Bayesian Probability·Advanced·4 lessons·~300 min

Bayesian Probability: Updating Belief with Evidence

How priors, likelihoods, and evidence interact in rational belief revision

Start the first lesson Back to curriculum

What you'll learn

By the end of this unit, you can…

Identify priors and likelihoods.
Avoid base rate neglect.
Compute posterior from natural frequencies.
Compare posteriors across hypotheses.

Lessons

Lesson sequence

1
Priors, Evidence, and Posterior Belief
Introduces the central components of Bayesian reasoning and why evidence updates rather than replaces prior belief.
15 activities5 worked examples
Open →
2
Base Rates and Conditional Probability
Shows how to structure a probabilistic problem so that base rates and conditionals are not confused, and performs simple quantitative Bayesian updates using a natural-frequency format.
15 activities5 worked examples
Student Pro
3
Bayesian Comparison of Rival Hypotheses
Connects Bayesian updating to comparative reasoning between competing hypotheses using the Bayes factor and qualitative Bayesian comparison.
15 activities5 worked examples
Student Pro
4
Capstone: Bayesian Judgment on Real Evidence
An integrative lesson that asks students to apply Bayesian updating to mixed evidence scenarios: identify priors, compute likelihoods under rival hypotheses, update to a posterior, and communicate the result in calibrated language.
2 activities1 worked example
Student Pro

How to study

Three moves that work for this unit

Read the explanation

Each lesson opens with a guided walkthrough — read it before the activity.

Study the worked example

Look at why each step follows, not just what the answer is.

Practice with the target in mind

Know which rule applies and what would make the response weak before you start.

Reference materials

Optional context for the unit. Each lesson surfaces the concepts and rules it uses — these are here when you want the bigger picture.

Concept map (7 terms)

Prior Probability

The degree of confidence assigned to a hypothesis before the new evidence is taken into account.

Likelihood

The probability of the observed evidence on the assumption that a given hypothesis is true, written P(E | H).

Posterior Probability

The revised degree of confidence in a hypothesis after incorporating new evidence, written P(H | E).

Base Rate

The background prevalence or prior frequency relevant to the hypothesis being assessed.

Bayes Factor

The ratio of likelihoods under two rival hypotheses, P(E | H1) / P(E | H2), which captures how strongly evidence favors one over the other.

False Positive Rate

The probability that a test or indicator yields a positive result when the hypothesis is false, written P(E | not-H).

Calibration

The property of having stated confidence match long-run accuracy — 70%-confident predictions should come true about 70% of the time.

Rules and standards (4)

Respect Base Rates. A probabilistic judgment should not ignore background prevalence or prior probability when the context makes it relevant. Common failures: A striking test result is treated as if it overrides the base rate automatically.; Rare-event contexts are assessed as though all hypotheses started equally likely..
Distinguish P(E | H) from P(H | E). A likelihood is not the same thing as a posterior probability. Swapping them is the 'prosecutor's fallacy'. Common failures: The probability of evidence given a hypothesis is mistaken for the probability of the hypothesis given the evidence.; Diagnostic accuracy is confused with posterior certainty..
Update Proportionately to Evidence. Belief revision should reflect both prior plausibility and the relative explanatory weight of the evidence — not the vividness or novelty of the evidence. Common failures: A small piece of evidence causes an excessive revision.; Strong contrary evidence produces almost no change in confidence..