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The updated probability distribution of a parameter after incorporating observed data via Bayes’ theorem.
DESCRIPTION
In Bayesian statistics, the posterior distribution represents beliefs about parameters after observing data. It combines the prior distribution with the likelihood through Bayes’ theorem. Posterior summaries include mean, variance, and credible intervals. Analytical posteriors exist for conjugate priors; otherwise, MCMC or variational inference approximates them. Posterior predictive distributions forecast new data. MAP estimation uses the posterior mode for regularization. Posterior analysis supports decision-making under uncertainty.
ALTERNATIVE DEFINITIONS
Wikipedia
In Bayesian statistics, the posterior probability of a random event or an uncertain proposition is the conditional probability assigned after the relevant evidence or background is taken into account.
Emphasizes conditional probability updated with evidence.
Statlect
The posterior distribution is the probability distribution of an unknown quantity, conditional on the evidence obtained.
Concise focus on conditional distribution given evidence.
RELATED TERMS
Likelihood Bayes’ Theorem Bayesian Inference Credible Interval Conjugate Prior Evidence Maximum A Posteriori Estimation Markov Chain Monte Carlo Bayesian Updating
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KNOWLEDGE AREAS
TARGET AUDIENCE
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EXTERNAL SOURCES
- Wikipedia - Posterior probability
- Statlect - Detailed Bayesian fundamentals
- Encyclopaedia Britannica - Context on Bayes’ theorem
TOP QUESTIONS
What is the Posterior in Bayesian inference?
It’s the distribution of a parameter after observing data, combining prior beliefs with the likelihood via Bayes’ theorem.
Category: Definition
How is the Posterior computed?
It’s proportional to prior times likelihood; normalize with evidence. Analytical forms exist for conjugate priors, else use sampling or approximation.
Category: Requirements
What is the difference between Posterior and Prior?
The prior encodes beliefs before data; the posterior updates those beliefs after incorporating observed evidence.
Category: Misunderstanding
What is a credible interval for a Posterior?
A credible interval is a Bayesian interval within which the parameter lies with a specified posterior probability.
Category: Definition
What are common mistakes when interpreting a Posterior?
Confusing credible intervals with frequentist confidence intervals and ignoring prior sensitivity are typical errors.
Category: Mistakes
How does the Posterior support predictions?
Use the posterior predictive distribution by integrating over the posterior to forecast new observations with uncertainty.
Category: Example
When is MAP estimation used with a Posterior?
When preferring the posterior mode as a regularized estimate, especially in high-dimensional or sparse problems.
Category: Opinion
What is the role of conjugate priors in Posterior analysis?
They yield closed-form posteriors, simplifying computations and interpretation.
Category: Overview
How do MCMC methods approximate a Posterior?
They sample from the posterior by constructing a Markov chain with the posterior as its stationary distribution.
Category: Requirements