Maximum Probability of a Correct Decision

As only one model can describe any given set of data (the models are mutually exclusive), the probability of being correct P c P c for distinguishing two models is given by P c =Pr say   ℳ 0   when   ℳ 0   true +Pr say   ℳ 1   when   ℳ 1   true P c say   ℳ 0   when   ℳ 0   true say   ℳ 1   when   ℳ 1   true We wish to determine the optimum decision region placement Expressing the probability correct in terms of the likelihood functions pr| ℳ i r p r ℳ i r , the a priori probabilities, and the decision regions, P c =∫ ℜ 0 π 0 pr| ℳ 0 rdr+∫ ℜ 1 π 1 pr| ℳ 1 rdr P c r ℜ 0 π 0 p r ℳ 0 r r ℜ 1 π 1 p r ℳ 1 r We want to maximize P c P c by selecting the decision regions ℜ 0 ℜ 0 and ℜ 0 ℜ 0 . The probability correct is maximized by associating each value of r r with the largest term in the expression for P c P c . Decision region ℜ 0 ℜ 0 , for example, is defined by the collection of values of r r for which the first term is largest. As all of the quantities involved are non-negative, the decision rule maximizing the probability of a correct decision is

Simple manipulations lead to the likelihood ratio test. pr| ℳ 1 rpr| ℳ 0 r ≷ ℳ 0 ℳ 1 π 0 π 1 p r ℳ 1 r p r ℳ 0 r ≷ ℳ 0 ℳ 1 π 0 π 1 Note that if the Bayes' costs were chosen so that C i i =0 C i i 0 and C i j =C C i j C , ( i≠j i j ), we would have the same threshold as in the previous section.

To evaluate the quality of the decision rule, we usually compute the probability of error P e P e rather than the probability of being correct. This quantity can be expressed in terms of the observations, the likelihood ratio, and the sufficient statistic.

When the likelihood ratio is non-monotonic, the first expression is most difficult to evaluate. When monotonic, the middle expression proves the most difficult. Furthermore, these expressions point out that the likelihood ratio and the sufficient statistic can be considered a function of the observations r r; hence, they are random variables and have probability densities for each model. Another aspect of the resulting probability of error is that no other decision rule can yield a lower probability of error. This statement is obvious as we minimized the probability of error in deriving the likelihood ratio test. The point is that these expressions represent a lower bound on performance (as assessed by the probability of error). This probability will be non-zero if the conditional densities overlap over some range of values of r r, such as occurred in the previous example. In this region of overlap, the observed values are ambiguous: either model is consistent with the observations. Our "optimum" decision rule operates in such regions by selecting that model which is most likely (has the highest probability) of generating any particular value.

Neyman-Pearson Criterion

Situations occur frequently where assigning or measuring the a priori probabilities P i P i is unreasonable. For example, just what is the a priori probability of a supernova occurring in any particular region of the sky? We clearly need a model evaluation procedure which can function without a priori probabilities. This kind of test results when the so-called Neyman-Pearson criterion is used to derive the decision rule. The ideas behind and decision rules derived with the Neyman-Pearson criterion (Neyman and Pearson) will serve us well in sequel; their result is important!

Using nomenclature from radar, where model ℳ 1 ℳ 1 represents the presence of a target and ℳ 0 ℳ 0 its absence, the various types of correct and incorrect decisions have the following names (Woodward, pp. 127-129).1

The remaining probability Pr say   ℳ 0 | ℳ 0   true ℳ 0   true say   ℳ 0 has historically been left nameless and equals 1- P F 1 P F . We should also note that the detection and miss probabilities are related by P M =1- P D P M 1 P D . As these are conditional probabilities, they do not depend on the a priori probabilities and the two probabilities P F P F and P D P D characterize the errors when any decision rule is used.

These two probabilities are related to each other in an interesting way. Expressing these quantities in terms of the decision regions and the likelihood functions, we have P F =∫ ℜ 1 pr| ℳ 0 rdr P F r ℜ 1 p r ℳ 0 r P D =∫ ℜ 1 pr| ℳ 1 rdr P D r ℜ 1 p r ℳ 1 r As the region ℜ 1 ℜ 1 shrinks, both of these probabilities tend toward zero; as ℜ 1 ℜ 1 expands to engulf the entire range of observation values, they both tend toward unity. This rather direct relationship between P D P D and P F P F does not mean that they equal each other; in most cases, as ℜ 1 ℜ 1 expands, P D P D increases more rapidly than P F P F (we had better be right more often than we are wrong!). However, the "ultimate" situation where a rule is always right and never wrong ( P D =1 P D 1 , P F =0 P F 0 ) cannot occur when the conditional distributions overlap. Thus, to increase the detection probability we must also allow the false-alarm probability to increase. This behavior represents the fundamental tradeoff in hypothesis testing and detection theory.

One can attempt to impose a performance criterion that depends only on these probabilities with the consequent decision rule not depending on the a priori probabilities. The Neyman-Pearson criterion assumes that the false-alarm probability is constrained to be less than or equal to a specified value α α while we attempt to maximize the detection probability P D P D . ∀ P F , P F ≤α:max ℜ 1 { P D } P F P F α ℜ 1 P D A subtlety of the succeeding solution is that the underlying probability distribution functions may not be continuous, with the result that P F P F can never equal the constraining value α α. Furthermore, an (unlikely) possibility is that the optimum value for the false-alarm probability is somewhat less than the criterion value. Assume, therefore, that we rephrase the optimization problem by requiring that the false-alarm probability equal a value α ′ α that is less than or equal to α α.

This optimization problem can be solved using Lagrange multipliers (see Constrained Optimization); we seek to find the decision rule that maximizes F= P D +λ P F - α ′ F P D λ P F α where λ λ is the Lagrange multiplier. This optimization technique amounts to finding the decision rule that maximizes F F, then finding the value of the multiplier that allows the criterion to be satisfied. As is usual in the derivation of optimum decision rules, we maximize these quantities with respect to the decision regions. Expressing P D P D and P F P F in terms of them, we have

To maximize this quantity with respect to ℜ 1 ℜ 1 , we need only to integrate over those regions of r r where the integrand is positive. The region ℜ 1 ℜ 1 thus corresponds to those values of r r where pr| ℳ 1 r>-λpr| ℳ 0 r p r ℳ 1 r λ p r ℳ 0 r and the resulting decision rule is pr| ℳ 1 rpr| ℳ 0 r ≷ ℳ 0 ℳ 1 -λ p r ℳ 1 r p r ℳ 0 r ≷ ℳ 0 ℳ 1 λ The ubiquitous likelihood ratio test again appears; it is indeed the fundamental quantity in hypothesis testing. Using the logarithm of the likelihood ratio or the sufficient statistic, this result can be expressed as either lnΛr ≷ ℳ 0 ℳ 1 ln-λ Λ r ≷ ℳ 0 ℳ 1 λ or ϒr ≷ ℳ 0 ℳ 1 γ ϒ r ≷ ℳ 0 ℳ 1 γ

We have not as yet found a value for the threshold. The false-alarm probability can be expressed in terms of the Neyman-Pearson threshold in two (useful) ways.

One of these implicit equations must be solved for the threshold by setting P F P F equal to α ′ α . The selection of which to use is usually based on pragmatic considerations: the easiest to compute. From the previous discussion of the relationship between the detection and false-alarm probabilities, we find that to maximize P D P D we must allow α ′ α to be as large as possible while remaining less than α α. Thus, we want to find the smallest value of -λ λ (note the minus sign) consistent with the constraint. Computation of the threshold is problem-dependent, but a solution always exists.