1. Read the model first
Each lesson opens with a guided explanation so the learner sees what the core move is before any saved response is required.
Bayesian Probability
Bayesian Probability: Updating Belief with Evidence
How priors, likelihoods, and evidence interact in rational belief revision
Students learn the logic of Bayesian reasoning, including priors, likelihoods, posterior updating, base rates, and the disciplined use of probabilistic evidence in inquiry and decision making. The unit emphasizes qualitative Bayesian discipline first and then builds toward simple quantitative updates.
Study Flow
How to work through this unit without overwhelm
2. Study an example on purpose
The examples are there to show what strong reasoning looks like and where the structure becomes clearer than ordinary language.
3. Practice with a target in mind
Activities work best when the learner already knows what the answer needs to show, what rule applies, and what mistake would make the response weak.
Lesson Sequence
What you will work through
Lesson 1
Priors, Evidence, and Posterior Belief
Introduces the central components of Bayesian reasoning and why evidence updates rather than replaces prior belief.
Start with a short reading sequence, study 1 worked example, then use 15 practice activitys to test whether the distinction is actually clear.
Guided reading1 worked example15 practice activitys
Lesson 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.
Read for structure first, inspect how the example turns ordinary language into cleaner form, then complete 15 formalization exercises yourself.
Guided reading2 worked examples15 practice activitystranslation support
Lesson 3
Bayesian Comparison of Rival Hypotheses
Connects Bayesian updating to comparative reasoning between competing hypotheses using the Bayes factor and qualitative Bayesian comparison.
Use the reading and examples to learn what the standards demand, then practice applying those standards explicitly in 15 activitys.
Guided reading2 worked examples15 practice activitysstandards focus
Lesson 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.
Each lesson now opens with guided reading, then moves through examples and 2 practice activitys so you are not dropped into the task cold.
Guided reading1 worked example2 practice activitys
Rules And Standards
What counts as good reasoning here
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
How arguments get translated into structure
Bayesian Update Schema
Input form
evidence_assessment_problem
Output form
prior_likelihood_posterior_analysis
Steps
- 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.
Common errors
- Skipping the prior entirely.
- Using only one-sided likelihood information.
- Treating the posterior as certainty rather than a revised degree of support.
Qualitative Bayesian Comparison
Input form
competing_hypotheses_with_evidence
Output form
relative_support_judgment
Steps
- 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.
Common errors
- Assuming the hypothesis with the most vivid story automatically gets the higher posterior.
- Ignoring that weaker priors can sometimes be overcome by much stronger evidence.
Concept Map
Key ideas in the unit
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.
Assessment
How to judge your own work
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.
Mastery requirements
- Identify Priors And Likelihoods
- Avoid Base Rate Neglect
- Compute Posterior From Natural Frequencies
- Compare Posteriors Across Hypotheses
History Links
How earlier logicians shaped modern tools
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.
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.
Amos Tversky and Daniel Kahneman
Documented systematic failures in intuitive probability reasoning, including base-rate neglect, the conjunction fallacy, and representativeness-driven errors.