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

have exactly the same effect on P(e|~h.b), which cancels out, so the natural improbability
of such evidence (i.e. the unlikelihood of a trusted colleague just happening to be in a
position to witness the event) can be ignored. Of course, that’s not so if other evidence
makes such a coincidence suspicious (i.e. you have evidence your colleague was set up),
but that’s a much more complex example.
Lesson 2: When there is good evidence for a hypothesis, it is ‘good’ simply because it
makes other hypotheses less likely, because those hypotheses have a harder time
explaining how that evidence came about. Therefore, evidence supporting h lowers the
probability of ~h (more specifically, it lowers the probability of e on ~h). When there is
evidence against a hypothesis, it is ‘against’ it simply because it makes other hypotheses
more likely, because the tested hypothesis has a harder time explaining that evidence than
they do. Therefore, evidence against h lowers the probability of h (more specifically, it
lowers the probability of e on h). Which also means evidence for an alternative
hypothesis is evidence against h, but only if that evidence is relevantly unexpected on h.
If, instead, it is just as predicted by h as by the alternative, then it is no longer evidence
for the alternative—it is then inconclusive evidence. Likewise, evidence believed to
support h is also, in fact, inconclusive (and thus doesn’t actually support h in any notable
way) if that same evidence is just as likely on an alternative hypothesis (~h).
Fully understanding both lessons, and why they are true, is one of the keys to
understanding why Bayes’ Theorem works and how to think like a Bayesian even
without the math. Equally key is a full understanding of why prior probability is not just
the raw frequency of such things happening (i.e. such things happening as are proposed
by h), but the relative frequency of different explanations of the same evidence.
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Advancing to Increasingly Objective Estimates
This has been a simple example of the mechanics of Bayes’ Theorem. In reality you will
want a more informed and carefully considered estimate for each of the four probability
values that must go into any Bayesian analysis. In other words, the question must be
asked, “How do you get those values? Do you just pull them out of your ass?” In a sense,
yes, but in a more important sense, no.
You aren’t just blindly making up numbers. You have some reason for preferring
a low number to a high one, for example (or a very low one to one that’s merely
somewhat low, and so on), and the strength of that reason will be the strength of any
conclusion derived from it—by the weakest link principle, i.e. any conclusion from
Bayes’ Theorem will only be as strong as the weakest premise in it (i.e. the least
defensible probability estimate you employ). So to make the conclusions of a Bayesian
analysis stronger, you must make history more scientific, and the degree to which you fall