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

Bayesian Reasoning Flowchart
- P(e|~h.b) is same or higher = low P(h|e.b) = h is probably false
-- low P(e|h.b) ---|
| - P(e|~h.b) is lower = low P(h|e.b) unless prior is high enough
|
low P(h|b) -------- |
|
| - P(e|~h.b) is lower = low P(h|e.b) unless prior is high enough
-- high P(e|h.b) ---|
- P (e|~h.b) is same or higher = low P(h|e.b) = h is probably false
- P(e|~h.b) is same or lower = high P(h|e.b) = h is probably true
-- low P(e|h.b) ---|
| - P(e|~h.b) is higher = low P(h|e.b) unless prior is high enough
|
high P(h|b) ------- |
|
| - P (e|~h.b) is higher = low P(h|e.b) unless prior is high enough
-- high P(e|h.b) ---|
- P(e|~h.b) is same or lower = high P(h|e.b) = h is probably true
“Low” and “High” = lower than 0.5 (50%) and higher than 0.5 (50%), respectively; when P(h|b)
= 0.5, so does P(~h|b): then the hypothesis with the higher P(e|b) is probably true.
“Prior is high enough” = when P(h|b) is higher than the Bayesian ratio between either P(e|~h.b) and
P(e|h.b) or vice versa, enough to overcome the gap and thus produce the opposite result.
B. The Application of Probability Theory to Bayesian Reasoning
1. All the rules of probability theory and statistical analysis apply to Bayesian
calculations, so the more you know about the former, the more you can do with
the latter.
For example:
2. Probability estimates in history are usually intuitive and thus inexact, therefore to
ensure a high confidence level (i.e. a strong enough certainty to warrant
designating your conclusion an objective result of historical inquiry), you must
adopt a wide margin of error.
Since in probability theory (e.g. in scientific polls) the wider the margin of error,
the higher the confidence level, if you widen your margin of error as far as you
can reasonably believe it possible to be, given the evidence available to all expert
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