Handbook of Psychology, Volume 4: Experimental Psychology

Psychophysical Methods 101

There are two ways to manipulate an observer’s guessing
rate: (a) Manipulate the probability of a catch trial or (b) use
a payoff matrix. The observers’ goal is to guess whether a
signal or a catch trial occurred; according to high-threshold
theory all they know is they were in Dstate or Dstate (see
Table 4.2).
Observers do not know which type of trial (t 1 ort 2 ) caused
the state they are presently experiencing (which we denote
). Assuming that they know the probabilities of the types of
trial and the probabilities of the states they could be in, they
can use Bayes’s rule for the probability of causes (Feller,
1968, p. 124) to determine the conditional probability of the
cause (type of trial) given the evidence (their state), which is
called the posterior probabilityof the cause. For example,

p(t 1 
) , (2)

wherep(t 1
) is the posterior probability of t 1 ,p(
t 1 ) and
p(
t 2 ) are likelihoods (of their state given the type of trial),
andp(t 1 ) and p(t 2 ) are the prior probabilities of the types of
trial. In a different form,

 ,

wherepp((

tt^12 ))is the likelihood ratio, pp((tt 21 ))is the prior odds, and

pp((tt^12 

))is the posterior odds in favor of t 1.

In high-threshold theory, the posterior odds in favor of
signal is

.

Suppose that the observers are in a Dstate, and they believe
that half the trials are catch trials [p(signal)=p(catch)=.5],
and that the threshold happens to be at the median of the dis-
tribution of energies [p(D
signal)=p(D

signal)=.5], then

.

.5



.5

.5



0

p(signal
D)



p(catch
D)

p(signal)



p(catch)

p(D
signal)



p(D
catch)

p(signal
D)



p(catch
D)

p(t 1 )



p(t 2 )

p(
t 1 )



p(
t 2 )

p(t 1 
)



p(t 2 
)

p(
t^1 )p(t^1 )

p(
t 1 )p(t 1 )p(
t 2 )p(t 2 )

Because the posterior odds in favor of signal are infinite, ob-

servers have no reason to guess. But if they are in a D
state

and hold the same beliefs, then the posterior odds are

.5,

orp(signal
D
)=^13 , that is, they should believe that one third

of the D
trials will be signal trials, and they will increase the

number of correct responses by guessing.

We can use two methods to induce observers to change

their guessing rate.

Prior Probabilities. In the example above, the prior

odds were 1. If we change these odds, that is, increase or

decrease the frequency of signal trials, the posterior probabil-

ity of signal in D
state will increase or decrease correspond-

ingly. As a result the observer’s guessing rate will increase or

decrease.

Payoff Matrix. We can also award our observers points

(which may correspond to tangible rewards) for each of the

outcomes of a trial (Table 4.1), as the examples in Table 4.3

show.

We could reward them for correct responses by giving

themB(h) or B(cr) points for hits or correct rejections, and

punish them for errors by subtracting C(fa) or C(m) points for

false alarms or a misses. To simplify Table 4.3 we set C(fa)=

C(m)=0. It is easy to see that when we bias the observer to-

ward “Yes,” the guessing rate will increase, and when we bias

the observer toward “No,” it will decrease.

The ROC curve is a particular case of a general framework

for thinking about perception—the Bayesian approach to

perception. It is summarized in Figure 4.4 (Mamassian,

Landy, & Maloney, in press).

This diagram represents a prescriptive framework: how

one should make decisions. Bayes’s rule is the correct way to

combine background information with present data. Further-

more, there is a considerable body of work on the correct way

to combine the resulting posterior distribution with informa-

tion about costs and benefits of the possible decisions (the

.5

.5

.5

1

p(signal
D
)

p(catch
D
)

TABLE 4.2 The Observer’s Decision Problem in High-Threshold
Theory

Observer State

Stimulus DD

Signal p(D
signal) p(DD

signal)
Catch p(D
catch) p(D

catch)

TABLE 4.3 Payoff Tables for Responses to Signal and Catch

Trials (in points)

General Bias Toward

Case “Yes” “No” No Bias

Response

Stimulus “Yes” “No” “Yes” “No” “Yes” “No” “Yes” “No”

Signal B(h) C(m)201010

Catch C(fa) B(cr) 010201