Bayesian posteriors and prior sensitivity

Write the likelihood as the probability of observed data given a parameter, then multiply by a prior density and normalize over the allowed parameter domain. The likelihood alone is not a density over the parameter. Its dimensions, normalization and transformation law serve a different role. A posterior mean, median or mode summarizes different features of the resulting distribution; none is automatically the answer to every loss function or scientific question. Reporting the full distribution or useful quantiles often preserves more information.

A prior must be specified before inspecting the information it is meant to precede. Earlier calibration data can inform it, provided those same data are not included again as an independent likelihood factor. A density uniform in one coordinate generally becomes nonuniform after a nonlinear reparameterization because of its Jacobian. An improper prior requires checking that the posterior is normalizable and can create additional difficulties for model probabilities. Prior sensitivity is part of the analysis rather than a test of personal confidence.

Let k successes occur in n independent trials sharing efficiency p. A beta prior with positive shape parameters a and b has support between zero and one, with density proportional to p to power a minus one times one minus p to power b minus one. Multiplication by the binomial likelihood updates the two exponents. The result is another beta distribution. The prior parameters can encode assumptions or previous evidence, but should not be described literally as extra observed trials without stating that interpretation.

For a small calibration sample the posterior can change appreciably under plausible alternative shape parameters. Compare posterior summaries relevant to the scientific use, such as an efficiency correction or a tail probability, rather than only comparing the modes. A bounded posterior is not generally symmetric, and replacing it with an unbounded Gaussian may assign impossible efficiencies. A concentrated posterior can also be misleading if the trials have different efficiencies, dependence, misclassified outcomes or unmodeled selection.

For a nonnegative Poisson rate r, choose a gamma prior with shape alpha and rate beta, not scale beta. Its density is proportional to r to power alpha minus one times the exponential of minus beta times r. If exposure has time units, beta also has time units, so its product with r is dimensionless. An observed count over known exposure increases the shape by the count and increases the prior rate by the exposure. Confusing rate and scale conventions reverses the relevant denominator.

A credible interval contains a declared posterior probability under the chosen model. It does not by its definition have that same frequentist coverage for every fixed true parameter. Future-data prediction additionally integrates the sampling distribution against the posterior, which generally adds predictive variation. Prior checks and posterior model checks address different questions, and neither is replaced by obtaining an analytic posterior. The examples assume exactly known trials or exposure and no background; uncertainty in these quantities requires a larger joint model.