##  [Posterior Distribution](/posterior-distribution-0) 

 Definition

A probability distribution over unknown parameters or latent quantities obtained by updating a prior distribution with observed data via the likelihood; it assigns to each parameter value its plausibility conditional on the observed dataset.

 

 

 

 

 

 





## Principle

Principle

Update prior beliefs by weighting parameter values according to how well the observed data are predicted by those values (Bayes' rule): posterior ∝ prior × likelihood, normalized to integrate to one.

 

 

 

 

 





## Demonstration

Demonstration

With a Beta(α,β) prior for a Bernoulli success probability and observing k successes in n trials, the posterior is Beta(α+k, β+n−k), giving updated credible mass for the probability.

 

 

 

 

## Misapplication

Misapplication

Reporting only a point summary such as the mode as if it represented the whole posterior, or interpreting the posterior as a frequency statement about repeated sampling without conditioning on the same observed data and prior.

 

 

 

 

 





## Consequence

Consequence

A posterior distribution provides a coherent quantified uncertainty for parameters that permits propagation through predictions, decision rules under loss functions, and model comparison when combined with marginal likelihoods.

 

 

 

 

## Reversal

Reversal

The prior is the distribution before conditioning on data; ignoring data yields the prior, whereas ignoring prior information yields only the likelihood, which is not a probability distribution over parameters.

 

 

 

 

 





## Boundary

Boundary

Requires a specified prior and likelihood; improper priors or misspecified likelihoods can produce improper or misleading posteriors; results depend on both data and the chosen prior model class.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often contrasted with frequentist confidence intervals or point estimators: a posterior is a conditional probability distribution expressing degree of belief, whereas frequentist constructs are sampling-procedure guarantees over hypothetical repetitions.

 

 

 

 

 





## Synthesis

Synthesis

A posterior distribution is the normalized update of a prior by the likelihood that quantifies the conditional plausibility of parameter values given the observed data.