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Beta Conjugate Priors worksheets

Normalize integer-shape Beta densities, interpret strength, update Bernoulli/binomial/geometric data, and predict.

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Add successes to a and failures to b. For geometric F, include one stopping success and F preceding failures.

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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 Beta conjugacy with explicit success/failure counting, including failures before the first success.

Worked example

Beta(2,3) prior and two successes, one failure. Find posterior.

Beta(4,4).

Add the observed counts to the corresponding shapes.

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  • 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 SheetNameScoreTeacherDateBeta Conjugate PriorsUse positive integer Beta shapes. Interpret a+b as prior strength and a/(a+b) as the mean. For geometric data, F is the number of failures before the first success, so observing F=f contributesone success and f failures.1.Start with Beta(2,6). Observe one Bernoulli outcome success. Find the posterior.Given: The likelihood contributes one theta factor per success and one (1-theta) factor per failure.2.Find c so f(theta)=c theta^2(1-theta)^5 is a PDF on [0,1]. Identify its Beta family.Given: Shapes a and b are positive integers.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateBeta Conjugate PriorsUse positive integer Beta shapes. Interpret a+b as prior strength and a/(a+b) as the mean. For geometric data, F is the number of failures before the first success, so observing F=f contributesone success and f failures.3.For a Beta(4,6) prior, find its mean and prior strength, and describe whether it favors theta below, near, or above 1/2.Given: Use mean a/(a+b) and strength a+b.4.A Beta(5,6) prior is updated by 3 successes and 1 failures. Compare prior and posterior means and give the one-step predictive success probability.Given: For a Bernoulli parameter, the posterior mean is the next-success predictive probability.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateBeta Conjugate PriorsUse positive integer Beta shapes. Interpret a+b as prior strength and a/(a+b) as the mean. For geometric data, F is the number of failures before the first success, so observing F=f contributesone success and f failures.5.Find c so f(theta)=c theta^5(1-theta)^5 is a PDF on [0,1]. Identify its Beta family.Given: Shapes a and b are positive integers.6.For a Beta(1,1) prior, find its mean and prior strength, and describe whether it favors theta below, near, or above 1/2.Given: Use mean a/(a+b) and strength a+b.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateBeta Conjugate PriorsUse positive integer Beta shapes. Interpret a+b as prior strength and a/(a+b) as the mean. For geometric data, F is the number of failures before the first success, so observing F=f contributesone success and f failures.7.A Beta(2,1) prior is updated by 2 successes and 4 failures. Compare prior and posterior means and give the one-step predictive success probability.Given: For a Bernoulli parameter, the posterior mean is the next-success predictive probability.8.Start with Beta(3,1). Observe one Bernoulli outcome failure. Find the posterior.Given: The likelihood contributes one theta factor per success and one (1-theta) factor per failure.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 4 of 4

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