When a receiver detects a signal stimulus, the first task is to assign it to one of several possible categories. Once this assignment has been made, a receiver can use the prior probabilities and the relevant signal coding matrix to update the probabilities of alternative conditions being true, compare expected values of alternative actions, and make a decision on how to respond to this signal.
If the patterns in the signal are completely non-overlapping with those of alternative signals, we say that the signals are discrete. We assumed that signals were in fact discrete in outlining the red line decision process in Web Topic 8.1. But what if signals are not discrete, either because senders emit signals with at least partially overlapping patterns, or because initially discrete signals become distorted and more overlapping during propagation between the sender and the receiver? Can a receiver faced with overlapping signals still define an optimal red line to use in decision making?
The answer is yes, and the method that describes this process is called signal detection theory. To see how a red line can be defined with overlapping signals, consider a female bird trying to assess the health of a potential mate by listening to his courtship song. Suppose that in this species, sick and parasitized males tend to sing slower songs, and healthy males tend to sing faster songs. However, the signals are not discrete and there is considerable overlap in song speeds between males in different states of health:
This plot shows the conditional probability that a male will sing a song at a given speed w depending on his health. There are two distributions shown: one for sick males, P(w|sick), and one for healthy males, P(w|well). We assume here that the two distributions are normal (bell-shaped), but that is for computational convenience and the general conclusions below do not depend on that assumption.
Suppose the female draws a red line on this plot: whenever she hears a song at a speed lower than the red line value, she will reject the male; whenever she hears a song at a speed higher than the red line value, she will accept that male as a mate. Where is the optimal place to draw this line?
Looking at the same graph with a red line on it, we can see that the red line divides each of the two song speed distributions into two parts. The area under the sick male distribution bounded on the right by the red line and on the bottom by the X axis is the total probability that the female will correctly reject a male when he is in fact sick. We shall denote this probability by P(Correct Rejection). Similarly, the area under the well male distribution that is bounded on the left by the red line and on the bottom by the X axis is the total probability that a female will correctly accept a male when he is healthy. We shall denote this by P(Hit).
With the line in this location, the female cannot avoid making two kinds of errors. The area in the dark blue region to the right of the red line defines the overall probability that the female will erroneously accept a sick male as a mate. This type of error is denoted as P(False Alarm). The area of the dark red region to the left of the red line defines the overall probability that the female will erroneously reject a well male and is denoted by P(Miss).
It should be obvious by looking at this graph that moving the red line to the right will reduce P(false alarm) but it will increase P(Miss). Similarly, moving the red line to the left will reduce P(Miss) but increase P(False Alarm). Since the female cannot reduce the total probability of making some errors, the only way to find an optimal location for the red line is by minimizing the costs of the errors. For example, if false alarms are more costly than misses, then the optimal location for the red line will be at faster song speeds; if misses are more costly than false alarms, then she should set the red line at a lower song speed. To find the optimal location, we therefore need to consider the relative payoffs of each outcome, and her estimated probabilities that a given male is sick or healthy after hearing him sing.
Suppose that the payoffs to a female of accepting or rejecting well versus sick males can be summarized in the following payoff matrix:
Condition | ||
---|---|---|
Action | Well Male | Sick Male |
Accept as mate | R11 | R12 |
Reject as mate | R21 | R22 |
Suppose that on average, a fraction P of the males in the population are well and (1–P) are sick. When the female hears a male sing at song speed w, she will update her estimate that he is healthy from P to P(well|w) and that he is sick from (1–P) to P(sick|w). She can now combine these updated probabilities with the relevant payoffs to compute expected values (average payoffs) for each action. The expected value for accepting this male as a mate will be
the expected value for rejecting this male will be
The optimal redline will occur at that w for which the expected value of accepting a male is equal to that for rejecting him; at higher song speeds, the female should accept males, and at slower song speeds, she should reject males. Setting the two expected values equal to each other and rearranging, we get that the critical song speed, wc, is that for which
We can simplify this further by assuming that the female used Bayesian methods (Web Topic 8.3) to update the probabilities that the male was well after hearing him sing. Specifically, she could have updated using the following formula:Plugging the right hand side of the Bayesian equation into the left side of the previous equation and rearranging, we get that the critical song speed, wc, is the one for which
The left side of this equation is simply the ratio of the Y axis values for the well versus sick distributions at wc. It is called the likelihood ratio and is usually denoted by b. If we increase wc, the likelihood ratio becomes larger since we move more into the well distribution and out of the sick distribution:
The right side of this equation includes the ratio of the prior probabilities (odds ratio) and the ratio of the differences in payoffs between right and wrong choices in the two conditions (payoff ratio). The entire right side is called the operating level in signal detection theory. All of these numbers are fixed before the male begins to sing or the female begins to make a decision. One can also think of the operating level as the ratio of the costs of the two types of errors noted earlier, each discounted by the prior probability that it will occur. As sick males become more common ((1–P) increases), or the cost of false alarms increases (R12 versus R22), the numerator of the operating ratio increases relative to the denominator, and the appropriate value of wc on the left side of the equation has to increase. If healthy males become more common (P increases), and/or the relative cost of misses increases (R21 versus R11), the right side of the equation decreases, and the optimal location for the red line, wc, moves to lower song speeds.
The strategies for drawing red lines on meters when signals are discrete (Web Topic 8.1), and on pattern axes when signals overlap (this Web Topic), use the same ingredients. Both approaches depend on the differences in the payoffs of right versus wrong decisions, and not on the absolute values of individual payoffs. Both approaches require access to the signal coding scheme and the prior probabilities. Both invoke Bayesian updating upon receipt of a signal to define the optimal probability estimates before computing expected values of alternative actions. And both permit shortcuts to decision making if prior probabilities and payoff differences remain sufficiently stable for reasonable periods. In the case of overlapping signals, a receiver need only compare perceived signal properties to threshold values to make a quick decision. They do not even have to compute a Bayesian update since the ingredients for that update are incorporated into the determination of the optimal red line.
Macmillan, N. A. and C.D. Creelman. 2004. Detection Theory: A User’s Guide. 2^{nd} Edition. New York: Cambridge University Press.
Wiley, R.H. 1994. Errors, exaggeration, and deception in animal communication. In, Behavioral Mechanisms in Evolutionary Biology, (L.A. Real, ed.). Chicago: University of Chicago Press. pp. 157–189.