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In Bayesian statistics, the **posterior probability** of a random event or an uncertain proposition^{[ clarification needed ]} is the conditional probability that is assigned^{[ clarification needed ]} after the relevant evidence or background is taken into account. Similarly, the **posterior probability distribution** is the probability distribution of an unknown quantity, treated as a random variable, conditional on the evidence obtained from an experiment or survey. "Posterior", in this context, means after taking into account the relevant evidence related to the particular case being examined. For instance, there is a ("non-posterior") probability of a person finding buried treasure if they dig in a random spot, and a posterior probability of finding buried treasure if they dig in a spot where their metal detector rings.

**Bayesian statistics** is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a *degree of belief* in an event. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event. This differs from a number of other interpretations of probability, such as the frequentist interpretation that views probability as the limit of the relative frequency of an event after a large number of trials.

In probability theory, **conditional probability** is a measure of the probability of an event occurring given that another event has occurred. If the event of interest is *A* and the event *B* is known or assumed to have occurred, "the conditional probability of *A* given *B*", or "the probability of *A* under the condition *B*", is usually written as P(*A* | *B*), or sometimes *P*_{B}(*A*) or P(*A* / *B*). For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person has a cold, then they are much more likely to be coughing. The conditional probability of coughing by the unwell might be 75%, then: P(Cough) = 5%; P(Cough | Sick) = 75%

The posterior probability is the probability of the parameters given the evidence : .

It contrasts with the likelihood function, which is the probability of the evidence given the parameters: .

In statistics, the **likelihood function** expresses how likely particular values of statistical parameters are for a given set of observations. It is equal to the joint probability distribution of the random sample evaluated at the given observations, and it is, thus, solely a function of parameters that index the family of those probability distributions.

The two are related as follows:

Let us have a prior belief that the probability distribution function is and observations with the likelihood , then the posterior probability is defined as

In Bayesian statistical inference, a **prior probability distribution**, often simply called the **prior**, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable.

A **probability distribution function** is some function that may be used to define a particular probability distribution. Depending upon which text is consulted, the term may refer to:

^{ [1] }

The posterior probability can be written in the memorable form as

- .

Suppose there is a school having 60% boys and 40% girls as students. The girls wear trousers or skirts in equal numbers; all boys wear trousers. An observer sees a (random) student from a distance; all the observer can see is that this student is wearing trousers. What is the probability this student is a girl? The correct answer can be computed using Bayes' theorem.

The event is that the student observed is a girl, and the event is that the student observed is wearing trousers. To compute the posterior probability , we first need to know:

- , or the probability that the student is a girl regardless of any other information. Since the observer sees a random student, meaning that all students have the same probability of being observed, and the percentage of girls among the students is 40%, this probability equals 0.4.
- , or the probability that the student is not a girl (i.e. a boy) regardless of any other information ( is the complementary event to ). This is 60%, or 0.6.
- , or the probability of the student wearing trousers given that the student is a girl. As they are as likely to wear skirts as trousers, this is 0.5.
- , or the probability of the student wearing trousers given that the student is a boy. This is given as 1.
- , or the probability of a (randomly selected) student wearing trousers regardless of any other information. Since (via the law of total probability), this is .

In probability theory, the **law****of total probability** is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events—hence the name.

Given all this information, the **posterior probability** of the observer having spotted a girl given that the observed student is wearing trousers can be computed by substituting these values in the formula:

An intuitive way to solve this is to assume the school has N students. Number of boys = 0.6N and number of girls = 0.4N. If N is sufficiently large, total number of trouser wearers = 0.6N+ 50% of 0.4N. And number of girl trouser wearers = 50% of 0.4N. Therefore, in the population of trousers, girls are (50% of 0.4N)/(0.6N+ 50% of 0.4N) = 25%. In other words, if you separated out the group of trouser wearers, a quarter of that group will be girls. Therefore, if you see trousers, the most you can deduce is that you are looking at a single sample from a subset of students where 25% are girls. And by definition, chance of this random student being a girl is 25%. Every Bayes theorem problem can be solved in this way .

The posterior probability distribution of one random variable given the value of another can be calculated with Bayes' theorem by multiplying the prior probability distribution by the likelihood function, and then dividing by the normalizing constant, as follows:

In probability and statistics, a **random variable**, **random quantity**, **aleatory variable**, or **stochastic variable** is described informally as a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space whose outcomes are typically real numbers.

In probability theory and statistics, **Bayes' theorem** describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes' theorem, a person's age can be used to more accurately assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person's age.

The concept of a **normalizing constant** arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one.

gives the posterior probability density function for a random variable given the data , where

In probability theory, a **probability density function** (**PDF**), or **density** of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a *relative likelihood* that the value of the random variable would equal that sample. In other words, while the *absolute likelihood* for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

- is the prior density of ,
- is the likelihood function as a function of ,
- is the normalizing constant, and
- is the posterior density of given the data .

Posterior probability is a conditional probability conditioned on randomly observed data. Hence it is a random variable. For a random variable, it is important to summarize its amount of uncertainty. One way to achieve this goal is to provide a credible interval of the posterior probability.

In classification, posterior probabilities reflect the uncertainty of assessing an observation to particular class, see also Class membership probabilities. While statistical classification methods by definition generate posterior probabilities, Machine Learners usually supply membership values which do not induce any probabilistic confidence. It is desirable to transform or re-scale membership values to class membership probabilities, since they are comparable and additionally more easily applicable for post-processing.

**Bayesian inference** is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability".

In statistics, **maximum likelihood estimation** (**MLE**) is a method of estimating the parameters by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the **maximum likelihood estimate**. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

In statistics, a statistic is *sufficient* with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter". In particular, a statistic is **sufficient** for a family of probability distributions if the sample from which it is calculated gives no additional information than does the statistic, as to which of those probability distributions is that of the population from which the sample was taken.

A **Bayesian network**, **Bayes network**, **belief network**, **decision network**, **Bayes(ian) model** or **probabilistic directed acyclic graphical model** is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.

In mathematical statistics, the **Kullback–Leibler divergence** is a measure of how one probability distribution is different from a second, reference probability distribution. Applications include characterizing the **relative**(Shannon) **entropy** in information systems, **randomness** in continuous time-series, and **information gain** when comparing statistical models of inference. In contrast to variation of information, it is a distribution-wise *asymmetric* measure and thus does not qualify as a statistical *metric* of spread. In the simple case, a Kullback–Leibler divergence of 0 indicates that the two distributions in question are identical. In simplified terms, it is a measure of surprise, with diverse applications such as applied statistics, fluid mechanics, neuroscience and machine learning.

In numerical analysis and computational statistics, **rejection sampling** is a basic technique used to generate observations from a distribution. It is also commonly called the **acceptance-rejection method** or "accept-reject algorithm" and is a type of exact simulation method. The method works for any distribution in with a density.

In statistics, the **score** is the gradient of the log-likelihood function with respect to the parameter vector. Evaluated at a particular point, the score indicates the steepness of the log-likelihood function and thereby the sensitivity to infinitesimal changes to the parameter values. If the log-likelihood function is continuous over the parameter space, the score will vanish at a local maximum or minimum; this fact is used in maximum likelihood estimation to find the parameter values that maximize the likelihood function.

In mathematical statistics, the **Fisher information** is a way of measuring the amount of information that an observable random variable *X* carries about an unknown parameter *θ* of a distribution that models *X*. Formally, it is the variance of the score, or the expected value of the observed information. In Bayesian statistics, the asymptotic distribution of the posterior mode depends on the Fisher information and not on the prior. The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized by the statistician Ronald Fisher. The Fisher information is also used in the calculation of the Jeffreys prior, which is used in Bayesian statistics.

In statistics, the **generalized linear model** (**GLM**) is a flexible generalization of ordinary linear regression that allows for response variables that have error distribution models other than a normal distribution. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a *link function* and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.

In Bayesian probability theory, if the posterior distributions *p*(*θ* | *x*) are in the same probability distribution family as the prior probability distribution *p*(θ), the prior and posterior are then called **conjugate distributions,** and the prior is called a **conjugate prior** for the likelihood function. For example, the Gaussian family is conjugate to itself with respect to a Gaussian likelihood function: if the likelihood function is Gaussian, choosing a Gaussian prior over the mean will ensure that the posterior distribution is also Gaussian. This means that the Gaussian distribution is a conjugate prior for the likelihood that is also Gaussian. The concept, as well as the term "conjugate prior", were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory. A similar concept had been discovered independently by George Alfred Barnard.

In statistics, a **marginal likelihood function**, or **integrated likelihood**, is a likelihood function in which some parameter variables have been marginalized. In the context of Bayesian statistics, it may also be referred to as the **evidence** or **model evidence**.

In Bayesian statistics, a **maximum a posteriori probability** (**MAP**) **estimate** is an estimate of an unknown quantity, that equals the mode of the posterior distribution. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior distribution over the quantity one wants to estimate. MAP estimation can therefore be seen as a regularization of ML estimation.

In mathematics, a **π-system** on a set Ω is a collection *P* of certain subsets of Ω, such that

**Bayesian econometrics** is a branch of econometrics which applies Bayesian principles to economic modelling. Bayesianism is based on a degree-of-belief interpretation of probability, as opposed to a relative-frequency interpretation.

A **product distribution** is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables *X* and *Y*, the distribution of the random variable *Z* that is formed as the product

In probability and statistics, a **compound probability distribution** is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with the parameters of that distribution themselves being random variables.

**Thompson sampling**, named after William R. Thompson, is a heuristic for choosing actions that addresses the exploration-exploitation dilemma in the multi-armed bandit problem. It consists in choosing the action that maximizes the expected reward with respect to a randomly drawn belief.

**Bayesian hierarchical modelling** is a statistical model written in multiple levels that estimates the parameters of the posterior distribution 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. The result of this integration is the posterior distribution, also known as the updated probability estimate, as additional evidence on the prior distribution is acquired.

- ↑ Christopher M. Bishop (2006).
*Pattern Recognition and Machine Learning*. Springer. pp. 21–24. ISBN 978-0-387-31073-2.

- Lancaster, Tony (2004).
*An Introduction to Modern Bayesian Econometrics*. Oxford: Blackwell. ISBN 1-4051-1720-6. - Lee, Peter M. (2004).
*Bayesian Statistics : An Introduction*(3rd ed.). Wiley. ISBN 0-340-81405-5.

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