Suppose, conditional on state $\theta \in \{0, 1\}$, there is a signal $x \sim F(x)$ which satisfy MLRP property. That is $\frac{f_1(x)}{f_0(x)}$ is increasing in x
Now, we add another level of signal. That is conditional on x, there is a signal g which also satisfies MLRP, that is $\frac{g(s|x_1)}{g(s|x_2)}$ is increasing in s for all $x_1 > x_2$
I am trying to show that the unconditional distribution of $g | \theta$ will also satisfy MLRP.
But $\frac{g(s|\theta = 1)}{g(s|\theta = 0)} = \frac{\int_X g(s|x) f_1(x) dx}{\int_X g(s|x) f_0(x) dx}$
Assuming absolute continuity of f and g (to make things simpler) how can I do that?
