About subjective probability and Bayesian inference

I hope to have been able to show that it is possible to build a powerful theory of measurement uncertainty starting from subjective probability and the rules of logics, from which the Bayes' theorem follows. Subjective probability is based on the natural concept of probability, as degree of belief, related to a status of uncertainty, whilst Bayes' theorem is the logical tool to update the probability in the light of new pieces of information.

The main advantages the Bayesian approach has over the others are (in addition to the non-negligible fact that it is able to treat problems on which the others fail):

• the recovery of the intuitive idea of probability as a valid concept for treating scientific problems;
• the simplicity and naturalness of the basic tool;
• the capability of combining prior knowledge and experimental information;
• the automatic updating property as soon as new information is available;
• the transparency of the method which allows the different assumptions on which the inference may depend to be checked and changed;
• the high degree of awareness that it gives to its user.

When employed on the problem of measurement errors, as a special application of conditional probabilities, it allows all possible sources of uncertainties to be treated in the most general way.

When the problems get complicated and the general method becomes too heavy to handle, it is often possible to use approximate methods based on the linearization to evaluate average and standard deviation of the distribution, while the central limit theorem makes the final distributions approximately Gaussian. Nevertheless, there are some cases in which the linearization may cause severe problems, as shown in Section . In such cases one needs to go back to the general method or to apply other kinds of approximations which are not just blind use of the covariance matrix.

Many conventional (frequentistic) methods can be easily recovered, like maximum likelihood or $\chi^2$ fitting procedures, as approximation of Bayesian methods, when the (implicit) assumptions on which they are based are reasonable.