Posterior

Definition

A posterior is the updated probability after observing evidence.

In Bayes theorem:

P(H | E)

is the posterior probability of hypothesis H given evidence E.

Posterior = probability after evidence.

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Prior To Posterior

Bayesian updating moves from:

prior -> evidence -> posterior

Example:

P(spam) -> observe message words -> P(spam | words)

The posterior is the revised probability after using the evidence.

The posterior is the result of updating.

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Simple Example

Let:

H = message is spam
E = message contains suspicious phrase

Posterior:

P(spam | suspicious phrase)

This is the probability the message is spam after seeing the phrase.

Posterior probability depends on both prior and evidence.

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Posterior Uses More Than Likelihood

A high likelihood does not automatically mean a high posterior.

Bayes theorem combines:

prior
likelihood
evidence probability

If the prior is very low, the posterior may still be moderate.

Posterior is updated belief, not evidence fit alone.

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Posterior As A Distribution

The posterior can be a full distribution over hypotheses.

Example:

H1: 0.7
H2: 0.2
H3: 0.1

This says evidence made H1 most likely, but uncertainty remains.

Posterior distributions preserve uncertainty after evidence.

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Why Posterior Matters For Learning

Learning can be viewed as updating beliefs from data.

Before data:

many explanations are possible

After data:

some explanations become more likely

The posterior captures this updated uncertainty.

A posterior represents what the data has taught the system.

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Posterior vs Final Certainty

A posterior does not always mean certainty.

Example:

P(H | E) = 0.65

The hypothesis is more likely than before, but not guaranteed.

Posterior probability can still represent uncertainty.

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Final Summary

• A posterior is probability after observing evidence.
• It is written as P(H | E).
• It results from updating a prior with evidence.
• It is not the same as likelihood.
• A posterior can be a distribution over hypotheses.
• Learning can be interpreted as updating beliefs from data.