🔁 Posterior

Posterior#

The distribution of parameters given the data; the central object of Bayesian inference.

Important

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What it is#

The posterior is the updated probability distribution of a parameter after observing data — the new belief formed by combining the prior (what you thought before) with the likelihood (what the data say). The one-line version: posterior = prior updated with evidence.

Bayes’ theorem#

\[P(\theta \mid D) = \frac{P(D \mid \theta)\, P(\theta)}{P(D)} \;\propto\; \underbrace{P(D \mid \theta)}_{\text{likelihood}} \times \underbrace{P(\theta)}_{\text{prior}},\]

with the marginal likelihood \(P(D)\) as the normalising constant. The proportional form is what you actually work with: the posterior shape is just prior times likelihood.

What you get from it#

The posterior is a distribution, so it supports direct probability statements — “a 95% probability the conversion rate is between 4% and 6%” — which a frequentist confidence interval cannot make. Point summaries (the posterior mean or mode/MAP) and credible intervals are all read off it.

Example — coin toss#

Uniform prior \(\text{Beta}(1,1)\), a Binomial likelihood, and 7 heads in 10 tosses give

\[P(p \mid \text{data}) \propto p^7 (1-p)^3 \cdot 1 = \text{Beta}(8, 4),\]

a posterior centred near 0.67.

The prior washes out#

A defining property: the more data you collect, the more the likelihood dominates and the less the prior matters. With small samples the prior shapes the answer; with large ones the posterior is driven almost entirely by the data — which is why honest priors are cheap insurance, not permanent bias. The posterior is the object all Bayesian decisions are based on.

Theme: Bayesian Inference · All terminology

Hint

Mind map — connected ideas

Prior Belief (or Prior Probability) · Posterior belief · Bayes’ Theorem · Posterior Probability · Marginal Likelihood (also called The Model Evidence or Integrated Likelihood) · Beta Distribution

Hint

More in Bayesian Inference

Bayes’ Theorem · Bayesian Correction · Bayesian Decision Theory (BDT) · Bayesian Inference. · Bayesian Neural Networks (BNNs) · Binomial Likelihood · Gaussian Processes (GPs) · Marginal Likelihood (also called The Model Evidence or Integrated Likelihood) · MCMC (Markov Chain Monte Carlo) · Parameter(s) of Interest · Posterior belief · Posterior Probability · Posterior probability of uplift · Prior Belief (or Prior Probability)