The constellation model is a probabilistic, generative model for category-level object recognition in computer vision. Like other part-based models, the constellation model attempts to represent an object class by a set of N parts under mutual geometric constraints. Because it considers the geometric relationship between different parts, the constellation model differs significantly from appearance-only, or "bag-of-words" representation models, which explicitly disregard the location of image features. The problem of defining a generative model for object recognition is difficult. The task becomes significantly complicated by factors such as background clutter, occlusion, and variations in viewpoint, illumination, and scale. Ideally, we would like the particular representation we choose to be robust to as many of these factors as possible. In category-level recognition, the problem is even more challenging because of the fundamental problem of intra-class variation. Even if two objects belong to the same visual category, their appearances may be significantly different. However, for structured objects such as cars, bicycles, and people, separate instances of objects from the same category are subject to similar geometric constraints. For this reason, particular parts of an object such as the headlights or tires of a car still have consistent appearances and relative positions. The Constellation Model takes advantage of this fact by explicitly modeling the relative location, relative scale, and appearance of these parts for a particular object category. Model parameters are estimated using an unsupervised learning algorithm, meaning that the visual concept of an object class can be extracted from an unlabeled set of training images, even if that set contains "junk" images or instances of objects from multiple categories. It can also account for the absence of model parts due to appearance variability, occlusion, clutter, or detector error. == History == The idea for a "parts and structure" model was originally introduced by Fischler and Elschlager in 1973. This model has since been built upon and extended in many directions. The Constellation Model, as introduced by Dr. Perona and his colleagues, was a probabilistic adaptation of this approach. In the late '90s, Burl et al. revisited the Fischler and Elschlager model for the purpose of face recognition. In their work, Burl et al. used manual selection of constellation parts in training images to construct a statistical model for a set of detectors and the relative locations at which they should be applied. In 2000, Weber et al. made the significant step of training the model using a more unsupervised learning process, which precluded the necessity for tedious hand-labeling of parts. Their algorithm was particularly remarkable because it performed well even on cluttered and occluded image data. Fergus et al. then improved upon this model by making the learning step fully unsupervised, having both shape and appearance learned simultaneously, and accounting explicitly for the relative scale of parts. == The method of Weber and Welling et al. == In the first step, a standard interest point detection method, such as Harris corner detection, is used to generate interest points. Image features generated from the vicinity of these points are then clustered using k-means or another appropriate algorithm. In this process of vector quantization, one can think of the centroids of these clusters as being representative of the appearance of distinctive object parts. Appropriate feature detectors are then trained using these clusters, which can be used to obtain a set of candidate parts from images. As a result of this process, each image can now be represented as a set of parts. Each part has a type, corresponding to one of the aforementioned appearance clusters, as well as a location in the image space. === Basic generative model === Weber & Welling here introduce the concept of foreground and background. Foreground parts correspond to an instance of a target object class, whereas background parts correspond to background clutter or false detections. Let T be the number of different types of parts. The positions of all parts extracted from an image can then be represented in the following "matrix," X o = ( x 11 , x 12 , ⋯ , x 1 N 1 x 21 , x 22 , ⋯ , x 2 N 2 ⋮ x T 1 , x T 2 , ⋯ , x T N T ) {\displaystyle X^{o}={\begin{pmatrix}x_{11},x_{12},{\cdots },x_{1N_{1}}\\x_{21},x_{22},{\cdots },x_{2N_{2}}\\\vdots \\x_{T1},x_{T2},{\cdots },x_{TN_{T}}\end{pmatrix}}} where N i {\displaystyle N_{i}\,} represents the number of parts of type i ∈ { 1 , … , T } {\displaystyle i\in \{1,\dots ,T\}} observed in the image. The superscript o indicates that these positions are observable, as opposed to missing. The positions of unobserved object parts can be represented by the vector x m {\displaystyle x^{m}\,} . Suppose that the object will be composed of F {\displaystyle F\,} distinct foreground parts. For notational simplicity, we assume here that F = T {\displaystyle F=T\,} , though the model can be generalized to F > T {\displaystyle F>T\,} . A hypothesis h {\displaystyle h\,} is then defined as a set of indices, with h i = j {\displaystyle h_{i}=j\,} , indicating that point x i j {\displaystyle x_{ij}\,} is a foreground point in X o {\displaystyle X^{o}\,} . The generative probabilistic model is defined through the joint probability density p ( X o , x m , h ) {\displaystyle p(X^{o},x^{m},h)\,} . === Model details === The rest of this section summarizes the details of Weber & Welling's model for a single component model. The formulas for multiple component models are extensions of those described here. To parametrize the joint probability density, Weber & Welling introduce the auxiliary variables b {\displaystyle b\,} and n {\displaystyle n\,} , where b {\displaystyle b\,} is a binary vector encoding the presence/absence of parts in detection ( b i = 1 {\displaystyle b_{i}=1\,} if h i > 0 {\displaystyle h_{i}>0\,} , otherwise b i = 0 {\displaystyle b_{i}=0\,} ), and n {\displaystyle n\,} is a vector where n i {\displaystyle n_{i}\,} denotes the number of background candidates included in the i t h {\displaystyle i^{th}} row of X o {\displaystyle X^{o}\,} . Since b {\displaystyle b\,} and n {\displaystyle n\,} are completely determined by h {\displaystyle h\,} and the size of X o {\displaystyle X^{o}\,} , we have p ( X o , x m , h ) = p ( X o , x m , h , n , b ) {\displaystyle p(X^{o},x^{m},h)=p(X^{o},x^{m},h,n,b)\,} . By decomposition, p ( X o , x m , h , n , b ) = p ( X o , x m | h , n , b ) p ( h | n , b ) p ( n ) p ( b ) {\displaystyle p(X^{o},x^{m},h,n,b)=p(X^{o},x^{m}|h,n,b)p(h|n,b)p(n)p(b)\,} The probability density over the number of background detections can be modeled by a Poisson distribution, p ( n ) = ∏ i = 1 T 1 n i ! ( M i ) n i e − M i {\displaystyle p(n)=\prod _{i=1}^{T}{\frac {1}{n_{i}!}}(M_{i})^{n_{i}}e^{-M_{i}}} where M i {\displaystyle M_{i}\,} is the average number of background detections of type i {\displaystyle i\,} per image. Depending on the number of parts F {\displaystyle F\,} , the probability p ( b ) {\displaystyle p(b)\,} can be modeled either as an explicit table of length 2 F {\displaystyle 2^{F}\,} , or, if F {\displaystyle F\,} is large, as F {\displaystyle F\,} independent probabilities, each governing the presence of an individual part. The density p ( h | n , b ) {\displaystyle p(h|n,b)\,} is modeled by p ( h | n , b ) = { 1 ∏ f = 1 F N f b f , if h ∈ H ( b , n ) 0 , for other h {\displaystyle p(h|n,b)={\begin{cases}{\frac {1}{\textstyle \prod _{f=1}^{F}N_{f}^{b_{f}}}},&{\mbox{if }}h\in H(b,n)\\0,&{\mbox{for other }}h\end{cases}}} where H ( b , n ) {\displaystyle H(b,n)\,} denotes the set of all hypotheses consistent with b {\displaystyle b\,} and n {\displaystyle n\,} , and N f {\displaystyle N_{f}\,} denotes the total number of detections of parts of type f {\displaystyle f\,} . This expresses the fact that all consistent hypotheses, of which there are ∏ f = 1 F N f b f {\displaystyle \textstyle \prod _{f=1}^{F}N_{f}^{b_{f}}} , are equally likely in the absence of information on part locations. And finally, p ( X o , x m | h , n ) = p f g ( z ) p b g ( x b g ) {\displaystyle p(X^{o},x^{m}|h,n)=p_{fg}(z)p_{bg}(x_{bg})\,} where z = ( x o x m ) {\displaystyle z=(x^{o}x^{m})\,} are the coordinates of all foreground detections, observed and missing, and x b g {\displaystyle x_{bg}\,} represents the coordinates of the background detections. Note that foreground detections are assumed to be independent of the background. p f g ( z ) {\displaystyle p_{fg}(z)\,} is modeled as a joint Gaussian with mean μ {\displaystyle \mu \,} and covariance Σ {\displaystyle \Sigma \,} . === Classification === The ultimate objective of this model is to classify images into classes "object present" (class C 1 {\displaystyle C_{1}\,} ) and "object absent" (class C 0 {\displaystyle C_{0}\,} ) given t
OpenIO
OpenIO offered object storage for a wide range of high-performance applications. OpenIO was founded in 2015 by Laurent Denel (CEO), Jean-François Smigielski (CTO) and five other co-founders; it leveraged open source software, developed since 2006, based on a grid technology that enabled dynamic behaviour and supported heterogenous hardware. In October 2017 OpenIO was completed a $5 million funding rounds. In July 2020 OpenIO had been acquired by OVH and withdrawn from the market to become the core technology of OVHcloud object storage offering. == Software == OpenIO is a software-defined object store that supports S3 and can be deployed on-premises, cloud-hosted or at the edge, on any hardware mix. It has been designed from the beginning for performance and cost-efficiency at any scale, and it has been optimized for Big Data, HPC and AI. OpenIO stores objects within a flat structure within a massively distributed directory with indirections, which allows the data query path to be independent of the number of nodes and the performance not to be affected by the growth of capacity. Servers are organized as a grid of nodes massively distributed, where each node takes part in directory and storage services, which ensures that there is no single point of failure and that new nodes are automatically discovered and immediately available without the need to rebalance data. The software is built on top of a technology that ensures optimal data placement based on real-time metrics and allows the addition or removal of storage devices with automatic performance and load impact optimization. For data protection OpenIO has synchronous and asynchronous replication with multiple copies, and an erasure coding implementation based on Reed-Solomon that can be deployed in one data center or geo-distributed or stretched clusters. The software has a feature that catches all events that occur in the cluster and can pass them up in the stack or to applications running on OpenIO nodes. This enables event-driven computing directly into the storage infrastructure. The open source code is available on Github and it is licensed under AGPL3 for server code and LGPL3 for client code. == Performance == OpenIO claimed in 2019 to have reached 1.372 Tbit/s write speed (171 GB/s) on a cluster of 350 physical machines. The benchmark scenario, conducted under production conditions with standard hardware (commodity servers with 7200 rpm HDDs), consisted in backing up a 38 PB Hadoop datalake via the DistCp command. This level of performance marked, according to analysts, the arrival of a new generation of object storage technologies oriented toward high performance and hyper-scalability.
Probably approximately correct learning
In computational learning theory, probably approximately correct (PAC) learning is a framework for mathematical analysis of machine learning. It was proposed in 1984 by Leslie Valiant. In this framework, the learner receives samples and must select a generalization function (called the hypothesis) from a certain class of possible functions. The goal is that, with high probability (the "probably" part), the selected function will have low generalization error (the "approximately correct" part). The learner must be able to learn the concept given any arbitrary approximation ratio, probability of success, or distribution of the samples. The model was later extended to treat noise (misclassified samples). An important innovation of the PAC framework is the introduction of computational complexity theory concepts to machine learning. In particular, the learner is expected to find efficient functions (time and space requirements bounded to a polynomial of the example size), and the learner itself must implement an efficient procedure (requiring an example count bounded to a polynomial of the concept size, modified by the approximation and likelihood bounds). == Definitions and terminology == In order to give the definition for something that is PAC-learnable, we first have to introduce some terminology. For the following definitions, two examples will be used. The first is the problem of character recognition given an array of n {\displaystyle n} bits encoding a binary-valued image. The other example is the problem of finding an interval that will correctly classify points within the interval as positive and the points outside of the range as negative. Let X {\displaystyle X} be a set called the instance space or the encoding of all the samples. In the character recognition problem, the instance space is X = { 0 , 1 } n {\displaystyle X=\{0,1\}^{n}} . In the interval problem the instance space, X {\displaystyle X} , is the set of all bounded intervals in R {\displaystyle \mathbb {R} } , where R {\displaystyle \mathbb {R} } denotes the set of all real numbers. A concept is a subset c ⊂ X {\displaystyle c\subset X} . One concept is the set of all patterns of bits in X = { 0 , 1 } n {\displaystyle X=\{0,1\}^{n}} that encode a picture of the letter "P". An example concept from the second example is the set of open intervals, { ( a , b ) ∣ 0 ≤ a ≤ π / 2 , π ≤ b ≤ 13 } {\displaystyle \{(a,b)\mid 0\leq a\leq \pi /2,\pi \leq b\leq {\sqrt {13}}\}} , each of which contains only the positive points. A concept class C {\displaystyle C} is a collection of concepts over X {\displaystyle X} . This could be the set of all subsets of the array of bits that are skeletonized 4-connected (width of the font is 1). Let EX ( c , D ) {\displaystyle \operatorname {EX} (c,D)} be a procedure that draws an example, x {\displaystyle x} , using a probability distribution D {\displaystyle D} and gives the correct label c ( x ) {\displaystyle c(x)} , that is 1 if x ∈ c {\displaystyle x\in c} and 0 otherwise. Now, given 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} , assume there is an algorithm A {\displaystyle A} and a polynomial p {\displaystyle p} in 1 / ϵ , 1 / δ {\displaystyle 1/\epsilon ,1/\delta } (and other relevant parameters of the class C {\displaystyle C} ) such that, given a sample of size p {\displaystyle p} drawn according to EX ( c , D ) {\displaystyle \operatorname {EX} (c,D)} , then, with probability of at least 1 − δ {\displaystyle 1-\delta } , A {\displaystyle A} outputs a hypothesis h ∈ C {\displaystyle h\in C} that has an average error less than or equal to ϵ {\displaystyle \epsilon } on X {\displaystyle X} with the same distribution D {\displaystyle D} . Further if the above statement for algorithm A {\displaystyle A} is true for every concept c ∈ C {\displaystyle c\in C} and for every distribution D {\displaystyle D} over X {\displaystyle X} , and for all 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} then C {\displaystyle C} is (efficiently) PAC learnable (or distribution-free PAC learnable). We can also say that A {\displaystyle A} is a PAC learning algorithm for C {\displaystyle C} . == Equivalence == Under some regularity conditions these conditions are equivalent: The concept class C is PAC learnable. The VC dimension of C is finite. C is a uniformly Glivenko-Cantelli class. C is compressible in the sense of Littlestone and Warmuth
Information Harvesting
Information Harvesting (IH) was an early data mining product from the 1990s. It was invented by Ralphe Wiggins and produced by the Ryan Corp, later Information Harvesting Inc., of Cambridge, Massachusetts. Wiggins had a background in genetic algorithms and fuzzy logic. IH sought to infer rules from sets of data. It did this first by classifying various input variables into one of a number of bins, thereby putting some structure on the continuous variables in the input. IH then proceeds to generate rules, trading off generalization against memorization, that will infer the value of the prediction variable, possibly creating many levels of rules in the process. It included strategies for checking if overfitting took place and, if so, correcting for it. Because of its strategies for correcting for overfitting by considering more data, and refining the rules based on that data, IH might also be considered to be a form of machine learning. The advantage of IH, as compared with other data mining products of its time and even later, was that it provided a mechanism for finding multiple rules that would classify the data and determining, according to set criteria, the best rules to use.
Principal component analysis
Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data are linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified. The principal components of a collection of points in a real coordinate space are a sequence of p {\displaystyle p} unit vectors, where the i {\displaystyle i} -th vector is the direction of a line that best fits the data while being orthogonal to the first i − 1 {\displaystyle i-1} vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science. == Overview == When performing PCA, the first principal component of a set of p {\displaystyle p} variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through p {\displaystyle p} iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The i {\displaystyle i} -th principal component can be taken as a direction orthogonal to the first i − 1 {\displaystyle i-1} principal components that maximizes the variance of the projected data. For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. Robust and L1-norm-based variants of standard PCA have also been proposed. == History == PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (invented in the last quarter of the 19th century), eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's Principal Component Analysis), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics. == Intuition == PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small. To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the covariance matrix of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must normalize each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues. Biplots and scree plots (degree of explained variance) are used to interpret findings of the PCA. == Details == PCA is defined as an orthogonal linear transformation on a real inner product space that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Consider an n × p {\displaystyle n\times p} data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the n rows represents a different repetition of the experiment, and each of the p columns gives a particular kind of feature (say, the results from a particular sensor). Mathematically, the transformation is defined by a set of size l {\displaystyle l} (where l {\displaystyle l} is usually selected to be strictly less than p {\displaystyle p} to reduce dimensionality) of p {\displaystyle p} -dimensional vectors of weights or coefficients w ( k ) = ( w 1 , … , w p ) ( k ) {\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}} that map each row vector x ( i ) = ( x 1 , … , x p ) ( i ) {\displaystyle \mathbf {x} _{(i)}=(x_{1},\dots ,x_{p})_{(i)}} of X to a new vector of principal component scores t ( i ) = ( t 1 , … , t l ) ( i ) {\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}} , given by t k ( i ) = x ( i ) ⋅ w ( k ) f o r i = 1 , … , n k = 1 , … , l {\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l} in such a way that the individual variables t 1 , … , t l {\displaystyle t_{1},\dots ,t_{l}} of t considered over the data set successively inherit the maximum possible variance from X, with each coefficient vector w constrained to be a unit vector. The above may equivalently be written in matrix form as T = X W {\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} } where T i k = t k ( i ) {\displaystyle {\mathbf {T} }_{ik}={t_{k}}_{(i)}} , X i j = x j ( i ) {\displaystyle {\mathbf {X} }_{ij}={x_{j}}_{(i)}} , and W j k = w j ( k ) {\displaystyle {\mathbf {W} }_{jk}={w_{j}}_{(k)}} . === First component === In order to maximize variance, the first weight vector w(1) thus has to satisfy w ( 1 ) = arg max ‖ w ‖ = 1 { ∑ i ( t 1 ) ( i ) 2 } = arg max ‖ w ‖ = 1 { ∑ i ( x ( i ) ⋅ w ) 2 } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}} Equivalently, writing this in matrix form gives w ( 1 ) = arg max ‖ w ‖ = 1 { ‖ X w ‖ 2 } = arg max ‖ w ‖ = 1 { w T X T X w } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}} Since w(1) has been defined to be a unit vector, it equivalently also satisfies w ( 1 ) = arg max { w T X T X w w T w } {\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}} The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as XTX is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector. With w(1) found, the first principal component of a data vector
Algorithmic inference
Algorithmic inference gathers new developments in the statistical inference methods made feasible by the powerful computing devices widely available to any data analyst. Cornerstones in this field are computational learning theory, granular computing, bioinformatics, and, long ago, structural probability (Fraser 1966). The main focus is on the algorithms which compute statistics rooting the study of a random phenomenon, along with the amount of data they must feed on to produce reliable results. This shifts the interest of mathematicians from the study of the distribution laws to the functional properties of the statistics, and the interest of computer scientists from the algorithms for processing data to the information they process. == The Fisher parametric inference problem == Concerning the identification of the parameters of a distribution law, the mature reader may recall lengthy disputes in the mid 20th century about the interpretation of their variability in terms of fiducial distribution (Fisher 1956), structural probabilities (Fraser 1966), priors/posteriors (Ramsey 1925), and so on. From an epistemology viewpoint, this entailed a companion dispute as to the nature of probability: is it a physical feature of phenomena to be described through random variables or a way of synthesizing data about a phenomenon? Opting for the latter, Fisher defines a fiducial distribution law of parameters of a given random variable that he deduces from a sample of its specifications. With this law he computes, for instance "the probability that μ (mean of a Gaussian variable – omeur note) is less than any assigned value, or the probability that it lies between any assigned values, or, in short, its probability distribution, in the light of the sample observed". == The classic solution == Fisher fought hard to defend the difference and superiority of his notion of parameter distribution in comparison to analogous notions, such as Bayes' posterior distribution, Fraser's constructive probability and Neyman's confidence intervals. For half a century, Neyman's confidence intervals won out for all practical purposes, crediting the phenomenological nature of probability. With this perspective, when you deal with a Gaussian variable, its mean μ is fixed by the physical features of the phenomenon you are observing, where the observations are random operators, hence the observed values are specifications of a random sample. Because of their randomness, you may compute from the sample specific intervals containing the fixed μ with a given probability that you denote confidence. === Example === Let X be a Gaussian variable with parameters μ {\displaystyle \mu } and σ 2 {\displaystyle \sigma ^{2}} and { X 1 , … , X m } {\displaystyle \{X_{1},\ldots ,X_{m}\}} a sample drawn from it. Working with statistics S μ = ∑ i = 1 m X i {\displaystyle S_{\mu }=\sum _{i=1}^{m}X_{i}} and S σ 2 = ∑ i = 1 m ( X i − X ¯ ) 2 , where X ¯ = S μ m {\displaystyle S_{\sigma ^{2}}=\sum _{i=1}^{m}(X_{i}-{\overline {X}})^{2},{\text{ where }}{\overline {X}}={\frac {S_{\mu }}{m}}} is the sample mean, we recognize that T = S μ − m μ S σ 2 m − 1 m = X ¯ − μ S σ 2 / ( m ( m − 1 ) ) {\displaystyle T={\frac {S_{\mu }-m\mu }{\sqrt {S_{\sigma ^{2}}}}}{\sqrt {\frac {m-1}{m}}}={\frac {{\overline {X}}-\mu }{\sqrt {S_{\sigma ^{2}}/(m(m-1))}}}} follows a Student's t distribution (Wilks 1962) with parameter (degrees of freedom) m − 1, so that f T ( t ) = Γ ( m / 2 ) Γ ( ( m − 1 ) / 2 ) 1 π ( m − 1 ) ( 1 + t 2 m − 1 ) m / 2 . {\displaystyle f_{T}(t)={\frac {\Gamma (m/2)}{\Gamma ((m-1)/2)}}{\frac {1}{\sqrt {\pi (m-1)}}}\left(1+{\frac {t^{2}}{m-1}}\right)^{m/2}.} Gauging T between two quantiles and inverting its expression as a function of μ {\displaystyle \mu } you obtain confidence intervals for μ {\displaystyle \mu } . With the sample specification: x = { 7.14 , 6.3 , 3.9 , 6.46 , 0.2 , 2.94 , 4.14 , 4.69 , 6.02 , 1.58 } {\displaystyle \mathbf {x} =\{7.14,6.3,3.9,6.46,0.2,2.94,4.14,4.69,6.02,1.58\}} having size m = 10, you compute the statistics s μ = 43.37 {\displaystyle s_{\mu }=43.37} and s σ 2 = 46.07 {\displaystyle s_{\sigma ^{2}}=46.07} , and obtain a 0.90 confidence interval for μ {\displaystyle \mu } with extremes (3.03, 5.65). == Inferring functions with the help of a computer == From a modeling perspective the entire dispute looks like a chicken-egg dilemma: either fixed data by first and probability distribution of their properties as a consequence, or fixed properties by first and probability distribution of the observed data as a corollary. The classic solution has one benefit and one drawback. The former was appreciated particularly back when people still did computations with sheet and pencil. Per se, the task of computing a Neyman confidence interval for the fixed parameter θ is hard: you do not know θ, but you look for disposing around it an interval with a possibly very low probability of failing. The analytical solution is allowed for a very limited number of theoretical cases. Vice versa a large variety of instances may be quickly solved in an approximate way via the central limit theorem in terms of confidence interval around a Gaussian distribution – that's the benefit. The drawback is that the central limit theorem is applicable when the sample size is sufficiently large. Therefore, it is less and less applicable with the sample involved in modern inference instances. The fault is not in the sample size on its own part. Rather, this size is not sufficiently large because of the complexity of the inference problem. With the availability of large computing facilities, scientists refocused from isolated parameters inference to complex functions inference, i.e. re sets of highly nested parameters identifying functions. In these cases we speak about learning of functions (in terms for instance of regression, neuro-fuzzy system or computational learning) on the basis of highly informative samples. A first effect of having a complex structure linking data is the reduction of the number of sample degrees of freedom, i.e. the burning of a part of sample points, so that the effective sample size to be considered in the central limit theorem is too small. Focusing on the sample size ensuring a limited learning error with a given confidence level, the consequence is that the lower bound on this size grows with complexity indices such as VC dimension or detail of a class to which the function we want to learn belongs. === Example === A sample of 1,000 independent bits is enough to ensure an absolute error of at most 0.081 on the estimation of the parameter p of the underlying Bernoulli variable with a confidence of at least 0.99. The same size cannot guarantee a threshold less than 0.088 with the same confidence 0.99 when the error is identified with the probability that a 20-year-old man living in New York does not fit the ranges of height, weight and waistline observed on 1,000 Big Apple inhabitants. The accuracy shortage occurs because both the VC dimension and the detail of the class of parallelepipeds, among which the one observed from the 1,000 inhabitants' ranges falls, are equal to 6. == The general inversion problem solving the Fisher question == With insufficiently large samples, the approach: fixed sample – random properties suggests inference procedures in three steps: === Definition === For a random variable and a sample drawn from it a compatible distribution is a distribution having the same sampling mechanism M X = ( Z , g θ ) {\displaystyle {\mathcal {M}}_{X}=(Z,g_{\boldsymbol {\theta }})} of X with a value θ {\displaystyle {\boldsymbol {\theta }}} of the random parameter Θ {\displaystyle \mathbf {\Theta } } derived from a master equation rooted on a well-behaved statistic s. === Example === You may find the distribution law of the Pareto parameters A and K as an implementation example of the population bootstrap method as in the figure on the left. Implementing the twisting argument method, you get the distribution law F M ( μ ) {\displaystyle F_{M}(\mu )} of the mean M of a Gaussian variable X on the basis of the statistic s M = ∑ i = 1 m x i {\textstyle s_{M}=\sum _{i=1}^{m}x_{i}} when Σ 2 {\displaystyle \Sigma ^{2}} is known to be equal to σ 2 {\displaystyle \sigma ^{2}} (Apolloni, Malchiodi & Gaito 2006). Its expression is: F M ( μ ) = Φ ( m μ − s M σ m ) , {\displaystyle F_{M}(\mu )=\Phi {\left({\frac {m\mu -s_{M}}{\sigma {\sqrt {m}}}}\right)},} shown in the figure on the right, where Φ {\displaystyle \Phi } is the cumulative distribution function of a standard normal distribution. Computing a confidence interval for M given its distribution function is straightforward: we need only find two quantiles (for instance δ / 2 {\displaystyle \delta /2} and 1 − δ / 2 {\displaystyle 1-\delta /2} quantiles in case we are interested in a confidence interval of level δ symmetric in the tail's probabilities) as indicated on the left in the diagram showing the behavior of
Stress majorization
Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of n {\displaystyle n} m {\displaystyle m} -dimensional data items, a configuration X {\displaystyle X} of n {\displaystyle n} points in r {\displaystyle r} ( ≪ m ) {\displaystyle (\ll m)} -dimensional space is sought that minimizes the so-called stress function σ ( X ) {\displaystyle \sigma (X)} . Usually r {\displaystyle r} is 2 {\displaystyle 2} or 3 {\displaystyle 3} , i.e. the ( n × r ) {\displaystyle (n\times r)} matrix X {\displaystyle X} lists points in 2 − {\displaystyle 2-} or 3 − {\displaystyle 3-} dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot). The function σ {\displaystyle \sigma } is a cost or loss function that measures the squared differences between ideal ( m {\displaystyle m} -dimensional) distances and actual distances in r-dimensional space. It is defined as: σ ( X ) = ∑ i < j ≤ n w i j ( d i j ( X ) − δ i j ) 2 {\displaystyle \sigma (X)=\sum _{i