Sufficient Statistics and the Neyman-Fisher Factorization Theorem

A risk system that has logged three years of daily P&L does not need to keep the full tick history to calibrate a $\mathcal N(\mu,\sigma^2)$ model — it needs only the running sum $\sum_i x_i$ and sum of squares $\sum_i x_i^2$ (equivalently $\bar x$ and $s^2$). Any two datasets with the same two numbers give the exact same likelihood curve, the exact same maximum-likelihood estimate, and the exact same inference about $(\mu,\sigma^2)$, however differently the individual daily returns were sequenced. This is not an approximation or a convenient simplification — it is an exact statement that the data can be losslessly compressed down to a low-dimensional statistic without discarding any information relevant to $\theta$. The formal name for this property is sufficiency, and the practical tool for finding a sufficient statistic without wading through a conditional-distribution computation is the Neyman-Fisher factorization theorem, the central result of this note.

Learning objectives. By the end of this note you should be able to: