Parameter uncertainty and Bayesian updates
What this concept is
Bayesian parameter inference combines a prior with a likelihood and normalizes. A Beta(a,b) prior for a probability p has density proportional to p^(a−1)(1−p)^(b−1), with a,b>0. Its normalizer is the integral B(a,b) over [0,1].
Why it matters here
Event-level Bayes reasoning and uncertainty about an unknown rate are different modeling tasks. A stated prior makes the extra assumption inspectable.
Concrete example
Given k successes in n conditionally independent Bernoulli trials at a common p, multiplying kernels gives a Beta(a+k,b+n−k) posterior. With a=b=1, k=2,n=5, this is Beta(3,4). Its mean is 3/7, also the probability of success for one new exchangeable trial under this model. The MLE is 2/5; different answers reflect different inferential rules.
Common confusion / misconception
A posterior mean is not a credible interval. A credible interval is a posterior probability statement conditional on prior and likelihood; a confidence interval has repeated-sampling coverage for a procedure. Neither framework repairs biased sampling or duplicated data.
How to use it
The Beta mean is B(a+1,b)/B(a,b)=a/(a+b); this identity follows from integration by parts and B(a,b)=B(a+1,b)+B(a,b+1). Update only genuinely new observations. Sequential updates agree with a pooled update if each observation appears exactly once. Compare plausible priors fixed before seeing the result. With prior Beta(2,8) and the same data, posterior mean is 4/15, illustrating prior sensitivity at small n.
Check yourself
Under a uniform prior and zero successes in four trials, why is the posterior predictive probability positive?
Mini drill or application
The posterior is Beta(1,5), density 5(1−p)^4, mean 1/6. Integrating gives CDF 1−(1−p)^5. Derive the one-sided 95% credible upper limit 1−0.05^(1/5), about 0.451. State the prior and common-p assumption every time you report it; do not relabel it as the Wilson interval.
Read this only if stuck
Primary source support. See the extension guide for practice, scope and evidence requirements. Local examples and lab code are original.