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Discrete Bayesian Updating worksheets

Complete prior-likelihood-posterior tables, compare sequential and batch evidence, and handle zero priors or impossible evidence.

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Multiply likelihood by prior, sum to find evidence, then normalize. Never divide by zero evidence.

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Keep every inferential role visible

Problems label parameters, observed data, likelihoods, priors, evidence, and posteriors before calculation. Bounded counts and printed reference values keep the work reproducible on paper.

Learning objective

What this activity builds

Use Bayesian update tables with two to four hypotheses while preserving the direction P(data|hypothesis).

Worked example

P(H1)=P(H2)=1/2 and likelihoods are 1/4 and 3/4. Find the posterior.

Posterior (1/4,3/4).

The numerators are 1/8 and 3/8, whose total is 1/2.

Teaching notes

Make the practice count

  • Label data, hypothesis, prior, and likelihood before calculating.
  • Normalize only after forming every Bayes numerator.
  • Interpret the posterior in the parameter context.

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Math SheetNameScoreTeacherDateDiscrete Bayesian UpdatingLabel hypotheses and data before multiplying likelihood by prior. Normalize the Bayes numerators to obtain posterior probabilities. For sequential evidence, use each posterior as the next priorand verify that it matches the batch likelihood.1.Complete the Bayesian update table after observing data D, and find every P(H_i | D).Given: The listed hypotheses are mutually exclusive and exhaustive. Likelihood means P(D | H_i).HypothesisPriorLikelihoodPrior x likelihoodPosteriorH10.200.700.1400?H20.200.300.0600?H30.300.600.1800?H40.300.200.0600?2.A three-hypothesis prior is (0.00, 0.10, 0.90), and likelihoods of D are (0.80, 0.50, 0.15). Find the posterior and explain H1.Given: Ordinary Bayes updating multiplies prior by likelihood before normalization.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateDiscrete Bayesian UpdatingLabel hypotheses and data before multiplying likelihood by prior. Normalize the Bayes numerators to obtain posterior probabilities. For sequential evidence, use each posterior as the next priorand verify that it matches the batch likelihood.3.A sensor has one of three fixed reliability levels. Its next 4 reports are recorded. Identify the hypothesis, data, prior, likelihood, and posterior roles.Given: P(data | hypothesis) and P(hypothesis | data) reverse the conditioning direction.4.Two hypotheses give success probabilities and 0.7. Starting from equal priors, observe 0 success(es) and 1 failure(s), for 1 observation(s) total. Updatesequentially and verify the batch posterior.Given: Observations are conditionally independent given the fixed hypothesis.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateDiscrete Bayesian UpdatingLabel hypotheses and data before multiplying likelihood by prior. Normalize the Bayes numerators to obtain posterior probabilities. For sequential evidence, use each posterior as the next priorand verify that it matches the batch likelihood.5.A model has 2 exhaustive hypotheses with prior (0.444, 0.556). Every hypothesis assigns likelihood 0 to observed data D. Can a posterior be formed byordinary conditioning?Given: The evidence is the prior-weighted sum of likelihoods.6.A sensor has one of three fixed reliability levels. Its next 7 reports are recorded. Identify the hypothesis, data, prior, likelihood, and posterior roles.Given: P(data | hypothesis) and P(hypothesis | data) reverse the conditioning direction.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateDiscrete Bayesian UpdatingLabel hypotheses and data before multiplying likelihood by prior. Normalize the Bayes numerators to obtain posterior probabilities. For sequential evidence, use each posterior as the next priorand verify that it matches the batch likelihood.7.Complete the Bayesian update table after observing data D, and find every P(H_i | D).Given: The listed hypotheses are mutually exclusive and exhaustive. Likelihood means P(D | H_i).HypothesisPriorLikelihoodPrior x likelihoodPosteriorH10.200.600.1200?H20.200.200.0400?H30.300.500.1500?H40.300.800.2400?8.Two hypotheses give success probabilities and 0.8. Starting from equal priors, observe 0 success(es) and 2 failure(s), for 2 observation(s) total. Updatesequentially and verify the batch posterior.Given: Observations are conditionally independent given the fixed hypothesis.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 4 of 4

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