Meta AI is a research division of Meta (formerly Facebook) that develops artificial intelligence and augmented reality technologies. == History == Meta AI was founded in 2013 as Facebook Artificial Intelligence Research (FAIR). It has workspaces in Menlo Park, London, New York City, Paris, Seattle, Pittsburgh, Tel Aviv, and Montreal as of 2025. In 2016, FAIR partnered with Google, Amazon, IBM, and Microsoft in creating the Partnership on Artificial Intelligence to Benefit People and Society. Meta AI was directed by Yann LeCun until 2018, when Jérôme Pesenti succeeded the role. Pesenti is formerly the CTO of IBM's big data group. FAIR's research includes self-supervised learning, generative adversarial networks, document classification and translation, and computer vision. FAIR released Torch deep-learning modules as well as PyTorch in 2017, an open-source machine learning framework, which was subsequently used in several deep learning technologies, such as Tesla's autopilot and Uber's Pyro. That same year, a pair of chatbots were falsely rumored to be discontinued for developing a language that was unintelligible to humans. FAIR clarified that the research had been shut down because they had accomplished their initial goal to understand how languages are generated by their models, rather than out of fear. FAIR was renamed Meta AI following the rebranding that changed Facebook, Inc. to Meta Platforms Inc. On October 1, 2025, Facebook announced "We will soon use your interactions with AI at Meta to personalize the content and ads you see". == Virtual assistant == Meta AI is also the name of the virtual assistant developed by the team, now integrated as a chatbot into Meta's social networking products. It is also available as a subscription-based stand-alone app. The virtual assistant was pre-installed on the second generation of Ray-Ban Meta smartglasses, and can incorporate inputs from the glasses' cameras after an update. It is also available on Quest 2 and newer HMDs. Since May 2024, the chatbot has summarized news from various outlets without linking directly to original articles, including in Canada, where news links are banned on its platforms. This use of news content without compensation and attribution has raised ethical and legal concerns, especially as Meta continues to reduce news visibility on its platforms. == Current research == === Natural language processing and chatbot === Natural language processing is the ability for machines to understand and generate natural language. The team is also researching unsupervised machine translation and multilingual chatbots. ==== Galactica ==== Galactica is a large language model (LLM) designed for generating scientific text. It was available for three days from 15 November 2022, before being withdrawn for generating racist and inaccurate content. ==== Llama ==== Llama is an LLM released in February 2023. As of January 2026, the most recent release is the Llama 4. === Hardware === Meta used CPUs and in-house custom chips before 2022; they switched to Nvidia GPUs since then. MTIA v1, one of their early chips, is designed for the company's content recommendation algorithms. It was fabricated on TSMC's 7 nm process technology and consumed 25W, capable of 51.2 TFlops FP16. == Controversy == The French media outlet Mediapart reports that in 2022, Facebook's parent company illegally used works accumulated by the pirate site LibGen to train its artificial intelligence.
Kuki AI
Kuki is an embodied AI bot designed for usage in the metaverse. Formerly known as Mitsuku, Kuki is a chatbot created from the Pandorabots framework. The bot has won the Loebner Prize 5 times. == Features == Kuki claims to be an 18-year-old female chatbot from the Metaverse, and the developers have stated she has been worked on since 2005. Early work by one of the company's co-founders inspired the Spike Jonze movie Her. As of 2015, she conversed, on average, in excess of a quarter of a million times daily, and it was estimated 5 million unique users had interacted with her between 2016 and 2020. == Virtual talent, model, and influencer == Kuki has appeared as a Virtual Model in Vogue Business and at Crypto Fashion Week where she modelled NFTs and spoke about the future of digital fashion. In 2021, Kuki modelled five digital looks from emerging Vogue Talents designers for Italian Vogue, that sold out as NFTs in under an hour. Kuki has also modeled for H&M on Instagram in a digital campaign that resulted in an "11x increase in ad recall" per a case study by Meta. == Awards == As of 2019, Kuki had been awarded the Loebner Prize five times, more than any other entrant. In 2020, Kuki competed against Facebook AI's Blenderbot in a 24/7 verbal sparring match called "Bot Battle", winning 79% of the audience vote.
Google Research
Google Research (also known as Research at Google) is the research division of Google, a subsidiary of Alphabet Inc.. According to its official website, Google Research publishes findings, releases open-source software, and applies research results within Google products and services as well as within the wider scientific community. == Notable contributions == The 2017 landmark paper Attention Is All You Need, which introduced the Transformer architecture, which has subsequently been used to build modern large language models. Advances in neural machine translation powering Google Translate. Time series forecasting. Development of scalable learning systems and infrastructure for large-model training. Flood forecasting. Research into computational discovery via Google Accelerated Science including demonstrating the first below-threshold quantum calculations.
Cognitive computing
Cognitive computing refers to technology platforms that, broadly speaking, are based on the scientific disciplines of artificial intelligence and signal processing. These platforms encompass machine learning, reasoning, natural language processing, speech recognition and vision (object recognition), human–computer interaction, dialog and narrative generation, among other technologies. == Definition == At present, there is no widely agreed upon definition for cognitive computing in either academia or industry. In general, the term cognitive computing has been used to refer to new hardware and/or software that mimics the functioning of the human brain (2004). In this sense, cognitive computing is a new type of computing with the goal of more accurate models of how the human brain/mind senses, reasons, and responds to stimulus. Cognitive computing applications link data analysis and adaptive page displays (AUI) to adjust content for a particular type of audience. As such, cognitive computing hardware and applications strive to be more affective and more influential by design. The term "cognitive system" also applies to any artificial construct able to perform a cognitive process where a cognitive process is the transformation of data, information, knowledge, or wisdom to a new level in the DIKW Pyramid. While many cognitive systems employ techniques having their origination in artificial intelligence research, cognitive systems, themselves, may not be artificially intelligent. For example, a neural network trained to recognize cancer on an MRI scan may achieve a higher success rate than a human doctor. This system is certainly a cognitive system but is not artificially intelligent. Cognitive systems may be engineered to feed on dynamic data in real-time, or near real-time, and may draw on multiple sources of information, including both structured and unstructured digital information, as well as sensory inputs (visual, gestural, auditory, or sensor-provided). == Cognitive analytics == Cognitive computing-branded technology platforms typically specialize in the processing and analysis of large, unstructured datasets. == Applications == Education Even if cognitive computing can not take the place of teachers, it can still be a heavy driving force in the education of students. Cognitive computing being used in the classroom is applied by essentially having an assistant that is personalized for each individual student. This cognitive assistant can relieve the stress that teachers face while teaching students, while also enhancing the student's learning experience over all. Teachers may not be able to pay each and every student individual attention, this being the place that cognitive computers fill the gap. Some students may need a little more help with a particular subject. For many students, Human interaction between student and teacher can cause anxiety and can be uncomfortable. With the help of Cognitive Computer tutors, students will not have to face their uneasiness and can gain the confidence to learn and do well in the classroom. While a student is in class with their personalized assistant, this assistant can develop various techniques, like creating lesson plans, to tailor and aid the student and their needs. Healthcare Numerous tech companies are in the process of developing technology that involves cognitive computing that can be used in the medical field. The ability to classify and identify is one of the main goals of these cognitive devices. This trait can be very helpful in the study of identifying carcinogens. This cognitive system that can detect would be able to assist the examiner in interpreting countless numbers of documents in a lesser amount of time than if they did not use Cognitive Computer technology. This technology can also evaluate information about the patient, looking through every medical record in depth, searching for indications that can be the source of their problems. Commerce Together with Artificial Intelligence, it has been used in warehouse management systems to collect, store, organize and analyze all related supplier data. All these aims at improving efficiency, enabling faster decision-making, monitoring inventory and fraud detection Human Cognitive Augmentation In situations where humans are using or working collaboratively with cognitive systems, called a human/cog ensemble, results achieved by the ensemble are superior to results obtainable by the human working alone. Therefore, the human is cognitively augmented. In cases where the human/cog ensemble achieves results at, or superior to, the level of a human expert then the ensemble has achieved synthetic expertise. In a human/cog ensemble, the "cog" is a cognitive system employing virtually any kind of cognitive computing technology. Other use cases Speech recognition Sentiment analysis Face detection Risk assessment Fraud detection Behavioral recommendations == Industry work == Cognitive computing in conjunction with big data and algorithms that comprehend customer needs, can be a major advantage in economic decision making. The powers of cognitive computing and artificial intelligence hold the potential to affect almost every task that humans are capable of performing. This can negatively affect employment for humans, as there would be no such need for human labor anymore. It would also increase the inequality of wealth; the people at the head of the cognitive computing industry would grow significantly richer, while workers without ongoing, reliable employment would become less well off. The more industries start to use cognitive computing, the more difficult it will be for humans to compete. Increased use of the technology will also increase the amount of work that AI-driven robots and machines can perform. The influence of competitive individuals in conjunction with artificial intelligence/cognitive computing has the potential to change the course of humankind.
Video Super Resolution
RTX Video Super Resolution (RTX VSR) is a video scaling feature by Nvidia. It was released on February 28, 2023. == History == The feature was first unveiled during CES 2023 as RTX Video Super Resolution. It uses the on-board Tensor Cores to upscale browser video content in real time. Video Super Resolution was initially only available on RTX 30 and 40 series GPUs, while support for 20 series GPUs was added afterwards; it is now available on all Nvidia RTX-branded GPUs. The feature supports input resolutions from 360p to 1440p and a max output of 4K and comes without support for HDR content although that could be likely added in the future. Nvidia released RTX Video Super Resolution 1.5 with improved video quality and RTX 20 series support on October 17, 2023. == Reception == According to ComputerBase, although "the algorithm is not yet working flawlessly", the feature is "overall recommendable".
Empirical risk minimization
In statistical learning theory, the principle of empirical risk minimization defines a family of learning algorithms based on evaluating performance over a known and fixed dataset. The core idea is based on an application of the law of large numbers; more specifically, we cannot know exactly how well a predictive algorithm will work in practice (i.e. the "true risk") because we do not know the true distribution of the data, but we can instead estimate and optimize the performance of the algorithm on a known set of training data. The performance over the known set of training data is referred to as the "empirical risk". == Background == The following situation is a general setting of many supervised learning problems. There are two spaces of objects X {\displaystyle X} and Y {\displaystyle Y} and we would like to learn a function h : X → Y {\displaystyle \ h:X\to Y} (often called hypothesis) which outputs an object y ∈ Y {\displaystyle y\in Y} , given x ∈ X {\displaystyle x\in X} . To do so, there is a training set of n {\displaystyle n} examples ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle \ (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} where x i ∈ X {\displaystyle x_{i}\in X} is an input and y i ∈ Y {\displaystyle y_{i}\in Y} is the corresponding response that is desired from h ( x i ) {\displaystyle h(x_{i})} . To put it more formally, assuming that there is a joint probability distribution P ( x , y ) {\displaystyle P(x,y)} over X {\displaystyle X} and Y {\displaystyle Y} , and that the training set consists of n {\displaystyle n} instances ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle \ (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} drawn i.i.d. from P ( x , y ) {\displaystyle P(x,y)} . The assumption of a joint probability distribution allows for the modelling of uncertainty in predictions (e.g. from noise in data) because y {\displaystyle y} is not a deterministic function of x {\displaystyle x} , but rather a random variable with conditional distribution P ( y | x ) {\displaystyle P(y|x)} for a fixed x {\displaystyle x} . It is also assumed that there is a non-negative real-valued loss function L ( y ^ , y ) {\displaystyle L({\hat {y}},y)} which measures how different the prediction y ^ {\displaystyle {\hat {y}}} of a hypothesis is from the true outcome y {\displaystyle y} . For classification tasks, these loss functions can be scoring rules. The risk associated with hypothesis h ( x ) {\displaystyle h(x)} is then defined as the expectation of the loss function: R ( h ) = E [ L ( h ( x ) , y ) ] = ∫ L ( h ( x ) , y ) d P ( x , y ) . {\displaystyle R(h)=\mathbf {E} [L(h(x),y)]=\int L(h(x),y)\,dP(x,y).} A loss function commonly used in theory is the 0-1 loss function: L ( y ^ , y ) = { 1 if y ^ ≠ y 0 if y ^ = y {\displaystyle L({\hat {y}},y)={\begin{cases}1&{\mbox{ if }}\quad {\hat {y}}\neq y\\0&{\mbox{ if }}\quad {\hat {y}}=y\end{cases}}} . The ultimate goal of a learning algorithm is to find a hypothesis h ∗ {\displaystyle h^{}} among a fixed class of functions H {\displaystyle {\mathcal {H}}} for which the risk R ( h ) {\displaystyle R(h)} is minimal: h ∗ = a r g m i n h ∈ H R ( h ) . {\displaystyle h^{}={\underset {h\in {\mathcal {H}}}{\operatorname {arg\,min} }}\,{R(h)}.} For classification problems, the Bayes classifier is defined to be the classifier minimizing the risk defined with the 0–1 loss function. == Formal definition == In general, the risk R ( h ) {\displaystyle R(h)} cannot be computed because the distribution P ( x , y ) {\displaystyle P(x,y)} is unknown to the learning algorithm. However, given a sample of iid training data points, we can compute an estimate, called the empirical risk, by computing the average of the loss function over the training set; more formally, computing the expectation with respect to the empirical measure: R emp ( h ) = 1 n ∑ i = 1 n L ( h ( x i ) , y i ) . {\displaystyle \!R_{\text{emp}}(h)={\frac {1}{n}}\sum _{i=1}^{n}L(h(x_{i}),y_{i}).} The empirical risk minimization principle states that the learning algorithm should choose a hypothesis h ^ {\displaystyle {\hat {h}}} which minimizes the empirical risk over the hypothesis class H {\displaystyle {\mathcal {H}}} : h ^ = a r g m i n h ∈ H R emp ( h ) . {\displaystyle {\hat {h}}={\underset {h\in {\mathcal {H}}}{\operatorname {arg\,min} }}\,R_{\text{emp}}(h).} Thus, the learning algorithm defined by the empirical risk minimization principle consists in solving the above optimization problem. == Properties == Guarantees for the performance of empirical risk minimization depend strongly on the function class selected as well as the distributional assumptions made. In general, distribution-free methods are too coarse, and do not lead to practical bounds. However, they are still useful in deriving asymptotic properties of learning algorithms, such as consistency. In particular, distribution-free bounds on the performance of empirical risk minimization given a fixed function class can be derived using bounds on the VC complexity of the function class. For simplicity, considering the case of binary classification tasks, it is possible to bound the probability of the selected classifier, ϕ n {\displaystyle \phi _{n}} being much worse than the best possible classifier ϕ ∗ {\displaystyle \phi ^{}} . Consider the risk L {\displaystyle L} defined over the hypothesis class C {\displaystyle {\mathcal {C}}} with growth function S ( C , n ) {\displaystyle {\mathcal {S}}({\mathcal {C}},n)} given a dataset of size n {\displaystyle n} . Then, for every ϵ > 0 {\displaystyle \epsilon >0} : P ( L ( ϕ n ) − L ( ϕ ∗ ) > ϵ ) ≤ 8 S ( C , n ) exp { − n ϵ 2 / 32 } {\displaystyle \mathbb {P} \left(L(\phi _{n})-L(\phi ^{})>\epsilon \right)\leq {\mathcal {8}}S({\mathcal {C}},n)\exp\{-n\epsilon ^{2}/32\}} Similar results hold for regression tasks. These results are often based on uniform laws of large numbers, which control the deviation of the empirical risk from the true risk, uniformly over the hypothesis class. === Impossibility results === It is also possible to show lower bounds on algorithm performance if no distributional assumptions are made. This is sometimes referred to as the No free lunch theorem. Even though a specific learning algorithm may provide the asymptotically optimal performance for any distribution, the finite sample performance is always poor for at least one data distribution. This means that no classifier can improve on the error for a given sample size for all distributions. Specifically, let ϵ > 0 {\displaystyle \epsilon >0} and consider a sample size n {\displaystyle n} and classification rule ϕ n {\displaystyle \phi _{n}} , there exists a distribution of ( X , Y ) {\displaystyle (X,Y)} with risk L ∗ = 0 {\displaystyle L^{}=0} (meaning that perfect prediction is possible) such that: E L n ≥ 1 / 2 − ϵ . {\displaystyle \mathbb {E} L_{n}\geq 1/2-\epsilon .} It is further possible to show that the convergence rate of a learning algorithm is poor for some distributions. Specifically, given a sequence of decreasing positive numbers a i {\displaystyle a_{i}} converging to zero, it is possible to find a distribution such that: E L n ≥ a i {\displaystyle \mathbb {E} L_{n}\geq a_{i}} for all n {\displaystyle n} . This result shows that universally good classification rules do not exist, in the sense that the rule must be low quality for at least one distribution. === Computational complexity === Empirical risk minimization for a classification problem with a 0-1 loss function is known to be an NP-hard problem even for a relatively simple class of functions such as linear classifiers. Nevertheless, it can be solved efficiently when the minimal empirical risk is zero, i.e., data is linearly separable. In practice, machine learning algorithms cope with this issue either by employing a convex approximation to the 0–1 loss function (like hinge loss for SVM), which is easier to optimize, or by imposing assumptions on the distribution P ( x , y ) {\displaystyle P(x,y)} (and thus stop being agnostic learning algorithms to which the above result applies). In the case of convexification, Zhang's lemma majors the excess risk of the original problem using the excess risk of the convexified problem. Minimizing the latter using convex optimization also allow to control the former. == Tilted empirical risk minimization == Tilted empirical risk minimization is a machine learning technique used to modify standard loss functions like squared error, by introducing a tilt parameter. This parameter dynamically adjusts the weight of data points during training, allowing the algorithm to focus on specific regions or characteristics of the data distribution. Tilted empirical risk minimization is particularly useful in scenarios with imbalanced data or when there is a need to emphasize errors in certain parts of the prediction space.
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