Other interesting limit cases are the following.
The prior has been left on purpose open in the above formulas, although we have already anticipated that usually a flat prior about all parameters gives the correct result in most 'healthy' cases, characterized by a sufficient number of data points. I cannot go here through an extensive discussion about the issue of the priors, often criticized as the weak point of the Bayesian approach and that are in reality one of its points of force. I refer to more extensive discussions available elsewhere (see e.g.  and references therein), giving here only a couple of advices. A flat prior is in most times a good starting point (unless one uses some packages, like BUGS , that does not like flat prior in the range to ; in this case one can mimic it with a very broad distribution, like a Gaussian with very large ). If the result of the inference `does not offend your physics sensitivity', it means that, essentially, flat priors have done a good job and it is not worth fooling around with more sophisticated ones. In the specific case we are looking closer, that of Eq. (53), the most critical quantity to watch is obviously , because it is positively defined. If, starting from a flat prior (also allowing negative values), the data constrain the value of in a (positive) region far from zero, and - in practice consequently - its marginal distribution is approximatively Gaussian, it means the flat prior was a reasonable choice. Otherwise, the next-to-simple modeling of is via the step function . A more technical choice would be a gamma distribution, with suitable parameters to `easily' accommodate all envisaged values of .
The easiest case, that happens very often if one has `many' data points (where `many' might be already as few as some dozens), is that obtained starting from flat priors is approximately a multi-variate Gaussian distribution, i.e. each marginal is approximately Gaussian. In this case the expected value of each variable is close to its mode, that, since the prior was a constant, corresponds to the value for which the likelihood gets its maximum. Therefore the parameter estimates derived by the maximum likelihood principle are very good approximations of the expected values of the parameters calculated directly from . In a certain sense the maximum likelihood principle best estimates are recovered as a special case that holds under particular conditions (many data points and vague priors). If either condition fails, the result the formulas derived from such a principle might be incorrect. This is the reason I dislike unneeded principles of this kind, once we have a more general framework, of which the methods obtained by `principles' are just special cases under well defined conditions.
The simple case in which
multi-variate Gaussian allows also to approximately
evaluate the covariance matrix of the fit parameters from
the Hessian of its logarithm.6This is due to a well known
property of the multi-variate Gaussian and it is not strictly
related to flat priors.
In fact it can easily proved that if the generic
is a multivariate Gaussian, then
An interesting feature of this approximated procedure is that, since it is based on the logarithm of the pdf, normalization factors are irrelevant. In particular, if the priors are flat, the relevant summaries of the inference can be obtained from the logarithm of the likelihood, stripped of all irrelevant factors (that become additive constants in the logarithm and vanish in the derivatives). Let us write down, for some cases of interest, the minus-log-likelihoods, stripped of constant terms and indicated by , i.e. .