Bayesian Probability·Advanced·4 lessons·47 practice activities·~300 min
Bayesian Probability: Updating Belief with Evidence
How priors, likelihoods, and evidence interact in rational belief revision
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
- 1Open →
Priors, Evidence, and Posterior Belief
Introduces the central components of Bayesian reasoning and why evidence updates rather than replaces prior belief.
- 2Student Pro
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.
- 3Student Pro
Bayesian Comparison of Rival Hypotheses
Connects Bayesian updating to comparative reasoning between competing hypotheses using the Bayes factor and qualitative Bayesian comparison.
- 4Student Pro
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.
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..
- Compare Evidence Under All Rival Hypotheses. The weight of evidence depends not only on how well it fits the favored hypothesis, but also on how well it fits the rivals. Common failures: Asking only whether the evidence fits H and ignoring whether it fits not-H equally well.; Treating evidence as strong because it 'supports' H without checking whether it also supports rival hypotheses..
Formalization patterns (2)
- Bayesian Update Schema. From evidence_assessment_problem to prior_likelihood_posterior_analysis — State the hypothesis under evaluation.; Identify the relevant prior probability or base rate.; State how likely the evidence would be if the hypothesis were true (the likelihood).; State how likely the evidence would be if the hypothesis were false (the false positive rate).; Compute or estimate the posterior proportionately..
- Qualitative Bayesian Comparison. From competing_hypotheses_with_evidence to relative_support_judgment — State the competing hypotheses.; Compare their priors qualitatively (which was more plausible before the evidence?).; Compare how strongly each predicts the evidence.; Multiply qualitatively: a higher prior and a higher likelihood both push the posterior up.; State the posterior ranking with appropriate caution..
Full mastery and assessment guidance
Mastery requirements
- Identify priors and likelihoods. Successful Analyses · 6_successful_analyses
- Avoid base rate neglect. Percent Consistent · 80_percent_consistent
- Compute posterior from natural frequencies. Successful Calculations · 6_successful_calculations
- Compare posteriors across hypotheses. Successful Comparisons · 4_successful_comparisons
Assessment advice
- What was the prior before the new evidence arrived?
- How strongly would this evidence have been expected under the hypothesis?
- How strongly would this evidence have been expected under the negation of the hypothesis?
- Collapsing the distinction between prior support and posterior support.
- Ignoring the denominator of the posterior calculation.
- What is the background prevalence?
- Am I mixing up P(E | H) with P(H | E)?
- Did I account for the false positive rate?
- Treating a high likelihood as the same thing as certainty about the hypothesis.
- Forgetting to include the non-H cases in the denominator.
- How plausible were the hypotheses before the evidence?
- Which hypothesis predicted the evidence better?
- Does the Bayes factor justify the size of the update I'm making?
- Treating Bayesian comparison as if it required certainty rather than differential support.
- Ignoring the prior because the evidence feels overwhelming.
- Did I write out the base rate before computing anything?
- Did I compute likelihoods for every rival?
- Is my verdict expressed in calibrated language?
- Letting the test's sensitivity alone decide the verdict.
- Converting a natural-frequency answer back into a probability without keeping the denominator visible.
Historical context (3)
- Thomas Bayes. In 'An Essay Towards Solving a Problem in the Doctrine of Chances' (published posthumously in 1763), laid the foundation for probabilistic belief updating in light of evidence. Bayesian inference, posterior updating, and evidence-sensitive probability judgments.
- Pierre-Simon Laplace. In his Theorie analytique des probabilites, extended probabilistic reasoning and helped turn Bayesian ideas into a general inferential framework applicable to science, astronomy, and decision-making. Applied probability, model comparison, and systematic uncertainty reasoning.
- Amos Tversky and Daniel Kahneman. Documented systematic failures in intuitive probability reasoning, including base-rate neglect, the conjunction fallacy, and representativeness-driven errors. The modern understanding of where and why intuitive Bayesian judgment goes wrong.