Bayesian statistics: learning by experience

**Example 1:**- Imagine some persons
listening to a common friend
having a phone conversation with an unknown person ,
and who
are trying to guess who is. Depending on the knowledge
they have about the friend, on the language spoken,
on the tone of voice, on the subject of conversation, etc.,
they will attribute some probability to several
possible persons. As the conversation goes on they begin
to consider some possible candidates for , discarding others,
then hesitating perhaps only between a couple of possibilities,
until
the state of information is such that they are
*practically sure*of the identity of . This experience has happened to most of us, and it is not difficult to recognize the Bayesian scheme:

We have put the initial state of information explicitly in () to remind us that likelihoods and initial probabilities depend on it. If we know nothing about the person, the final probabilities will be very*vague*, i.e. for many persons the probability will be different from zero, without necessarily favouring any particular person. **Example 2:**- A person meets an old friend in a pub.
proposes
that the drinks should be payed for by
whichever
of the two extracts
the card of lower value
from a pack
(according to some rule which is of no
interest to us). accepts and
wins. This situation happens again in the following days
and it is always who has to pay.
What is the probability that has become a cheat, as
the number of consecutive wins increases?
The two hypotheses are:

*cheat*() and*honest*(). is low because is an ``old friend'', but certainly not zero: let us__assume__. To make the problem simpler let us make the approximation that a cheat always wins (not very clever): . The probability of winning if he is honest is, instead, given by the rules of probability*assuming*that the chance of winning at each trial is (``why not?", we shall come back to this point later): . The result

is shown in the following table.

(%) (%) 0 5.0 95.0 1 9.5 90.5 2 17.4 82.6 3 29.4 70.6 4 45.7 54.3 5 62.7 37.3 6 77.1 22.9

- The answer is always probabilistic. can never reach
absolute certainty that is a cheat,
unless he catches cheating, or
confesses to having cheated. This is coherent
with the fact that we are dealing with random events
and with the fact that any sequence of outcomes has the
same probability (although there is only one possibility over
in which is
__always__luckier). Making__use__of , can make a__decision__about the next action to take:__continue__the game, with probability of__losing__with certainty the next time too;__refuse__to play further, with probability of__offending__the innocent friend.

- If the final probability will always remain zero: if fully trusts , then he has just to record the occurrence of a rare event when becomes large.

To better follow the process of updating the probability when new experimental data become available, according to the Bayesian scheme

*``the final probability of the present inference is the initial probability of the next one''*.

(3.22) (3.23)

where and are the probabilities of__each__win. The interesting result is that__exactly__the same values of of () are obtained (try to believe it!). - The answer is always probabilistic. can never reach
absolute certainty that is a cheat,
unless he catches cheating, or
confesses to having cheated. This is coherent
with the fact that we are dealing with random events
and with the fact that any sequence of outcomes has the
same probability (although there is only one possibility over
in which is

It is also instructive to see the dependence of the final probability on the initial probabilities, for a given number of wins .

24 | 91 | 99.7 | 99.99 | |

63 | 98 | 99.94 | 99.998 | |

97 | 99.90 | 99.997 | 99.9999 |