🔁 Posterior Probability

Posterior Probability#

The probability of a hypothesis or event evaluated under the posterior distribution.

Important

✨ AI-generated content. This page was written with the assistance of an AI language model and is provided as a learning aid. Despite careful review, it may still contain mistakes, omissions, or out-of-date information. Whether you are new to the topic, a team lead, or a senior practitioner, treat it as a starting point rather than an authoritative reference: read it critically and independently verify anything you act on (code, commands, figures, and factual claims) against official documentation and primary sources before relying on it.

What it is#

The posterior probability is the probability of a hypothesis (or parameter value) after observing data, computed with Bayes’ theorem by combining the prior (belief before) with the likelihood (evidence from the data):

\[P(H \mid \text{data}) = \frac{P(\text{data} \mid H)\, P(H)}{P(\text{data})}.\]

What it answers#

It answers a directly useful question — “given the data, how probable is this hypothesis?” — which is not what a frequentist p-value answers (“if \(H_0\) were true, how likely is data this extreme?”). The posterior is the quantity people usually think a p-value gives them.

Worked example#

Is a coin biased toward heads? Take two point hypotheses — \(H_0: p = 0.5\) and \(H_1: p = 0.7\) — with equal priors, and observe 7 heads in 10 tosses. The likelihoods are

\[P(\text{data} \mid H_0) = \binom{10}{7}(0.5)^{10} \approx 0.117, \qquad P(\text{data} \mid H_1) = \binom{10}{7}(0.7)^7 (0.3)^3 \approx 0.266,\]

so the posterior for the biased hypothesis is

\[P(H_1 \mid \text{data}) = \frac{0.266 \times 0.5}{0.266 \times 0.5 + 0.117 \times 0.5} \approx 0.69.\]

After 7 heads in 10, there’s about a 69% probability the coin is biased.

Parameter posteriors#

When the unknown is a continuous parameter rather than a discrete hypothesis, the posterior is a whole distribution, \(P(\theta \mid \text{data}) \propto P(\text{data} \mid \theta)\, P(\theta)\) — the object used to estimate means, regression coefficients or conversion rates. (A single posterior probability is then one number read off that distribution.)

Where it shows up#

Bayesian sequential testing (stop once \(P(H_1 \mid \text{data}) > 0.95\)), Bayesian estimation, ML models (Naive Bayes, Bayesian nets), and medical diagnosis (probability of disease given a test result).

Theme: Bayesian Inference · All terminology

Hint

Mind map — connected ideas

Bayes’ Theorem · Posterior · Prior Belief (or Prior Probability) · Bayesian Sequential Testing · Posterior belief · Frequentist

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 · Posterior belief · Posterior probability of uplift · Prior Belief (or Prior Probability)