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In probability theory, inverse probability is an obsolete term for the probability distribution of an unobserved variable. 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). The distribution p(x|θ) itself is called the direct probability.
The inverse probability problem (in the 1700s and 1800s) 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 twentieth century, when the terms "likelihood function" and "posterior distribution" became prevalent.
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