Sequential Probability Ratio Test

The Sequential Probability Ratio Test is a Likelihood ratio test for determining which of two hypotheses is more likely. It is appropriate for sequential independent and identically distributed (iid) data.

Assume we take a series of measurements, from time 1 up to time t ( $m_{1 : t}$ ), and that our state $s$ is either +1 or -1. To figure out what the state is given our measurements, we cam use a log likelihood ratio $L_{T}$ .

\begin{aligned} L_{T} & = l o g \frac{p (m_{1 : t} | s = + 1)}{p (m_{1 : t} | s = - 1)} \end{aligned}

Since our data is independent and identically distribution, the probability of all measurements given the state equals the product of the separate probabilities of each measurement given the state ( $p (m_{1 : t} | s) = \prod_{t = 1}^{T} p (m_{t} | s)$ ). We can substitute this in and use log properties to convert to a sum.

\begin{aligned} L_{T} & = l o g \frac{p (m_{1 : t} | s = + 1)}{p (m_{1 : t} | s = - 1)} \\ = l o g \frac{\prod_{t = 1}^{T} p (m_{t} | s = + 1)}{\prod_{t = 1}^{T} p (m_{t} | s = - 1)} \\ = \sum_{t = 1}^{T} l o g \frac{p (m_{t} | s = + 1)}{p (m_{t} | s = - 1)} \\ = \sum_{t = 1}^{T} Δ_{t} \end{aligned}

where $Δ_{t} = l o g \frac{p (m_{t} | s = + 1)}{p (m_{t} | s = - 1)}$ .

If $L_{T}$ is positive, then the state $s = + 1$ is more likely than $s = - 1$ , and vice versa.