= Solution
The <likelihood> is $L(\theta;d)=p(d\mid\theta)$ considered as a function of parameters at the observed data. It is not itself a probability distribution over $\theta$. <Bayes' theorem> gives
$$
\boxed{p(\theta\mid d)=\frac{p(d\mid\theta)\pi(\theta)}{\int p(d\mid\vartheta)\pi(\vartheta)\,d\vartheta}}.
$$
The <prior> represents parameter information before these data; the denominator normalizes the <posterior> and can compare fully specified models with compatible <prior> assumptions. Detector response, backgrounds, dependence and selection must be included in the data model.
For independent measurements $x_i\sim N(\theta,\sigma_i^2)$ with known errors, $\log L=\mathrm{constant}-\tfrac12\sum_i(x_i-\theta)^2/\sigma_i^2$. Maximum <likelihood> minimizes this <chi-squared> and gives $\widehat\theta=\sum_iw_ix_i/\sum_iw_i$, $w_i=\sigma_i^{-2}$. A <Gaussian> <prior> $N(\mu_0,s_0^2)$ instead yields
$$
\boxed{s_{\rm post}^2=\frac1{\sum_iw_i+s_0^{-2}},\qquad
\mu_{\rm post}=s_{\rm post}^2\left(\sum_iw_ix_i+\frac{\mu_0}{s_0^2}\right)}.
$$
This explicitly combines data and <prior> precision, approaching the <weighted mean> when <prior> information is weak. Unknown <calibration> parameters should be jointly inferred and marginalized, not silently fixed. <posterior> intervals are conditional on the assumed model and <prior>; <posterior predictive checks> and sensitivity to plausible <priors> help expose poor fits and weakly identified parameters. The maximum-likelihood estimate and a Bayesian <posterior> answer different inferential questions even when their numerical centres agree.
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