# Sufficient Statistics

A statistic $$t$$ is called a sufficient statistic for $$\theta$$ for a given $$\boldsymbol{y}$$ if:

$$p(\boldsymbol{y} | t, \theta)=p(\boldsymbol{y} | t)$$

Let $$Y_{i} \sim \text { Bernoulli }(\theta)$$ for $$i = 1, \dots, n$$, and $$T=\sum_{i=1}^{n} Y_{i}$$. Then it can be shown that $$t=\sum_{i=1}^{n} y_{i}$$ is a sufficient statistic for $$\theta$$ given $$y=\left(y_{1}, \ldots, y_{n}\right)$$.

## Fisher-Neyman Theorem

The Fisher-Neyman theorem, or the factorization theorem, helps us find sufficient statistics more readily. It states that:

A statistic $$t$$ is sufficient for $$\theta$$ if and only if there are functions $$f$$ and $$g$$ such that:

$$p(\boldsymbol{y} | \theta)=f(t, \theta) g(\boldsymbol{y})$$

where $$t=t(\boldsymbol{y})$$.