For any n, the sequence y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable. == Hierarchical models == === Components === Bayesian hierarchical modeling makes use of two important concepts in deriving the posterior distribution, namely: Hyperparameters: parameters of the prior distribution Hyperpriors: distributions of Hyperparameters Suppose a random variable Y follows a normal distribution with parameter θ {\displaystyle \theta } as the mean and 1 as the variance, that is Y ∣ θ ∼ N ( θ , 1 ) {\displaystyle Y\mid \theta \sim N(\theta ,1)} . The tilde relation ∼ {\displaystyle \sim } can be read as "has the distribution of" or "is distributed as". Suppose also that the parameter θ {\displaystyle \theta } has a distribution given by a normal distribution with mean μ {\displaystyle \mu } and variance 1, i.e. θ ∣ μ ∼ N ( μ , 1 ) {\displaystyle \theta \mid \mu \sim N(\mu ,1)} . Furthermore, μ {\displaystyle \mu } follows another distribution given, for example, by the standard normal distribution, N ( 0 , 1 ) {\displaystyle {\text{N}}(0,1)} . The parameter μ {\dis

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  IDistance

  In pattern recognition, iDistance is an indexing and query processing technique for k-nearest neighbor queries on point data in multi-dimensional metric spaces. The kNN query is one of the hardest problems on multi-dimensional data, especially when the dimensionality of the data is high. iDistance is designed to process kNN queries in high-dimensional spaces efficiently and performs extremely well for skewed data distributions, which usually occur in real-life data sets. iDistance employs a two-phase search strategy involving an initial filtering of candidate regions and a subsequent refinement of results, an approach aligned with the Filter and Refine Principle (FRP). This means that the index first prunes the search space to eliminate unlikely candidates, then verifies the true nearest neighbors in a refinement step, following the general FRP paradigm used in database search algorithms. The iDistance index can also be augmented with machine learning models to learn data distributions for improved searching and storage of multi-dimensional data. == Indexing == Building the iDistance index has two steps: A number of reference points in the data space are chosen. There are various ways of choosing reference points. Using cluster centers as reference points is the most efficient way. The data points are partitioned into Voronoi cells based on well-chosen reference points. The distance between a data point and its closest reference point is calculated. This distance plus a scaling value is called the point's iDistance. By this means, points in a multi-dimensional space are mapped to one-dimensional values, and then a B+-tree can be adopted to index the points using the iDistance as the key. The figure on the right shows an example where three reference points (O1, O2, O3) are chosen. The data points are then mapped to a one-dimensional space and indexed in a B+-tree. Various extensions have been proposed to make the selection of reference points for effective query performance, including employing machine learning to learn the identification of reference points. == Query processing == To process a kNN query, the query is mapped to a number of one-dimensional range queries, which can be processed efficiently on a B+-tree. In the above figure, the query Q is mapped to a value in the B+-tree while the kNN search ``sphere" is mapped to a range in the B+-tree. The search sphere expands gradually until the k NNs are found. This corresponds to gradually expanding range searches in the B+-tree. The iDistance technique can be viewed as a way of accelerating the sequential scan. Instead of scanning records from the beginning to the end of the data file, the iDistance starts the scan from spots where the nearest neighbors can be obtained early with a very high probability. == Applications == The iDistance has been used in many applications including Image retrieval Video indexing Similarity search in P2P systems Mobile computing Recommender system == Historical background == The iDistance was first proposed by Cui Yu, Beng Chin Ooi, Kian-Lee Tan and H. V. Jagadish in 2001. Later, together with Rui Zhang, they improved the technique and performed a more comprehensive study on it in 2005.

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  Facial age estimation

  Facial age estimation is the use of artificial intelligence to estimate the age of a person based on their facial features. Computer vision techniques are used to analyse the facial features in the images of millions of people whose age is known and then deep learning is used to create an algorithm that tries to predict the age of an unknown person. The key use of the technology is to prevent access to age-restricted goods and services. Examples include restricting children from accessing internet pornography, checking that they meet a mandatory minimum age when registering for an account on social media, or preventing adults from accessing websites, online chat or games designed only for use by children. The technology is distinct from facial recognition systems as the software does not attempt to uniquely identify the individual. Researchers have applied neural networks for age estimation since at least 2010. == Evaluation == An ongoing study by the National Institute of Standards and Technology (NIST) entitled 'Face Analysis Technology Evaluation' seeks to establish the technical performance of prototype age estimation algorithms submitted by academic teams and software vendors including Brno University of Technology, Czech Technical University in Prague, Dermalog, IDEMIA, Incode Technologies Inc, Jumio, Nominder, Rank One Computing, Unissey and Yoti. == Public sector use == The UK government has explored using facial age estimation at the UK border as an alternative to bone X-rays and MRI scans when determining child status of asylum seekers. == Commercial use == Commercial users of facial age estimation include Instagram and OnlyFans. In January 2025, John Lewis & Partners announced that had started using the technology to check the age of people shopping for knives on its website, to comply with UK legislation to limit knife crime. In the UK, several supermarket chains have taken part in Home Office trials of the technology to automate the checking of a customer's age when buying age-restricted goods such as alcohol. UK legislation introduced in January 2025 mandates robust forms of age verification hosting adult content viewable in the UK by July 2025. Allowable methods include facial age estimation. == Criticism == Adam Schwartz, a lawyer for the Electronic Frontier Foundation, criticized the use of facial age estimation software, noting its inaccuracy especially in cases of minorities and women, as was found in NIST's 2024 report. Twenty organisations jointly under European Digital Rights called the practice a "systematic and invasive processing of young people's data" that risks discriminatory profiling.

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  Semantic mapping (statistics)

  Semantic mapping (SM) is a statistical method for dimensionality reduction (the transformation of data from a high-dimensional space into a low-dimensional space). SM can be used in a set of multidimensional vectors of features to extract a few new features that preserves the main data characteristics. SM performs dimensionality reduction by clustering the original features in semantic clusters and combining features mapped in the same cluster to generate an extracted feature. Given a data set, this method constructs a projection matrix that can be used to map a data element from a high-dimensional space into a reduced dimensional space. SM can be applied in construction of text mining and information retrieval systems, as well as systems managing vectors of high dimensionality. SM is an alternative to random mapping, principal components analysis and latent semantic indexing methods.

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  Policy gradient method

  Policy gradient methods are a class of reinforcement learning algorithms and a sub-class of policy optimization methods. Unlike value-based methods which learn a value function to derive a policy, policy optimization methods directly learn a policy function π {\displaystyle \pi } that selects actions without consulting a value function. For policy gradient to apply, the policy function π θ {\displaystyle \pi _{\theta }} is parameterized by a differentiable parameter θ {\displaystyle \theta } . == Overview == In policy-based RL, the actor is a parameterized policy function π θ {\displaystyle \pi _{\theta }} , where θ {\displaystyle \theta } are the parameters of the actor. The actor takes as argument the state of the environment s {\displaystyle s} and produces a probability distribution π θ ( ⋅ ∣ s ) {\displaystyle \pi _{\theta }(\cdot \mid s)} . If the action space is discrete, then ∑ a π θ ( a ∣ s ) = 1 {\displaystyle \sum _{a}\pi _{\theta }(a\mid s)=1} . If the action space is continuous, then ∫ a π θ ( a ∣ s ) d a = 1 {\displaystyle \int _{a}\pi _{\theta }(a\mid s)\mathrm {d} a=1} . The goal of policy optimization is to find some θ {\displaystyle \theta } that maximizes the expected episodic reward J ( θ ) {\displaystyle J(\theta )} : J ( θ ) = E π θ [ ∑ t = 0 T γ t R t | S 0 = s 0 ] {\displaystyle J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\gamma ^{t}R_{t}{\Big |}S_{0}=s_{0}\right]} where γ {\displaystyle \gamma } is the discount factor, R t {\displaystyle R_{t}} is the reward at step t {\displaystyle t} , s 0 {\displaystyle s_{0}} is the starting state, and T {\displaystyle T} is the time-horizon (which can be infinite). The policy gradient is defined as ∇ θ J ( θ ) {\displaystyle \nabla _{\theta }J(\theta )} . Different policy gradient methods stochastically estimate the policy gradient in different ways. The goal of any policy gradient method is to iteratively maximize J ( θ ) {\displaystyle J(\theta )} by gradient ascent. Since the key part of any policy gradient method is the stochastic estimation of the policy gradient, they are also studied under the title of "Monte Carlo gradient estimation". == REINFORCE == === Policy gradient === The REINFORCE algorithm, introduced by Ronald J. Williams in 1992, was the first policy gradient method. It is based on the identity for the policy gradient ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t ∣ S t ) ∑ t = 0 T ( γ t R t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\;\sum _{t=0}^{T}(\gamma ^{t}R_{t}){\Big |}S_{0}=s_{0}\right]} which can be improved via the "causality trick" ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t ∣ S t ) ∑ τ = t T ( γ τ R τ ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau }){\Big |}S_{0}=s_{0}\right]} Thus, we have an unbiased estimator of the policy gradient: ∇ θ J ( θ ) ≈ 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ − t R τ , n ) ] {\displaystyle \nabla _{\theta }J(\theta )\approx {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau -t}R_{\tau ,n})\right]} where the index n {\displaystyle n} ranges over N {\displaystyle N} rollout trajectories using the policy π θ {\displaystyle \pi _{\theta }} . The score function ∇ θ ln ⁡ π θ ( A t ∣ S t ) {\displaystyle \nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})} can be interpreted as the direction in the parameter space that increases the probability of taking action A t {\displaystyle A_{t}} in state S t {\displaystyle S_{t}} . The policy gradient, then, is a weighted average of all possible directions to increase the probability of taking any action in any state, but weighted by reward signals, so that if taking a certain action in a certain state is associated with high reward, then that direction would be highly reinforced, and vice versa. === Algorithm === The REINFORCE algorithm is a loop: Rollout N {\displaystyle N} trajectories in the environment, using π θ t {\displaystyle \pi _{\theta _{t}}} as the policy function. Compute the policy gradient estimation: g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln ⁡ π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ R τ , n ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})\right]} Update the policy by gradient ascent: θ i + 1 ← θ i + α i g i {\displaystyle \theta _{i+1}\leftarrow \theta _{i}+\alpha _{i}g_{i}} Here, α i {\displaystyle \alpha _{i}} is the learning rate at update step i {\displaystyle i} . == Variance reduction == REINFORCE is an on-policy algorithm, meaning that the trajectories used for the update must be sampled from the current policy π θ {\displaystyle \pi _{\theta }} . This can lead to high variance in the updates, as the returns R ( τ ) {\displaystyle R(\tau )} can vary significantly between trajectories. Many variants of REINFORCE have been introduced, under the title of variance reduction. === REINFORCE with baseline === A common way for reducing variance is the REINFORCE with baseline algorithm, based on the following identity: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − b ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-b(S_{t})\right){\Big |}S_{0}=s_{0}\right]} for any function b : States → R {\displaystyle b:{\text{States}}\to \mathbb {R} } . This can be proven by applying the previous lemma. The algorithm uses the modified gradient estimator g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln ⁡ π θ ( A t , n | S t , n ) ( ∑ τ = t T ( γ τ R τ , n ) − b i ( S t , n ) ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}|S_{t,n})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})-b_{i}(S_{t,n})\right)\right]} and the original REINFORCE algorithm is the special case where b i ≡ 0 {\displaystyle b_{i}\equiv 0} . === Actor-critic methods === If b i {\textstyle b_{i}} is chosen well, such that b i ( S t ) ≈ ∑ τ = t T ( γ τ R τ ) = γ t V π θ i ( S t ) {\textstyle b_{i}(S_{t})\approx \sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })=\gamma ^{t}V^{\pi _{\theta _{i}}}(S_{t})} , this could significantly decrease variance in the gradient estimation. That is, the baseline should be as close to the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} as possible, approaching the ideal of: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − γ t V π θ ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-\gamma ^{t}V^{\pi _{\theta }}(S_{t})\right){\Big |}S_{0}=s_{0}\right]} Note that, as the policy π θ t {\displaystyle \pi _{\theta _{t}}} updates, the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} updates as well, so the baseline should also be updated. One common approach is to train a separate function that estimates the value function, and use that as the baseline. This is one of the actor-critic methods, where the policy function is the actor and the value function is the critic. The Q-function Q π {\displaystyle Q^{\pi }} can also be used as the critic, since ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ Q π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot Q^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} by a similar argument using the tower law. Subtracting the value function as a baseline, we find that the advantage function A π ( S , A ) = Q π ( S , A ) − V π ( S ) {\displaystyle A^{\pi }(S,A)=Q^{\pi }(S,A)-V^{\pi }(S)} can be used as the critic as well: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ A π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot A^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} In summary, there are many unbiased estimators for ∇ θ J θ {\textstyle \nabla _{\theta }J_{\theta }} , all in the form of: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ Ψ t | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\su

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