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.^{35}But 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
flavors^{36}
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 [ , and not `as little', for example, or ]. 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 Bayes factor initial oddswhere 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