 Statistical data compression by a sufficient statistic

A sufficient statistic allows the observed likelihood function to be stored through a shorter record, up to a parameter-independent factor. For the Bernoulli sample sum as a sufficient statistic, writeThe second likelihood function belongs to the observed count; the first belongs to the observed sequence. Their absolute values differ by the binomial coefficient, but their likelihood ratios between any two interior parameter values agree. A maximum-likelihood estimator can therefore use just $(n,t)$.

This compression discards which arrangement actually occurred. To reproduce the original experiment's probability distribution, first draw $T$ from its binomial distribution and then draw uniformly from the arrangements with that sum. The second step is independent of $p$, so it supplies no additional evidence about $p$. It does not recover the particular discarded sequence. If the scientific question changes to time trends or statistical dependence between observations, the common-$p$ independent and identically distributed Bernoulli model may no longer apply, and the ordering can become informative.