Change detection algorithm - likelihood ratio

Question

Consider a sequence of independent random variables $(y_k)_k$ with a probability density $p_{\theta}(y)$ depending upon only one scalar parameter. Before the unknown change time $t_0$, the parameter $\theta$ is equal to $\theta_0$, and after the change it is equal to $\theta_1 \neq \theta_0$. The problems are then to detect and estimate this change in the parameter.

snip... snip...

The tools for reaching this goal are as follows. First, our description of all the algorithms of this chapter is based on a concept that is very important in mathematical statistics, namely the logarithm of the likelihood ratio, defined by $$s(y) = \ln\frac{p_{\theta_1}(y)}{p_{\theta_0}(y)}$$ and referred to as the log-likelihood ratio. The key statistical property of this ratio is as follows : Let $E_{\theta_0}$ and $E_{\theta_1}$ denote the expectations of the random variables under the two distributions $p_{\theta_0}$ and $p_{\theta_1}$, respectively. Then,

$$E_{\theta_0}(s) < 0\space \text{and} \space E_{\theta_1}(s) > 0 $$

(This appears to be a quotation from Chand & Xiao, Change-Point Monitoring for Secure In-Network Aggregation in Wireless Sensor Networks, 2007.)

My question is HOW can we get the 2nd formula? What does $E_{\theta_0}(s)$ mean (I know $E$ is the expectation of $y$, but what about $(s)$ )? How can we get the > 0 or < 0?

Answer 1

Here $E_{\theta_0}(s)$ is the expectation of log likelihood ratio given that $\theta=\theta_0$, and similarly for $E_{\theta_1}(s)$. Now, if $\theta=\theta_0$ we expect $p_{\theta_0}(y)$ to be greater than $p_{\theta_1}(y)$ so $\frac{p_{\theta_1}(y)}{p_{\theta_0}(y)}<1$, and so $s<0$.