(Principle of sufficient statistic). The statistic
is sufficient if the distribution
conditionally on given value of
does not depend on
.

In other words the sufficient statistic
captures all the relevant information.

Theorem

(Factorization). Let
is a sufficient statistic for
if and only if there exist some functions
and
such
that

Proof

For simplicity we assume that the sample space is discrete. The
is a random variable, the
is a deterministic variable and possible realized value of
.
We prove direct statement. We assume that
is a sufficient
statistic:
By
(
Bayes_formula
),
According to the formula (
Bayes
formula
)
and the direct claim follows.

We now prove the inverse statement. Suppose that there is the factorization.
Then
We also have the
property
Hence,
Therefore, the ratio
does not depend on
This completes the proof.