Visualize Bayesian Updating From Prior to Posterior

Posterior ∝ prior × likelihood. That's Bayes' rule for an unknown probability θ: you start with a prior belief about θ, multiply by the likelihood of the data given θ, and the renormalized product is the posterior. The page works on Bernoulli/binomial data because the conjugate prior is Beta(α, β): with s successes and f failures, the posterior is Beta(α + s, β + f) in closed form. No MCMC needed.

The Beta parameters have an intuitive read. α acts like pseudo-successes, β like pseudo-failures, baked into the prior before any data arrives. Beta(1, 1) is uniform (no information). Beta(2, 2) is weak, gently bunched around 0.5. Beta(50, 50) is sharp, peaked at 0.5, and takes a lot of contrary data to budge. The credible interval (the central region containing 95% of the posterior mass) is the Bayesian analogue of a frequentist CI, and unlike a CI it can be read directly as "P(θ ∈ this range | data) = 0.95." That clean interpretation is the practical reason to go Bayesian for proportion problems when stakeholders ask the natural question.

The Beta distribution lives on [0, 1], making it a natural choice for probabilities. Its two parameters, α and β, shape the curve. Think of α as "pseudo-successes" and β as "pseudo-failures" baked into your initial belief. Beta(1, 1) is uniform—every probability is equally likely before you see data.

Higher α + β means a tighter distribution. Beta(2, 2) is gently mounded around 0.5, representing mild uncertainty. Beta(50, 50) is sharply peaked at 0.5, representing strong confidence that the probability is near 50%. The sum α + β acts like sample size—larger sums indicate firmer beliefs that take more data to move.

To encode historical knowledge, translate past data into α and β. If last quarter's conversion rate was 8% across 200 trials, a reasonable informative prior is Beta(16, 184), with mean 16/200 = 0.08. This anchors your analysis without ignoring what you already know.

The Beta-Binomial conjugate pair makes updating trivial. If your prior is Beta(α, β) and you observe s successes and f failures, the posterior is Beta(α + s, β + f). No integrals, no numerical approximations—just add the counts. This property is why the Beta prior is so popular for binary outcomes.

Sequential updating works identically: update after each batch of data, or wait until all data arrives—the final posterior is the same. Only total counts matter, not the order. This lets you monitor results in real time without penalty.

With little data, the posterior clings to the prior. With lots of data, the posterior shifts toward the observed rate and sharpens. Twenty observations barely move a Beta(100, 100) prior, but decisively reshape a Beta(1, 1) prior.

The posterior mean is (α + s) / (α + s + β + f), a weighted blend of your prior belief and the observed rate. When α + β is small relative to s + f, the data dominates. When α + β is large, the prior anchors the estimate.

A 95% credible interval marks the region where the true probability lies with 95% posterior probability. Unlike frequentist confidence intervals, this is a direct probability statement given your prior and data—not a long-run coverage guarantee.

As data accumulates, the credible interval shrinks. With 10 observations, you might have [0.25, 0.75]. With 1,000 observations, that narrows to something like [0.48, 0.52]. More data means more certainty.

Sensitivity analysis tests how much your conclusions depend on the prior. Run the same data through Beta(1, 1), Beta(5, 5), and Beta(50, 50). If all three posteriors agree, your inference is robust. If they diverge, you need more data or a better-justified prior.

A weak prior (low α + β) lets the data speak. A strong prior (high α + β) resists change. Neither is inherently right—the choice depends on how much you trust your prior information versus the new observations.

When sample size vastly exceeds prior strength, different priors converge to the same posterior. At n = 1,000 observations, Beta(1, 1) and Beta(10, 10) yield nearly identical results. At n = 10, they differ noticeably.

The model assumes independent, identically distributed Bernoulli trials with constant success probability. If your success rate drifts over time, trials are clustered, or outcomes depend on covariates, the simple Beta-Binomial breaks down.

Non-binary outcomes need different conjugate pairs. For counts without an upper bound, use Poisson-Gamma. For continuous measurements, use Normal-Normal. The Beta-Binomial is purpose-built for yes/no data.

Hierarchical models handle groups with different underlying rates. If you're comparing multiple variants or segments, each with its own probability, a hierarchical Beta-Binomial pools information across groups rather than treating them as one homogeneous population.