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

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Defining the Hypothesis
1. First you must define the hypothesis to be tested. In this case, the hypothesis being
tested is the claim “The dog ate my homework” (in that specific context, from that
specific student). Therefore:
h = “the dog ate the student’s homework”
(= the student’s claim is true)
2. Then define the negation of this hypothesis. This is what is the case if the hypothesis is
false, which must be what would explain the same evidence if that hypothesis were false.
There may of course be many alternative explanations. There are different ways to handle
that circumstance. We’ll bypass this by sticking with the artificial assumption that there is
only one alternative:
~h = “the student lied about the dog eating her homework”
(= the student’s claim is false)
3. Note that ~h could be generalized (to include all alternative explanations) as simply
(and obviously) “The dog did not eat the student’s homework,” but as this includes no
explicit explanation for any of the evidence (e.g. why the homework is missing, or how
any other evidence came about that the student may have presented in support of his
claim), it has limited explanatory value. This risks becoming a straw man. Specific
theories of the evidence are more robust as explanations and thus pose a greater challenge
to any hypothesis being tested. It is therefore always a good idea to think of what the
most plausible alternative explanation of the evidence may be (or several, if there are
several, and, say, rank them according to what you deem to be their relative plausibility).
Determining Prior Probabilities
1. The prior probability is a relative measure of what is typical. It is a measure of what is
typical in that it draws on what you know about the world, the context, etc. (and students
in general, this student in particular, etc.). It is a relative measure because its value is
determined relative to all other explanations, and not in isolation from them. Thus, all
possible explanations must be examined to determine the prior probability of any.