AI - a Guide to Intelligent Systems.pdf - Member of EEPIS

BIAS OF THE BAYESIAN METHOD
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Domain experts do not deal easily with conditional probabilities and quite
often deny the very existence of the hidden implicit probability (0.3 in our
example).
In our case, we would use available statistical information and empirical
studies to derive the following two rules:
IF the starter is bad
THEN the symptom is ‘odd noises’ {with probability 0.85}
IF the starter is bad
THEN the symptom is not ‘odd noises’ {with probability 0.15}
To use the Bayesian rule, we still need the prior probability, the probability
that the starter is bad if the car does not start. Here we need an expert judgement.
Suppose, the expert supplies us the value of 5 per cent. Now we can apply the
Bayesian rule (3.18) to obtain
pðstarter is bad j odd noisesÞ ¼
0:85 0:05
0:85 0:05 þ 0:15 0:95 ¼ 0:23
The number obtained is significantly lower than the expert’s estimate of 0.7
given at the beginning of this section.
Why this inconsistency? Did the expert make a mistake?
The most obvious reason for the inconsistency is that the expert made different
assumptions when assessing the conditional and prior probabilities. We may
attempt to investigate it by working backwards from the posterior probability
pðstarter is bad j odd noisesÞ to the prior probability pðstarter is badÞ. In our case,
we can assume that
pðstarter is goodÞ ¼1
pðstarter is badÞ
From Eq. (3.18) we obtain:
pðHÞ ¼
pðHjEÞpðEj:HÞ
pðHjEÞpðEj:HÞþpðEjHÞ½1
pðHjEÞŠ
where:
pðHÞ ¼pðstarter is badÞ;
pðHjEÞ ¼pðstarter is bad j odd noisesÞ;
pðEjHÞ ¼pðodd noises j starter is badÞ;
pðEj:HÞ ¼pðodd noises j starter is goodÞ.
If we now take the value of 0.7, pðstarter is badjodd noisesÞ, provided by the
expert as the correct one, the prior probability pðstarter is badÞ would have