Bayesian Inference
Treating unknown parameters as random variables with a prior distribution, then updating to a posterior as data arrives — the alternative to frequentist confidence intervals and p-values.
Bayesian inference treats an unknown quantity (a parameter, like a coin's true probability of heads) as something you have a degree of belief about, represented as a probability distribution — not a single fixed number. You start with a prior distribution (what you believe before seeing data), then use Bayes' theorem to update it into a posterior distribution once data arrives.
This contrasts with frequentist statistics, which treats parameters as fixed (but unknown) constants, and instead asks questions about the long-run behavior of estimators across hypothetical repeated samples.
Frequentist: estimate the coin's bias as the observed proportion of heads, and build a 95% confidence interval describing how that estimate would vary across many repeated experiments.
Bayesian: start with a prior belief about the bias (e.g. "probably close to fair"), then update it with each flip into a posterior distribution — directly answering "given what I've seen, how likely is it that the bias is above 0.6?"
Why can a Bayesian directly answer "what's the probability the true bias is above 0.6?" while a frequentist confidence interval technically cannot?
Solution
A Bayesian's posterior is a genuine probability distribution over the parameter itself, so "P(bias > 0.6)" is a well-defined area under that posterior curve. A frequentist confidence interval is a statement about the procedure (95% of intervals built this way would contain the true value across repeated experiments) — the true bias is treated as fixed, not random, so "the probability it's above 0.6" isn't a meaningful frequentist statement about a single, already-collected dataset.