The Likelihood Ratio Test

In a binary hypothesis testing problem, four possible outcomes can result. Model ℳ 0 ℳ 0 did in fact represent the best model for the data and the decision rule said it was (a correct decision) or said it wasn't (an erroneous decision). The other two outcomes arise when model ℳ 1 ℳ 1 was in fact true with either a correct or incorrect decision made. The decision process operates by segmenting the range of observation values into two disjoint decision regions ℜ 0 ℜ 0 and ℜ 1 ℜ 1 . All values of r r fall into either ℜ 0 ℜ 0 or ℜ 1 ℜ 1 . If a given r r lies in ℜ 0 ℜ 0 , for example, we will announce our decision "model ℳ 0 ℳ 0 was true"; if in ℜ 1 ℜ 1 , model ℳ 1 ℳ 1 would be proclaimed. To derive a rational method of deciding which model best describes the observations, we need a criterion to assess the quality of the decision process. Optimizing this criterion will specify the decision regions.

The Bayes' decision criterion seeks to minimize a cost function associated with making a decision. Let C i j C i j be the cost of mistaking model j j for model i i ( i≠j i j ) and C i i C i i the presumably smaller cost of correctly choosing model i i: C i j > C i i C i j C i i , i≠j i j . Let π i π i be the a priori probability of model i i. The so-called Bayes' cost C¯ C is the average cost of making a decision.

The Bayes' cost can be expressed as pr| ℳ i r p r ℳ i r is the conditional probability density function of the observed data r r given that model ℳ i ℳ i was true. To minimize this expression with respect to the decision regions ℜ 0 ℜ 0 and ℜ 1 ℜ 1 , ponder which integral would yield the smallest value if its integration domain included a specific observation vector. This selection process defines the decision regions; for example, we choose ℳ 0 ℳ 0 for those values of r r which yield a smaller value for the first integral. π 0 C 0 0 pr| ℳ 0 r+ π 1 C 0 1 pr| ℳ 1 r< π 0 C 1 0 pr| ℳ 0 r+ π 1 C 1 1 pr| ℳ 1 r π 0 C 0 0 p r ℳ 0 r π 1 C 0 1 p r ℳ 1 r π 0 C 1 0 p r ℳ 0 r π 1 C 1 1 p r ℳ 1 r We choose ℳ 1 ℳ 1 when the inequality is reversed. This expression is easily manipulated to obtain the decision rule known as the likelihood ratio test. The comparison relation means selecting model ℳ 1 ℳ 1 if the left-hand ratio exceeds the value on the right; otherwise, ℳ 0 ℳ 0 is selected. Thus, the likelihood ratio pr| ℳ 1 rpr| ℳ 0 r p r ℳ 1 r p r ℳ 0 r symbolically represented by Λr Λ r , is computed from the observed value of r r and then compared with a threshold η η equaling π 0 C 1 0 − C 0 0 π 1 C 0 1 − C 1 1 π 0 C 1 0 C 0 0 π 1 C 0 1 C 1 1 . Thus, when two models are hypothesized, the likelihood ratio test can be succinctly expressed as the comparison of the likelihood ratio with a threshold.

The data processing operations are captured entirely by the likelihood ratio pr| ℳ 1 rpr| ℳ 0 r p r ℳ 1 r p r ℳ 0 r . Furthermore, note that only the value of the likelihood ratio relative to the threshold matters; to simplify the computation of the likelihood ratio, we can perform any positively monotonic operations simultaneously on the likelihood ratio and the threshold without affecting the comparison. We can multiply the ratio by a positive constant, add any constant, or apply a monotonically increasing function which simplifies the expressions. We single one such function, the logarithm, because it simplifies likelihood ratios that commonly occur in signal processing applications. Known as the log-likelihood, we explicitly express the likelihood ratio test with it as

Useful simplifying transformations are problem-dependent; by laying bare that aspect of the observations essential to the model testing problem, we reveal the sufficient statistic ϒr ϒ r : the scalar quantity which best summarizes the data (Lehmann, pp. 18-22). The likelihood ratio test is best expressed in terms of the sufficient statistic. We will denote the threshold value by γ γ when the sufficient statistic is used or by η η when the likelihood ratio appears prior to its reduction to a sufficient statistic.

As we shall see, if we use a different criterion other than the Bayes' criterion, the decision rule often involves the likelihood ratio. The likelihood ratio is comprised of the quantities pr| ℳ i r p r ℳ i r , termed the likelihood function, which is also important in estimation theory. It is this conditional density that portrays the probabilistic model describing data generation. The likelihood function completely characterizes the kind of "world" assumed by each model; for each model, we must specify the likelihood function so that we can solve the hypothesis testing problem.

A complication, which arises in some cases, is that the sufficient statistic may not be monotonic. If monotonic, the decision regions ℜ 0 ℜ 0 and ℜ 1 ℜ 1 are simply connected (all portions of a region can be reached without crossing into the other region). If not, the regions are not simply connected and decision region islands are created (see this problem). Such regions usually complicate calculations of decision performance. Monotonic or not, the decision rule proceeds as described: the sufficient statistic is computed for each observation vector and compared to a threshold.