{"id":10,"date":"2022-06-20T01:35:02","date_gmt":"2022-06-20T01:35:02","guid":{"rendered":"https:\/\/ebd-2.net\/mkanders\/?page_id=10"},"modified":"2022-09-10T18:55:11","modified_gmt":"2022-09-10T18:55:11","slug":"home","status":"publish","type":"page","link":"https:\/\/ebd-2.net\/mkanders\/","title":{"rendered":"MKAnders"},"content":{"rendered":"\n

Probability Theory Website<\/h3>\n\n\n\n
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What did Thomas Bayes really say in An Essay towards solving a Problem in the Doctrine of Chances<\/a><\/em> (1763)?<\/p>\n\n\n\n

We all use “Bayes’s Rule” to calculate the posterior probability of A<\/em> given B<\/em> from the probability of B<\/em> given A<\/em>, the prior probability of A<\/em>, and the overall probability of B<\/em>:<\/p>\n\n\n\n$$P(A|B) = \\frac{P(B|A)P(A)}{P(B)} $$\n\n\n\n

But Bayes never wrote it.<\/p>\n\n\n\n

Instead, he said something more specific about the probability of \"k\" successes in \"n\" trials of an unknown binary event.  <\/p>\n\n\n\n$$P(k ; n) = \\frac{1}{n+1}, \\quad k = 0, 1, 2, …n.$$\n\n\n\n

Here is how he said it in words:<\/p>\n\n\n\n

… in the case of an event concerning the probability of which we absolutely know nothing antecedently, …I have no reason to think that, in a certain number of trials, it should rather happen any one possible number of times than another.<\/p>\n\n\n\n

And here is my “translation”:<\/p>\n\n\n\n

In independent trials of an unknown binary event, all possible success counts are equally likely.<\/p>\n\n\n\n

The mathematical rule that he highlighted was, in modern notation,<\/p>\n\n\n\n$$ \\int_{0}^{1} \\binom{n}{k} \\theta^{k}(1-\\theta)^{n-k}d\\theta = \\frac{1}{n+1} \\quad k = 0, 1, 2, …n.$$\n\n\n\n

The main purpose of this website is to present my article, What Did Bayes Really Say?<\/a> and provide links to my two other websites: EBD-2.net and sample-size.net . EBD-2.net<\/a> is the website for the textbook Evidence-Based Diagnosis, 2nd edition<\/em>, by Thomas B. Newman and Michael A. Kohn, illustrated by Martina A. Steurer. Sample-size.net<\/a> is a website of calculators for clinical investigators. It was was developed by Michael Kohn and Josh Senyak (Quicksilver Consulting<\/a>).<\/p>\n

— Michael Kohn, updated 7 September 2022<\/p>\n","protected":false},"excerpt":{"rendered":"

Probability Theory Website What did Thomas Bayes really say in An Essay towards solving a Problem in the Doctrine of Chances (1763)? We all use “Bayes’s Rule” to calculate the posterior probability of A given B from the probability of B given A, the prior probability of A, and the overall probability of B: $$P(A|B) … <\/p>\n

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