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 two tests are identical, so that
P[Ck | Ai ]
= P[Bk | Ai ]
and that the result of the first test does not affect the outcome of
the second test for the same true input, so that
P[Ck | Ai Bj ]
= P[Ck | Ai ]
Then the two tests are exchangeable.
However, even though the results of the two tests are identical and exchangeable, it follows that they are not independent of each other (except in the special case when the test is completely useless [see the additional note below]). A particular result from the first test enhances the probability of the same result occurring again in the repeated test and reduces the probability of a different result.
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 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:
Note about the dependence of Cj on Bj:
For the same outcome on both tests (k = j),
in general. The two tests are therefore not independent of each
other, unless
This will happen only when Bj is independent
of Ai for all (i, j), when
But the independence of all Bj from all
Ai means that the test is completely
useless
in determining whether or not event
Ai has occurred!