= Solution
<Bayes' theorem> updates a parameter distribution using measured data:
$$
\boxed{p(\theta\mid D)=\frac{p(D\mid\theta)p(\theta)}{\int p(D\mid\theta')p(\theta')\,d\theta'}}.
$$
The <likelihood> describes the measurement process, the <prior distribution> encodes information before the present data, and the denominator normalizes the <posterior distribution>. Include instrumental <noise> and known selection effects in the <likelihood>. <nuisance parameters>, such as an uncertain background, should be jointly inferred and marginalized, rather than silently fixed.
For $n$ independent readings with a <Gaussian distribution> with known common <variance> $\sigma^2$ and <prior distribution> $\theta\sim N(\mu_0,\tau_0^2)$, completing the square gives
$$
\tau_n^{-2}=\tau_0^{-2}+n\sigma^{-2},\qquad
\mu_n=\tau_n^2\big(\mu_0\tau_0^{-2}+n\bar x\,\sigma^{-2}\big).
$$
The <posterior distribution> is $N(\mu_n,\tau_n^2)$: <prior distribution> and data contribute in proportion to precision. A broad <prior distribution> recovers the usual estimate $\bar x$ with <standard deviation> $\sigma/\sqrt n$. Report a <posterior distribution> estimate and <credible interval>, specify the <prior distribution>, and examine sensitivity when the data are weak. A <credible interval> is a <posterior distribution> <probability> statement about the parameter conditional on the model; it is not defined by the repeated-sampling coverage of a <confidence interval>.
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