# Posterior probability

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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 PB(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%

## Definition

The posterior probability is the probability of the parameters ${\displaystyle \theta }$ given the evidence ${\displaystyle X}$: ${\displaystyle p(\theta |X)}$.

It contrasts with the likelihood function, which is the probability of the evidence given the parameters: ${\displaystyle p(X|\theta )}$.

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 ${\displaystyle p(\theta )}$ and observations ${\displaystyle x}$ with the likelihood ${\displaystyle p(x|\theta )}$, 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:

${\displaystyle p(\theta |x)={\frac {p(x|\theta )}{p(x)}}p(\theta ).}$ [1]

The posterior probability can be written in the memorable form as

${\displaystyle {\text{Posterior probability}}\propto {\text{Likelihood}}\times {\text{Prior probability}}}$.

## Example

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 ${\displaystyle G}$ is that the student observed is a girl, and the event ${\displaystyle T}$ is that the student observed is wearing trousers. To compute the posterior probability ${\displaystyle P(G|T)}$, we first need to know:

• ${\displaystyle P(G)}$, 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.
• ${\displaystyle P(B)}$, or the probability that the student is not a girl (i.e. a boy) regardless of any other information (${\displaystyle B}$ is the complementary event to ${\displaystyle G}$). This is 60%, or 0.6.
• ${\displaystyle P(T|G)}$, 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.
• ${\displaystyle P(T|B)}$, or the probability of the student wearing trousers given that the student is a boy. This is given as 1.
• ${\displaystyle P(T)}$, or the probability of a (randomly selected) student wearing trousers regardless of any other information. Since ${\displaystyle P(T)=P(T|G)P(G)+P(T|B)P(B)}$ (via the law of total probability), this is ${\displaystyle P(T)=0.5\times 0.4+1\times 0.6=0.8}$.

In probability theory, the lawof 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:

${\displaystyle P(G|T)={\frac {P(T|G)P(G)}{P(T)}}={\frac {0.5\times 0.4}{0.8}}=0.25.}$

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 .

## Calculation

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.

${\displaystyle f_{X\mid Y=y}(x)={f_{X}(x){\mathcal {L}}_{X\mid Y=y}(x) \over {\int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du}}}$

gives the posterior probability density function for a random variable ${\displaystyle X}$ given the data ${\displaystyle Y=y}$, 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.

• ${\displaystyle f_{X}(x)}$ is the prior density of ${\displaystyle X}$,
• ${\displaystyle {\mathcal {L}}_{X\mid Y=y}(x)=f_{Y\mid X=x}(y)}$ is the likelihood function as a function of ${\displaystyle x}$,
• ${\displaystyle \int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du}$ is the normalizing constant, and
• ${\displaystyle f_{X\mid Y=y}(x)}$ is the posterior density of ${\displaystyle X}$ given the data ${\displaystyle Y=y}$.

## Credible interval

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.

## Classification

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.

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## References

1. 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.