# Change detection algorithm - likelihood ratio

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.

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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?

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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 $s<0$.

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