## What is Inverse Probability?

Inverse probability is the probability of things that are unobserved; or, more technically, the probability distribution of an unobserved variable. It’s generally considered **an obsolete term. **

Nowadays, the basis of inverse probability (determining the unobserved variable) is usually called inferential statistics, and the main problem of inverse probability—finding a probability distribution for an unobserved variable—is usually called Bayesian probability.

## Inverse Probability, aka Bayesian Probability— What it Involves

Bayesian probability is often used when we want to calculate the likelihood of certain outcomes given a particular hypothesis. It’s a way of making logical inference problems into simple statistics problems, by looking at conditional probabilities and comparing outcomes given different hypothetical scenarios.

The Bayes Rule formula is:

The rule can be written with slightly different notation to illustrate the connection between a hypothesis and a condition:

Here P(H|D) is the likelihood that hypothesis H is true given a particular condition D, P(D|H) is the probability that the condition D is true given the hypothesis being true, and P(H) and P(D) are the probabilities of observing the hypothesis and the condition D, independently of each other.

Watch the video for a quick example of working a Bayes’ Theorem problem, or read a different example problem below:

## Simple Inverse Probability Example: Applying Bayes Rule

A screening test for a particular genetic abnormality is highly effective; it gives 99% true positive results for those who carry the abnormality, and 95% true negative results to those who don’t. Only a very small percentage of the general population, 0.001%, carry this genetic abnormality.

Inverse probability and Bayes rule allows us to calculate what the likelihood is that a random someone carries the genetic abnormality, given a positive test. The genetic abnormality is the hypothesis, and the positive test is our condition. In our formula above, we’ll want to plug in the values:

P(D|H) = 0.99

P(H) = 0.00001

P(D), or the probability of a positive test, is just the sum of two terms. The first term is the probability of a positive test given the genetic abnormality times the likelihood that the abnormality exists. The second term will be the probability of a positive test given no genetic abnormality, times the likelihood of no genetic abnormality. So:

P(D) = P(D|H) P(H) + P(D|~H) P(~H).

That equals 0.99*0.00001 + 0.01*0.99999, or 0.0100098.

Plugging these into the formula above for Bayes rule, we get:

P(H|D) = [0.99*0.00001] / 0.0100098

Or, with no regard for significant digits, 0.00098903074. We see that though the test may be fairly reliable, the genetic abnormality is rare enough that even a positive test only leaves a person with only about a 0.09 percent likelihood of having the abnormality.

**Next**: Bayes Theorem Problems.

## References

Olshausen, B. (2004). Bayesian Probability Theory. Retrieved December 5, 2017 from http://redwood.berkeley.edu/bruno/npb163/bayes.pdf.

Bayes Theorem, Stanford Encyclopedia of Philosophy

https://plato.stanford.edu/entries/bayes-theorem/

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