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Bayesian Prediction and Odds worksheets

Form predictive mixtures, convert odds and probability, update odds with Bayes factors, and interpret evidence direction.

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Use total probability for predictions and posterior odds = prior odds times Bayes factors.

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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

Connect posterior weights to prediction and use Bayes factors to update odds without treating a factor as a posterior probability.

Worked example

Prior odds are 2 and BF=3. Find posterior odds.

6.

Bayes factors multiply odds.

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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 SheetNameScoreTeacherDateBayesian Prediction, Odds, and Bayes FactorsUse total probability for prior and posterior predictive mixtures. Convert probability and odds carefully. Posterior odds equal prior odds times the Bayes factor; the factor measures evidencedirection and is not itself a posterior probability.1.The odds for H are 6:2. Convert to probability, then convert that probability back to decimal odds.Given: Odds a:b mean probability a/(a+b), not a/b as a probability.2.4 possible machines have success probabilities 1, 0.1, 0.4, 0.7 and exact prior weights 1/5, 3/10, 2/5, 1/10. Find the prior predictive success probability.After one success, find the posterior weights and posterior predictive probability for the next trial.Given: The machine identity stays fixed; trials are conditionally independent given it.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateBayesian Prediction, Odds, and Bayes FactorsUse total probability for prior and posterior predictive mixtures. Convert probability and odds carefully. Posterior odds equal prior odds times the Bayes factor; the factor measures evidencedirection and is not itself a posterior probability.3.A data set has Bayes factor BF_HK = 1.2. Interpret the evidence direction. Is 1.2 the posterior probability of H?Given: BF_HK=P(data|H)/P(data|K); posterior odds also require prior odds.4.Prior odds for H versus K are 3/4. 2 independent evidence item(s) have Bayes factors 1 and 1.3 for H versus K. Find posterior odds and probability of H.Given: Multiply odds by each likelihood ratio; a Bayes factor is not a probability.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateBayesian Prediction, Odds, and Bayes FactorsUse total probability for prior and posterior predictive mixtures. Convert probability and odds carefully. Posterior odds equal prior odds times the Bayes factor; the factor measures evidencedirection and is not itself a posterior probability.5.The odds for H are 4:3. Convert to probability, then convert that probability back to decimal odds.Given: Odds a:b mean probability a/(a+b), not a/b as a probability.6.Prior odds for H versus K are 5/4. 1 independent evidence item(s) have Bayes factors 2 for H versus K. Find posterior odds and probability of H.Given: Multiply odds by each likelihood ratio; a Bayes factor is not a probability.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateBayesian Prediction, Odds, and Bayes FactorsUse total probability for prior and posterior predictive mixtures. Convert probability and odds carefully. Posterior odds equal prior odds times the Bayes factor; the factor measures evidencedirection and is not itself a posterior probability.7.3 possible machines have success probabilities , 0.6, 0.1 and exact prior weights 3/8, 1/2, 1/8. Find the prior predictive success probability. After onesuccess, find the posterior weights and posterior predictive probability for the next trial.Given: The machine identity stays fixed; trials are conditionally independent given it.8.A data set has Bayes factor BF_HK = 4. Interpret the evidence direction. Is 4 the posterior probability of H?Given: BF_HK=P(data|H)/P(data|K); posterior odds also require prior odds.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 4 of 4

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