 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 givesThe 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, $\log L=\mathrm{constant}-\frac{1}{2}{\sum }_i(x_i-\theta {)}^2/{\sigma }_i^2$. Maximum likelihood minimizes this chi-squared and gives $\hat{\theta }={\sum }_i{w}_i{x}_i/{\sum }_i{w}_i$, $w_i={\sigma }_i^{-2}$. A Gaussian prior $N({\mu }_0,s_0^2)$ instead yieldsThis 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.