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Credible Intervals worksheets

Calculate discrete and continuous posterior interval mass, equal-tail intervals, interpretations, and quantile-based prior choices.

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Use PMF sums or CDF differences. Equal-tail intervals split omitted posterior mass equally.

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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

Summarize posterior belief with probability intervals and use subjective quantiles to describe a compatible prior.

Worked example

Find the central 80% equal-tail interval from quantiles q.10 and q.90.

[q.10,q.90].

Ten percent posterior mass lies in each tail.

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 SheetNameScoreTeacherDateCredible Intervals and Prior ElicitationA credible interval summarizes posterior belief about a parameter. Compute mass from the printed PMF or CDF, use equal tail quantiles when requested, and do not interpret posterior coverageas a repeated-sample confidence statement.1.An expert states prior q10=0.0400, median=0.2000, and q90=0.3600, so the central 80% interval is [0.0400,0.3600]. Choose the compatible parameterizedprior: F_1(theta)=(theta/0.4)^1 on [0,0.4], with (q10,q50,q90)=(0.0400,0.2000,0.3600); F_2(theta)=(theta/0.4)^2 on [0,0.4], with(q10,q50,q90)=(0.1265,0.2828,0.3795); F_3(theta)=(theta/0.4)^3 on [0,0.4], with (q10,q50,q90)=(0.1857,0.3175,0.3862).Given: For candidate CDF F_k(theta)=(theta/u)^k on [0,u], the p-quantile is u p^(1/k).2.A posterior PMF on 4 support values is shown. Find P(2 <= theta <= 3) and state the corresponding credible interval mass.Given: Posterior mass is summed over support values inside the closed interval.theta1234Posterior1/61/31/61/3Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateCredible Intervals and Prior ElicitationA credible interval summarizes posterior belief about a parameter. Compute mass from the printed PMF or CDF, use equal tail quantiles when requested, and do not interpret posterior coverageas a repeated-sample confidence statement.3.A posterior CDF is F(theta)=theta^3 on [0,1]. Find the 50% equal-tail credible interval.Given: An equal-tail interval leaves (1-coverage)/2 posterior probability in each tail.4.A 95% credible interval for theta is [4, 6]. Interpret this Bayesian statement and identify an incorrect confidence-style claim.Given: Theta is modeled as uncertain under the posterior distribution.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateCredible Intervals and Prior ElicitationA credible interval summarizes posterior belief about a parameter. Compute mass from the printed PMF or CDF, use equal tail quantiles when requested, and do not interpret posterior coverageas a repeated-sample confidence statement.5.A posterior on [0,1] has CDF F(theta)=theta^2. Find P(0.3 <= theta <= 0.8).Given: For a continuous posterior, interval mass is a CDF difference; endpoint point masses are zero.6.A posterior PMF on 5 support values is shown. Find P(3 <= theta <= 4) and state the corresponding credible interval mass.Given: Posterior mass is summed over support values inside the closed interval.theta23456Posterior1/72/71/72/71/7Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateCredible Intervals and Prior ElicitationA credible interval summarizes posterior belief about a parameter. Compute mass from the printed PMF or CDF, use equal tail quantiles when requested, and do not interpret posterior coverageas a repeated-sample confidence statement.7.A posterior CDF is F(theta)=theta^1 on [0,1]. Find the 90% equal-tail credible interval.Given: An equal-tail interval leaves (1-coverage)/2 posterior probability in each tail.8.A 90% credible interval for theta is [2, 4]. Interpret this Bayesian statement and identify an incorrect confidence-style claim.Given: Theta is modeled as uncertain under the posterior distribution.Keep likelihood, prior, evidence, and posterior roles distinct.math-sheet.comPage 4 of 4

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