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This is more of a history of science question, but I hope it's on-topic here.

I've read that Thomas Bayes only managed to discover Bayes' theorem for the special case of a uniform prior, and even then he struggled with it, apparently.

Considering how trivial the general Bayes' theorem is in modern treatment, why did it present a challenge for Bayes and other mathematicians at the time? For comparison, Isaac Newton's Philosophiæ Naturalis Principia Mathematica was published 36 years before Bayes' main work, which was published posthumously.

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    $\begingroup$ Because the prior knowledge at the time was not very informative for his theorem. $\endgroup$
    – Marc Claesen
    Jan 17, 2014 at 9:14

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Bayes' paper† begins:–

Given the number of times in which an unknown event has happened and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named.

Coming up with the theorem that now bears his name may not have been the most challenging part, nor his primary concern; rather he struggled with applying it to the problem of inference, & especially with justifying the assumption of a particular prior probability distribution. Argumentation on these issues continues into the 21st Century.

† Bayes (1763), "An Essay towards solving a Problem in the Doctrine of Chances", Philosophical Transactions of the Royal Society of London 53, pp370–418.

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Because everything is better understood now after great efforts by many people. As a result of that, it is much easier to teach these concepts in an understandable, intuitive manner. Imagine that you only know what was known at that time, instead of everything you've learnt now.

You can think of it as a puzzle: the more pieces are in place the easier it is to solve the remainder. The comparison may be a stretch, but discovering fire was no trivial matter either at the time it happened even if it may seem like one now.

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    $\begingroup$ I happened to be there at the time fire was discovered (so I can vouch for this analogy). It was not trivial at all; it was downright terrifying! $\endgroup$
    – Nick Stauner
    Jan 17, 2014 at 8:24

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