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Inverse probability relates strongly, or is synonymous to, Bayesian probability.

Thomas Bayes applied the idea in 'An Essay towards solving a Problem in the Doctrine of Chances' (published in 1763).

Who was the first to use the term 'inverse probability' and why did they call it 'inverse probability' instead of 'reverse probability'?

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Reading verbatim from Dale's History of Inverse Probability he mentions that the first occurrence of the term in English is due to Augustus de Morgan in the 1830's as for instance in his 1834 Encyclopædia Metropolitana but he considers that the first "inverse" perspective was to be in Abraham de Moivre's 1756 edition of his Doctrine of Chances, which contains as an Appendix his 1733 Approximatio ad summam terminorum binomii $\overline{a+b}^n$ in seriem expansi and uses the term conversely

...upon the Supposition of a certain determinate Law according to which any Event is to happen, we demonstrate that the Ratio of Happenings will continuously approach to that Law, as the Experiments or Observations are multiplied: so, conversely, if from numberless Observations we find the Ratio of the Events to converge to a certain quantity...[1756, p.251]

David Hartley in his 1749 Observations on Man, His Frame, His Duty and His Expectations called the above resolution by Mr. de Moivre, a "Solution of the inverse Problem" [p.338-339], mentioning in the same paragraph recovering the probability of " unknown Causes by a sufficient Observation of their Effects"

Dale also points out the use of a priori and a posteriori [in a Bayesian sense] by Jakob Bernoulli much earlier in Ars Conjectandi (posthumously published in 1713).

Richard Price in his 1763 introduction to Bayes' Essay also mentions the "converse problem".

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