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Bayesian updating: posterior ∝ likelihood × prior. Posterior distribution of θ summarizes belief after seeing data. Credible intervals [a,b] satisfy P(θ∈[a,b]|data)=0.95, directly answering 'where is θ?' Unlike frequentist CIs, these are probability statements about θ.
From Bayes' theorem, you know that P(A|B) = P(B|A)·P(A)/P(B). Bayesian statistics is what happens when A is a hypothesis about a parameter θ and B is the data you observed. Before seeing data, you have a prior distribution π(θ) that encodes your beliefs (or lack thereof) about θ. It is a full probability distribution — not just a point estimate, but a spread of plausibility over all possible values of θ. After observing data x, you update this belief using the data's likelihood L(θ; x) = f(x|θ), the probability of seeing your data if θ were the true value.
The update formula is the fundamental equation: posterior ∝ likelihood × prior, written π(θ|x) ∝ f(x|θ) · π(θ). The proportionality hides the normalizing constant (the marginal likelihood ∫ f(x|θ) π(θ) dθ), which makes the posterior integrate to 1 but plays no role in inference about θ. Intuitively: the prior says "here is where I thought θ lived before"; the likelihood says "here are the values of θ that make my data probable"; the posterior says "here is where I believe θ lives now, combining both signals." Regions where the prior is high and the likelihood is high become regions where the posterior is especially concentrated.
The posterior distribution is the complete answer to a Bayesian inference problem — not a number, but a distribution. From it you can extract any summary you want. The posterior mean E[θ|x] and posterior mode (MAP estimate) are both point summaries. A credible interval [a, b] with coverage 1 − α is any interval satisfying P(θ ∈ [a, b] | x) = 1 − α — a direct probability statement about where θ lies. This is what people often *want* from a confidence interval but cannot get: a 95% credible interval literally means "given my prior and this data, I am 95% sure θ is in [a, b]." A frequentist confidence interval means something more convoluted (it describes a procedure that covers the true θ 95% of the time across hypothetical repetitions), not a probability about this particular θ.
The practical power of the Bayesian framework is most visible when you have genuine prior information — domain knowledge about plausible parameter ranges, previous experiments, or regularizing constraints — and when you need to propagate uncertainty rather than just report a point estimate. The prior is often criticized as subjective, but this objection weakens as data accumulates: with sufficient data, the likelihood dominates the prior, and posteriors from different reasonable priors converge. The Bayesian framework also naturally handles hierarchical models (priors on priors), sequential updating (the posterior from one study becomes the prior for the next), and prediction via the posterior predictive distribution p(x̃|x) = ∫ f(x̃|θ) π(θ|x) dθ — integrating over all uncertainty in θ rather than plugging in a point estimate.