Consider a family of density functions $f(x|\theta)$ where the parameter space for $\theta$ is finite, that is, $\theta \in \{\theta_0, \cdots, \theta_p\}$. Assume that $\theta_0$ is such that $f(x|\theta_0) > 0$ for all $x$ on the support. Now consider the statistic $T(\mathbf{X}) = \left(\frac{f(\mathbf{X}|\theta_1)}{f(\mathbf{X}|\theta_0)}, \cdots, \frac{f(\mathbf{X}|\theta_p)}{f(\mathbf{X}|\theta_0)} \right)$. The question is to show that $T(X)$ is a minimal sufficient statistic.
I can show that this is the minimum sufficient statistics easily. However, I just wonder why we need the parameter space to be finite? Can we extend this result to infinite case? Why or why not? Thank you!