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Continuous-Prior Bayesian Updating worksheets

Normalize polynomial prior-likelihood products, integrate evidence, predict future outcomes, and distinguish density from point mass.

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Integrate over parameter support to find evidence and normalize the posterior density.

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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

Apply continuous Bayes' theorem on parameter support [0,1] using exact bounded polynomial integrals.

Worked example

A uniform prior observes one Bernoulli success. Find the posterior.

f(theta|data)=2theta on [0,1].

Prior times likelihood is theta and normalizes by its integral 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 SheetNameScoreTeacherDateContinuous-Prior Bayesian UpdatingThe parameter theta has support [0,1]. Multiply the prior density by the likelihood, integrate to find the evidence, and normalize to a posterior density. A density height is not probability at onepoint.1.Prior density f(theta)=3 theta^2 on [0,1]. Ordered Bernoulli data contain 3 success(es) and 1 failure(s). Find the normalized posterior density.Given: The ordered-data likelihood is theta^s(1-theta)^f; theta is a continuous hypothesis in [0,1].2.Prior density f(theta)=1 on [0,1]. Ordered Bernoulli data contain 1 success(es) and 0 failure(s). Find the posterior predictive probability that the next trialsucceeds.Given: Prediction averages theta over the posterior density.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateContinuous-Prior Bayesian UpdatingThe parameter theta has support [0,1]. Multiply the prior density by the likelihood, integrate to find the evidence, and normalize to a posterior density. A density height is not probability at onepoint.3.Prior density f(theta)=2 theta^1 on [0,1]. Ordered Bernoulli data contain 1 success(es) and 0 failure(s). Compute the evidence P(data) by integration.Given: The observations are an ordered sequence, so no binomial coefficient is included.4.Prior density f(theta)=3 theta^2 on [0,1]. Ordered Bernoulli data contain 1 success(es) and 0 failure(s). State the parameter support and findP(theta=0.75). Explain why the posterior density height at 0.75 is not that probability.Given: Theta has a continuous prior and posterior density.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateContinuous-Prior Bayesian UpdatingThe parameter theta has support [0,1]. Multiply the prior density by the likelihood, integrate to find the evidence, and normalize to a posterior density. A density height is not probability at onepoint.5.Prior density f(theta)=1 on [0,1]. Ordered Bernoulli data contain 2 success(es) and 0 failure(s). Find the normalized posterior density.Given: The ordered-data likelihood is theta^s(1-theta)^f; theta is a continuous hypothesis in [0,1].6.Prior density f(theta)=2 theta^1 on [0,1]. Ordered Bernoulli data contain 2 success(es) and 0 failure(s). Compute the evidence P(data) by integration.Given: The observations are an ordered sequence, so no binomial coefficient is included.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateContinuous-Prior Bayesian UpdatingThe parameter theta has support [0,1]. Multiply the prior density by the likelihood, integrate to find the evidence, and normalize to a posterior density. A density height is not probability at onepoint.7.Prior density f(theta)=3 theta^2 on [0,1]. Ordered Bernoulli data contain 2 success(es) and 0 failure(s). State the parameter support and findP(theta=0.75). Explain why the posterior density height at 0.75 is not that probability.Given: Theta has a continuous prior and posterior density.8.Prior density f(theta)=1 on [0,1]. Ordered Bernoulli data contain 3 success(es) and 0 failure(s). Find the posterior predictive probability that the next trialsucceeds.Given: Prediction averages theta over the posterior density.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 4 of 4

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