Suppose that an input signal is one of the mutually exclusive and collectively exhaustive set of signals { A1, A2, A3, ... , An } and that a test is available to determine which signal was input. However, the test is not perfect. It is possible that the test will claim that a signal other than the true one was input. The same test is applied twice to the same input signal.
Let Bj represent the claim from the
first application of the test and
let Ck represent the claim from the
second application of the test.
The probabilities P[Ai],
P[Bj | Ai ] and
P[Ck | Ai Bj ]
are all known.
Furthermore, we can assume that the results of the two tests are
independent of each other, so that
Then after the first application of the test, Bayes’ theorem tells us that
Extended to the pair of tests, this becomes (see the proof below)
But, because the tests are independent and identical (and therefore exchangeable),
which is the same as the single test result, except that the posterior
probabilities from the first test
{ P[ Ai | Bj] } replace
{ P[ Ai ] } as the prior probabilities in the second
test.
An application of this result to a system of three signals, { A1, A2, A3 } and a pair of tests is provided in an associated Excel file.
Proof: