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Source: p 224, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn.
I capitalised miniscules, which the author uses for variables.

I pursue only intuition; please do not answer with formal proofs. Discussing Bayes's Theorem
( $ \Pr(H|E)=\frac{\Pr(E|H)\Pr(H)}{\Pr(E)} $ ), the author abbreviates 'evidence' to E and 'hypothesis' to H.

$\qquad$ For interest, it is worth mentioning that there are quite orthodox methods of statistical inference that try to bypass Bayesian ideas. Much scientific research contents itself with ascertaining that some result would only occur by chance some small percentage of the time (less that 5 per cent, or less than 1 per cent, for example). But it THEN infers that probably the result is not due to chance -- that is, there is a significant causal factor or correlation of some kind involved. This prevalent reasoning is actually highly doubtful, and Bayes shows why.

$\color{green}{[1.]}$ If the antecedent probability that a result is due to anything else than chance is very, very low, then even enormously improbable results will not overturn it. [...]
[I omit the author's example using Scrabble of which I know little.]

$\color{green}{[2.]}$ In this setup ANY result is going to be very improbable, and we should not be able to infer back to say that anything other than chance is responsible for it.

I do not understand $\color{green}{1 \; \& \; 2.}$ About which probabilities is the author cautioning? What exactly is the problem, as (scientists would know that) any probability can be caused 100% by chance?

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Say you make yourself a six-sided die from homogeneous material with precision tools. You roll it three times, and you get the same number every time. The probability of that occurring by chance is $1/36\approx0.028$ ($6$ outcomes out of $6^3$). By the reasoning that the author is criticizing, you could now say that you have significant evidence, at significance level $p\lt0.05$, that your die is biased. But that's wrong; you took care to make it highly symmetrical, and you have high confidence in your machining abilities, and you find it much more likely that the result was due to chance than that the die is biased. So whether something occurring with probability $\lt0.05$ convinces you of a hypothesis depends on how likely you found that hypothesis a priori.

Some other questions and answers related to ignoring priors:

Tricky probability question involving false positives and negatives

What is the probability that the letter came from LONDON?

conditional probability on Truth and lies

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  • Thanks. Sorry for any ignorance, but is there a discrepancy between your 2nd and 3rd sentences? You wrote 'roll it three times', but then '$1/36' = \dfrac{1}{6^2}$? Should not the latter be $\dfrac{1}{6^3}$? –  Feb 07 '16 at 06:34
  • @LePressentiment: I added an explanation to the answer. – joriki Feb 07 '16 at 06:45
  • Thanks. I forgot that 6 was the possible number of outcomes, as you have 6 numbers and each has probability $1/6^2%. –  Feb 07 '16 at 06:53
  • @LePressentiment: No, each has probability $1/216$, for a total of $1/36$. – joriki Feb 07 '16 at 06:54
  • Yes, thank you! Another careless mistake! $1/216 = 1/6^3$ is the probability the each of the six numbers are produced for all three rolls. So the probability the same number is produced for all three rolls= $1/6^3$ for each number $\times 6$ numbers in total. –  Feb 07 '16 at 06:56