Comments and conclusions

Well, this was meant to be a short note. Obviously it is not just a comment to the New Scientist article, that could have been contained in a couple of sentences. In fact, discussing with several people, I felt that yet another introduction to Bayesian reasoning, not focused on physics issues, might be useful. So, at the end of the work, Columbo's cameras were just an excuse.

Let us now summarize what we have learned and make further comments on some important issues.

First, we have to be aware that often we do not see `a fact' (e.g. Galesco killing his wife), but we infer it from other facts, assuming a causal connection among them.³⁵But sometimes the observed effect can be attributed to several causes and, therefore, having observed an effect we cannot be sure about its cause. Fortunately, since our beliefs that each possible cause could produce that effect are not equal, the observation modifies our beliefs on the different causes. That is the essence of Bayesian reasoning. Since `Bayesian' has several flavors³⁶ in the literature, I summarize the points of view expressed here:

Probability simply states, in a quantitative way, how much we believe something. (If you like, you can reason the other way around, thinking that something is highly improbable if you would be highly surprised if it occurs.³⁷)
``Since the knowledge may be different with different persons or with the same person at different times, they may anticipate the same event with more or less confidence, and thus different numerical probabilities may be attached to the same event.'' [11] This is the subjective nature of probability.
Initial probabilities can be elicited, with all the vagueness of the case,³⁸ on a pure subjective base (see Appendix C). Virtual bets or comparisons with reference events can be useful `tools' to force ourselves or experts to provide quantitative statements of our/their beliefs. (See also Appendix C.)
Probabilities can (but need not) be evaluated by past frequencies and can even be expressed in terms of expected frequencies of `successes' in hypothetical trials. (See Appendix B.)
Probabilities of causes are not generated, but only updated by new pieces of evidence.
Evidence is not only the `bare fact', but also all available information about it (see Appendix D). This point is often overlooked, as in the criticisms to Columbo's episode raised by New Scientist [1].
The update depends on how differently we believe that the various causes might produce the same effect (see also Appendix G).
The probability of a single hypothesis cannot be updated, if there isn't at least a second hypothesis to compare with, unless the hypothesis is absolutely incompatible with the effect [ $P(E\,\vert\,H,I)=0$ , and not `as little', for example, $10^{-9}$ or $10^{-23}$ ]. Only in this special case an hypothesis is definitely falsified. (See Appendix G.)
In particular, if there is only one hypothesis in the game, the final probability of this hypothesis will be one, no matter if it could produce the effect with very small probability (but not zero). ³⁹
Initial probabilities depend on the information stored somehow in our brain; being, fortunately, each brain different from all others, it is quite natural to admit that, in lack of `experimental data',``quot capita, tot sententiae''. (See Appendix C.)
In the probabilistic inference (i.e. that stems from probability theory) the updating rule is univocally defined by Bayes' theorem (hence the adjective `Bayesian' related to these methods).
This objective updating rule makes final beliefs virtually independent from the initial ones, if rational people all share the same `solid' experimental information and are ready to change their opinion (the latter disposition has been named Cromwell's rule by Dennis Lindley [18]).
In the simple case that two hypotheses are involved, the most convenient way to express the Bayes' rule is

final odds $\displaystyle =$ Bayes factor $\displaystyle \times$ initial odds $\displaystyle ,$
where the Bayes factor can be seen as the odds due to a single piece of evidence, if the two hypotheses were considered otherwise equally likely. (See also examples in Appendices F and G, as well as Appendix H, for comments on statistical methods based on likelihood.)
In some cases - almost always in scientific applications - Bayes factors can be calculated exactly, or almost exactly, in the sense that all experts will agree. In many other real life cases their interpretation as `virtual' odds (in the sense stated above) allows to elicit them with the bet mechanism as any subjective probability. (See Appendix C.)
Bayes factors due to several independent pieces of evidence factorize.
The multiplicative updating rule can be turned into an additive one using logarithms of the factors. (See Appendix E.)
The base 10 logarithms has been preferred here because they are easily related to the orders of magnitudes of the odds and the name `judgement leanings' (JL) has been chosen to have no conflict with other terms already engaged in probability and statistics.
Each logarithmic addend has the meaning of weight of evidence, if the initial odds are taken as 0-th evidence.
Individual contribution to the judgement might be small in module and even somehow uncertain, but, nevertheless, their combination might result into strong convincingness. (See Appendix G.)
In most real life cases there are not just two alternative causes and two possible effects. Moreover, causes can be effects of other causes and effects can be themselves causes of other effects. All hypotheses in the game make up a complex `belief network'. Experts can certainly provide kinds of educated guesses to state how likely a cause can generate several effects, but the analysis of the full network goes well beyond human capabilities, as discussed more extensively in Appendix C and J.
A next to simply case is when the evidence is mediated by a testimony. The formal treatment in Appendix I shows that, although experts can easily assess the required ingredients, the conclusions are really not so obvious.
The question of the critical value of the judgement leaning, above which a suspected can be condemned, goes beyond the purpose of this notes, focused on belief. That is a delicate decision problem that inherits all issues of assessing beliefs, to which the evaluations of benefits and losses need to be added.

And Galesco? Come on, there is little to argue.

Nevertheless, the reading of the instructive New Scientist article is warmly recommended!

It is a pleasure to thank Pia and Maddalena, who introduced me Columbo, and Dino Esposito, Paolo Agnoli and Stefania Scaglia for having taken part to the post dinner jury that absolved him. The text has benefitted of the careful reading by Dino, Paolo and Enrico Franco (see in particular his interesting remark in footnote 44).

Giulio D'Agostini 2010-09-30