= Solution
<Bayes' theorem> updates prior knowledge using the <likelihood> of new observations. For alternative hypotheses $H_j$,
$$
\boxed{P(H_j\mid D)=\frac{P(D\mid H_j)P(H_j)}{\sum_kP(D\mid H_k)P(H_k)}.}
$$
The prior $P(H_j)$ describes information before the data $D$; $P(D\mid H_j)$ is the sampling <likelihood>, and the posterior $P(H_j\mid D)$ describes the updated probabilities. Posterior odds equal prior odds times the <likelihood> ratio. A <likelihood> is a <probability> model for data at specified parameters, not automatically a normalized <probability> distribution over those parameters.
For a continuous parameter $\theta$, the same rule uses densities: $\pi(\theta\mid D)=L(D\mid\theta)\pi(\theta)/\int L(D\mid u)\pi(u)du$. Priors can encode positivity, physical limits, previous independent measurements or calibration knowledge. Their parameterization matters: a density uniform in a rate is not uniform in its logarithm. Proper priors make model evidences well defined; an arbitrary normalization in an improper prior is especially problematic for comparing models.
A counting example uses <Poisson-gamma conjugacy>. If a detector registers $K$ events in exposure $t$ with constant rate $\lambda$, $L\propto\lambda^Ke^{-t\lambda}$. A gamma prior with shape $\alpha$ and rate parameter $\beta$, $\pi(\lambda)\propto\lambda^{\alpha-1}e^{-\beta\lambda}$, gives
$$
\boxed{\lambda\mid K\sim\operatorname{Gamma}(\alpha+K,\beta+t),\qquad \mathbb E[\lambda\mid K]=\frac{\alpha+K}{\beta+t}.}
$$
The <posterior mean> combines the prior rate $\alpha/\beta$ with the observed rate $K/t$, weighted by the prior and actual exposures. The parameters must use a consistent shape-rate convention. Small counts can leave substantial prior sensitivity; abundant informative data usually reduce it for regular identifiable models.
Independent old data can become the prior for a new experiment, giving the same result as joint updating under a consistent model. The new data must not also have been used to construct that prior, or the evidence is counted twice. A <credible interval> has a specified posterior <probability> under the chosen model and prior; this interpretation differs from repeated-sampling coverage of a <confidence interval>. Predictions for a future observation average its sampling distribution over the posterior, incorporating parameter uncertainty. Prior sensitivity checks and a defensible <likelihood> are therefore part of the inference, rather than <Bayes' theorem> itself supplying missing physical knowledge.
Back to article page
