Details
In modern terms, given a probability distribution p(x|θ) for an observable quantity x conditional on an unobserved variable θ, the "inverse probability" is the posterior distribution p(θ|x), which depends both on the likelihood function (the inversion of the probability distribution) and a prior distribution. The distribution p(x|θ) itself is called the direct probability.
The inverse probability problem (in the 18th and 19th centuries) was the problem of estimating a parameter from experimental data in the experimental sciences, especially astronomy and biology. A simple example would be the problem of estimating the position of a star in the sky (at a certain time on a certain date) for purposes of navigation. Given the data, one must estimate the true position (probably by averaging). This problem would now be considered one of inferential statistics.
The terms "direct probability" and "inverse probability" were in use until the middle part of the 20th century, when the terms "likelihood function" and "posterior distribution" became prevalent.
Read more about this topic: Inverse Probability
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