“Bayes' Theorem for Beginners: Formal Logic and ... - Richard Carrier

15. When you genuinely don’t know what is typical, or have no idea what would be
typical (i.e. if you had no idea whether it was common for students to lie or dogs to eat
homework), then the prior probabilities are equal, i.e. P(h|b) = 0.50 and P(~h|b) = 0.50, if
you would agree there is as much chance of one hypothesis being true as the other
(margins of error can still be assigned depending on how uncertain you are of even that).
16. All that aside, we will assume for our example that P(h|b) = 0.10 and P(~h|b) = 0.90
(our first estimates, i.e. we estimated that typically 90% of all ‘dog ate my homework’
stories are false, and only 10% are true). Therefore:
32
0.10 x P(e|h.b)
P(h|e.b) = __________________________________
[ 0.10 x P(e|h.b) ] + [ 0.90 x P(e|~h.b) ]
Determining Consequent Probabilities
1. There are two prior probabilities in the Bayesian equation (which together sum to 1,
covering a complete probability space) and two consequent probabilities (which do not have
to sum to 1 or any other value, as they are completely independent of each other). The
first consequent probability is that of having the evidence we do if h is true, the second consequent
probability is that of having the evidence we do if h is false. Those four values
complete the Bayesian equation. They are in effect four premises in a formal logical
argument, so if you can defend these premises, you can defend the conclusion.
2. If there is no reason to expect any evidence of the incident other than the missing
homework (and the student’s story), then the probability of having the evidence we do if
h (“the dog ate the student’s homework”) is true is 100% (because we have exactly what
we expect and nothing more nor less), as high as can be, and therefore:
P(e|h.b) = 1.00 (common case)
3. What would it take to lower this value? If any of the evidence we have were contrary
to expectation. For example, if you think there was a small chance (say, 20%) that your
student, given her story, could at least produce scraps of dog-mangled homework as
evidence, then your student’s inability to produce that evidence is somewhat contrary to
expectation, and therefore the absence of this expected evidence drops the probability of
all the evidence we have by as much as the missing evidence is expected, in other words
by 20% (or 0.20). Therefore:
P(e|h.b) = 1.00 – 0.20 = 0.80 (case with small incongruity)