Let be a parameter, and let X be discrete with p.f. fn(x|) conditional on . Let

Let θ be a parameter, and let X be discrete with p.f. fn(x|θ) conditional on θ. Let T = r(X) be a statistic. Prove that T is sufficient if and only if, for every t and every x such that t = r(x), the likelihood function from observing T = t is proportional to the likelihood function from observing X = x.