(52) | |||

(53) | |||

i.e. | |||

(54) |

The resulting

- Initial box compositions have probability 50% each, that was our assumption.
- The probability of white and black are the same for all extractions, with white a bit more probable than black (14/26 versus 12/26, that is 53.85% versus 46.15%).
- There is also higher probability that the `witness' reports
white, rather than black, but the difference is attenuated by
the `lie factors'.
^{65}In fact, calling and the reported colors we have

(55) (56)

More interesting is the case in which our inference is based on the reported color (figure 13).

The fact that the witness could lie reduces, with respect to the previous case, our confidence on and on white balls in future extractions. As an exercise on what we have learned in appendix H, we can evaluate the `effective' Bayes factor that takes into account the testimony. Applying Eq. (41) we get(57) | |||

(58) |

or JL, about a factor of two smaller than JL, that was 1.1 (this mean we need two pieces of evidence of this kind to recover the loss of information due to the testimony).

The network gives us also the probability that the witness
has really told us the truth, i.e.
, that is
*different* from
, the reason being that
white was initially a bit more probable than black.

Let us see now what happens if we get two concording testimonies (figure 14).

As expected, the probability of increases and becomes closer to the case of a direct observation of white. As usual, also the probabilities of future white balls increase.
The most interesting thing that comes from the
result of the network is how the probabilities
that the two witness lie change. First we see that they are the same,
about 95%, as expected for symmetry. But the surprise is that
the probability the the first witness said the truth has increased,
passing from 85% to 95%. We can justify the variation
because, in qualitative agreement with intuition, if we have
concordant witnesses, *we tend to believe to each of them more
than what we believed individually*. Once again, the result is,
perhaps after an initial surprise, in qualitative agreement with
intuition. The important point is that intuition is unable to
get quantitative estimates. Again, the message is that,
once we agree on the basic assumption and we check, whenever it
is possible, that the results are reasonable, it is better to rely on
automatic computation of beliefs.

Let's go on with the experiment and suppose the third witness says black (figure 15).

This last information reduces the probability of , but does not falsify this hypothesis, as if, instead, we had
The other interesting feature concerns the probability that each
witness has reported the truth. Our belief that the previous
two witnesses really saw what they said
is reduced to 83%. But, nevertheless we are more confident on
the first two witnesses than on the third one,
that we trust only at 76%,
although the lie factor is the same for
all of them. The result is again in agreement with intuition:
if many witnesses state something and fewer say the opposite,
*we tend to believe the majority*, if we initially consider
all witnesses equally reliable.
But a Bayesian network
tells us also
how much we have to believe the many more then the fewer.

Let us do, also in this case the exercise of calculating the
effective Bayes factor, using however the first formula
in footnote 63:
the effective odds
can be written as

(59) |

i.e. , smaller then 1 because they provide an evidence against box ( JL). It is also easy to check that the resulting probability of 75.7% of can be obtained summing up the three weights of evidence, two in favor of and two against it: JL, i.e. , that gives a probability of of 3.1/(1+3.1)=76%.

Finally, let us see what happens if we really see a black ball ( in figure 16).

Only in this case we become certain that the box is of the kind , and the game is, to say, finished. But, nevertheless, we still remain in a state on uncertainty with respect to several things. The first one is the probability of a white ball in future extractions, that, from now becomes 1/13, i.e. 7.7%, and does not change any longer. But we also remain uncertain on whether the witnesses told us the truth, because what they said is not incompatible with the box composition. But, and again in qualitative agreement with the intuition, we trust much more whom told black (1.6% he lied) than the two who told white (70.6% they lied).Another interesting way of analyzing the final network is to consider the probability of a black ball in the five extractions considered. The fourth is one, because we have seen it. The fifth is 92.3% () because we know the box composition. But in the first two extractions the probability is smaller than it (70.6%), while in the third is higher (98.4%). That is because in the two different cases we had an evidence respectively against and in favor of them.

Giulio D'Agostini 2010-09-30