Lecture Notes - 7/8/99
Lera Boroditsky (with parts adapted from David Heeger)

0. What is the ultimate goal of psychology?

• Psychologists aim for a complete understanding of human behavior. What does this mean?

• We can devide our perceptual experience into three components: the stimulus, the internal response caused by the stimulus, and the perception or behavior that results from this internal response.

  

• If we had a complete understanding of a how a system works, we would be able to predict all of these components given only one of them. For example, we would be able to back-derive the stimulus & predict the behavior given only the internal response.

• Here's an fMRI image of Alex's brain. What is he thinking?

• Although we can't quite do the kind of complete mind-reading or mind-control that you see in spy movies, we do have a good understanding of many perceptual systems. In this lecture we will talk about the kinds of quantitative tools psychologists have used to gain a complete understanding of some perceptual systems.

• Review

  

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  • Light of different wavelengths is absorbed in different proportions by our cones. Roughly, S-cones absorb blue light, M-cones absorb green light, and L-cones absorb red light.

  • The ratio of excitation of the three cone types predicts our perception of color.

  • There is an amazing correspondence between cone absorptions and color matching.
    • suppose we shine a test light on a screen

    • now we sit you down with three primary lights (whose wavelengths match the peak absorption wavelengths of your cones) and ask you to adjust the brightness knobs on the three lights until you have matched the test light

    • here's the amazing part: the ratio of brightness settings of the three knobs will match exactly the ratio of cone absorptions! there is a simple, predictable relationship between what our cones absorb and what color we see.

  • So, by looking at your knob settings, we can predict not only the wavelength of light that you're looking at, but also the internal response (the cone absorptions)!

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• This very simple, nice type of system is called a Linear System
  • Suppose you want to figure out the relationship between how much you pet your cat and how much it purrs. Step one is to understand how to represent possible inputs to systems. Imagine a picture that shows the structure of the physical stimulus (your petting). On the horizontal axis we have time, and on the vertical axis we will plot the pressure of your hand on your cat. Thus, we plot signal strength as a function of time. In the case of a simple touch, the stimulus is a short, transient burst and is aptly named an impulse. It looks like a single upwards blip on the graph: the pressure of your hand momentarily increases when you touch the cat. More complex stimuli look like more complex graphs on this kind of plot. This sort of graph offers a general way to describe all of the possible stimuli.

  • One possible way to characterize the cat's purring response to petting might be to build a look-up table: a table that shows the exact purring response for every possible tactile stimulus. Obviously, it would take an infinite amount of time to construct such a table, because there is an infinite number of ways to pet a cat.

  • However, if you're lucky, you might discover that your cat acts like a linear system in regard to petting. If so, you could completely characterize your cat's response with just a few measurements.

  • How do you know if your cat is a linear system? Linear systems follow 3 important rules:
    • Homogeneity: As we increase the strength of a simple input to a linear system, say we double it, then we predict that the output function will also be doubled. For example, if you pet your cat twice as hard, the cat should purr twice as much if it's a linear system. This is called homogeneity or sometimes the scalar rule of linear systems.

      

    • Additivity:Suppose you pet your cat once (stimulus S1), and we measure the cat's purring. Next, we present a second stimulus S2 that is a little different: a different kind of touch. The second stimulus also generates a set of responses which we measure and write down. Then, we present the sum of the two stimuli S1 + S2: we present both touches together and see what happens. If the system is linear, then the measured response will be just the sum of its responses to each of the two stimuli presented separately.

      

    • Shift-invariance: Suppose you touch your cat once (an impulse stimulus) and we measure the purring response. Then you touch your cat again the same way but at a different point in time, and again we measure the response. If you haven't damaged your cat with the first impulse, then we should expect that the response to the second impulse will be the same as the response to the first impulse. The only difference between them will be that the second impulse has occurred later in time, that is, it is shifted in time. When the responses to the identical stimulus presented shifted in time are the same, except for the corresponding shift in time, then we have a special kind of linear system called a shift-invariant linear system. Just as not all systems are linear, not all linear systems are shift-invariant.

  • Superposition: Systems that satisfy both homogeneity and additivity are considered to be linear systems. These two rules, taken together, are often referred to as the principle of superposition.

  • Because linear systems follow these three rules, they are really easy to study.

  • Characterizing your cat as a linear system. A system that follows homogeneity, additivity, and shift-invariance can be characterized completely with a single measurement!
    • The trick is to conceive of a complex stimulus (such as your favorite way of petting your cat) as a combination of impulses (a combination of touches). We can approximate any complex stimulus as if it were simply the sum of a number of impulses that are scaled copies of one another and shifted in time. (A digital compact disc, for example, stores whole complex pieces of music as lots of simple numbers representing very short impulses, and then the CD player adds all the impulses back together one after another to recreate the complex musical waveform.)

      

    • For shift-invariant linear systems, we can measure the system's response to an impulse and we will know how to predict the response to any stimulus (combinations of impulses) through the principle of superposition. To characterize shift-invariant linear systems, then, we need to measure only one thing: the way the system responds to an impulse of a particular intensity. This response is called the impulse response function of the system.

    • The problem of characterizing your cat (a complex system) has become simpler now. For shift-invariant linear systems, there is only a single impulse response function to measure. Once we've measured this function, we can predict how the system will respond to any other possible stimulus.

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• The reason we know so much about color encoding, is because it's so amazingly linear.
  • Lights are additive: Measure SPD separately for each of two light sources. Then turn both lights on together. SPD of the mixture (sum of 2 lights) equals the sum of the two SPDs. That is, lights obey the additivity rule. Lights also obey the homogeneity rule. Doubling the intensity of a light doubles the SPD at each wavelength.

  • Cone responses are additive: Measure the response to light 1 & light 2 separately. Shine them together, and you get the sum of the two responses.

  • Color-matching knob settings are additive: Settings to (test light 1 + test light 2) = (settings to test light 1) + (settings to test light 2).

  • amazing for such a complex system!

• Because we want to study not just neural responses, but also the perceptual experiences associated with these responses, we do psychophysics. What we need is a way to quantify perceptual phenomena.

• There are three basic experimental paradigms that we use in perceptual psychology experiments: magnitude estimation, matching, and detection.

• Let's take a simple example. What is the smallest amount of light that you can detect? How would we find out?

• Signal Detection theory
  • We are not perfect. Nearly all reasoning and decision making takes place in the presence of some uncertainty.

  • Let's begin with a medical scenario. Imagine that a radiologist is examining a CT scan, looking for evidence of a tumor. Interpreting CT images is hard and it takes a lot of training. Because the task is so hard, there is always some uncertainty as to what is there or not. Either there is a tumor (signal present) or there is not (signal absent). Either the doctor sees a tumor (they respond "yes'') or does not (they respond "no''). There are four possible outcomes: hit (tumor present and doctor says "yes''), miss (tumor present and doctor says "no''), false alarm (tumor absent and doctor says "yes"), and correct rejection (tumor absent and doctor says "no"). Hits and correct rejections are good. False alarms and misses are bad.

    

  • There are two main components to the decision-making process: information aquisition and criterion.
    • Information acquisition: First, there is information in the CT scan. For example, healthy lungs have a characteristic shape. The presence of a tumor might distort that shape. Tumors may have different image characteristics: brighter or darker, different texture, etc. With proper training a doctor learns what kinds of things to look for, so with more practice/training they will be able to acquire more (and more reliable) information. Running another test (e.g., MRI) can also be used to acquire more information. Regardless, acquiring more information is good. The effect of information is to increase the likelihood of getting either a hit or a correct rejection, while reducing the likelihood of an outcome in the two error boxes.

    • Criterion: The second component of the decision process is quite different. For, in addition to relying on technology/testing to provide information, the medical profession allows doctors to use their own judgement. Different doctors may feel that the different types of errors are not equal. For example, one doctor may feel that missing an opportunity for early diagnosis may mean the difference between life and death. A false alarm, on the other hand, may result only in a routine biopsy operation. They may chose to err toward ``yes'' (tumor present) decisions. Other doctors, however, may feel that unnecessary surgeries (even routine ones) are very bad (expensive, stress, etc.). They may chose to be more conservative and say ``no'' (no turmor) more often. They will miss more tumors, but they will be doing their part to reduce unnecessary surgeries. And they may feel that a tumor, if there really is one, will be picked up at the next check-up. These differences are not about information. Two doctors, with equally good training, looking at the same CT scan, will have the same information. But they may have a different bias/criteria.

  • Internal response and internal noise.
    • Remember that even in the absence of stimuli, our neurons are not silent. Instead they fire randomly once in a while at some baseline rate. That is, the neurons are noisy. Because their baseline firing is pretty random, at any given point in time, they might be firing a little more or a little less. There is a normal (bell-shaped) distribution of noise for any set of neurons.

      

    • When we turn on a stimulus, the neurons will on average fire more. The internal response to the stimulus can be represented as the normal noise plus the boost from the signal.

      

    • Notice that we never lose the noise. The internal response for the signal-plus-noise case is generally greater but there is still a distribution (a spread) of possible responses.

    • Notice also that the curves overlap, that is, the internal response for a noise-alone trial may exceed the internal response for a signal-plus-noise trial.

  • The role of the criterion: The simplest strategy that the doctor can adopt is to pick a criterion location along the internal response axis. Whenever the internal response is greater than this criterion they respond "yes''. Whenever the internal response is less than this criterion they respond "no''.
    • An example criterion is indicated by the vertical lines in the Figure below. The criterion line divides the graph into four sections that correspond to: hits, misses, false alarms, and correct rejections. On both hits and false alarms, the internal response is greater than the criterion, because the doctor is responding "yes''. Hits correspond to signal-plus-noise trials when the internal response is greater than criterion, as indicated in the figure. False alarms correspond to noise-alone trials when the internal response is greater than criterion, as indicated in the figure.

      

    • Suppose that the doctor chooses a low criterion (top part of Figure below), so that they respond "yes'' to almost everything. Then they will never miss a tumor when it is present and they will therefore have a very high hit rate. On the other hand, saying "yes'' to almost everything will greatly increase the number of false alarms (potentially leading to unnecessary surgeries). Thus, there is a clear cost to increasing the number of hits, and that cost is paid in terms of false alarms. If the doctor chooses a high criterion (bottom part of Figure below) then they respond "no'' to almost everything. They will rarely make a false alarm, but they will also miss many real tumors.

      

    • Notice that there is no way that the doctor can set their criterion to achieve only hits and no false alarms. The message that you should be taking home from this is that it is inevitable that some mistakes will be made. Because of the noise it is simply a true, undeniable, fact that the internal responses on noise-alone trials may exceed the internal responses on signal-plus-noise trials, in some instances. Thus the doctor cannot always be right. They can adjust the kind of errors that they make by manipulating their criterion, the one part of this diagram that is under their control.

  • The role of information: Aquiring more information makes the decision easier.
    • Running another test (e.g., MRI) can be used to acquire more information about the presence or absence of a tumor. Unfortunately, the radiologist does not have much control over how much information is available.

    • In a controlled perception experiment the experimenter has complete control over how much information is provided. Having this control allows for quite a different sort of outcome. If the experimenter chooses to present a stronger stimulus, then the subject's internal response strength will, on the average, be stronger.

    • Pictorially, this will have the effect of shifting the probability of occurrence curve for signal-plus-noise trials to the right, a bit further away from the noise-aloneprobability of occurrence curve.

      

  • Varying the noise: For stronger signals, the probability of occurrence curve for signal-plus-noise shifts right and detection is easier. There is another aspect of the probability of occurrence curves that also determines detectability: the spread of the curves. For example, consider the two probability of occurrence curves in the Figure below. The separation between the peaks is the same but the second set of curves are much skinnier. Clearly, the signal is much more discriminable when there is less spread (less noise) in the probability of occurrence curves. So the subject would have an easier time setting their criterion in order to be right nearly all the time.

    

  • What the heck is d' anyway? as we've seen, discriminability depends on both signal strength (separation between the two curves) and the amount of noise (the spread of the two curves). d' = separation / spread

  • Medical Malpractice Example: A study of doctors' performance was performed in Boston. 10,000 cases were analyzed by a special commission. The commission decided which were handled negligently and which well. They found that 100 were handled very badly and there is good cause for a malpractice suit. Of these 100, only 20 cases were pursued. What should we conclude? ..... Ralph Nader (and Naider*s Raiders) concluded that doctors are not being sued enough. But this conclusion was based on only partial information (hits and misses). I did not tell you what happened in the other 9900 cases. How many law suits were there in those cases? What if there were many (e.g., 9000 out of 9900) false alarms? The AMA concluded that doctors are being sued too much.

• Comparing Methods
  • Method of Adjustment: Simply ask the observer to adjust the intensity of the light until they judge it to be just barely detectable. Like what happens when you get fitted for a new eye glasses prescription. Typically the doctor drops in different lenses and asks you if this lens is better than that one. BUT, there's something very unsatisfying about the method of adjustment. Introspectionist/subjective. Asking the observer to tell us what they think their own threshold is. When subjects are very inexperienced - and indeed, even when they are quite experienced - this can be a dangerous strategy. It is not so much that the subject is lying or being dishonest. Rather, it is simply difficult to judge when a light is on the threshold of visibility. Moreover, it's kind of stressful for the subject - does this lens work better or does that one work better?

  • Yes/No method of constant stimuli: Observer is presented a series of lights of various intensities. Each intensity is presented several times (randomly intermixed). On each trial the observer is asked to report whether or not they saw it. We calculate the probability of detection (or percent of "yes" responses) for each light intensity.
    • There's something seriously wrong here: How do we know that what we're measuring is someone's perceptual threshold, and not just their propensity to say "yes"? This is a problem. We have to be able to separate an individual's senisitivity to a stimulus from how liberal or conservative they are in making a yes/no decision. How can we do this? We need to measure not only "signal" trials (trials where there is a light present), but also noise trials (trials where there is no light).

  • Forced Choice: The method of choice. Present signal on some trials, no signal on other trials. Subject is forced to respond on every trial either ``Yes the light was presented'' or ``No it wasn't''. If they're not sure then they must guess. Now since we have both types of trials (blank noise only trials some of the time and signal+noise trials at other times), we can count both the number of hits and the number of false alarms to get an estimate of d'.

  • Which method is best? Different experimental procedures lead to different estimates of threshold. For example, threshold settings from naive observers using the method of adjustment are usually about ten times higher than thresholds estimated by the forced choice procedure. With some practice this difference gets smaller. But even for experienced observers the adjustment thresholds and forced choice thresholds are different by about a factor of 3.

• Weber's law is a general law of threshold behavior that spans quite a large number of sensory phenomena.

• The basic point is that your ability to detect the presence of a light depends on how bright the background is.

• Weber used signal detection methods to figure out the difference threshold for lights given different backgrounds. A difference threshold is how different a light has to be from the background to be detected.

• let's call the base intensity (the background intensity) x and the difference threshold dx

  

• If we have a base intensity x=10, dx needs to be about 1. If we begin with a base of x=100, then dx needs to be about 10 (considerably larger than when we had a base of only 10).

• Generality of Weber's Law: These kinds of experiments have been performed in many different sensory modalities to measure our abilities to discriminate: intensities of 2 lights, intensities of two sounds, pressure on the skin, weight of two objects, intensity of electric shocks, and a whole host of other things. The result is always the same: The difference threhsold is proportional to the baseline/starting intensity. The generality of this observation proved so surprising that it has been called a psychological Law, and it is named after its discoverer, Weber.

• Why does this happen? Because our brains are not good at figuring out absolute intensities, and instead compute the intensity at a point relative to the local average intensity (a ratio of point intensity to surround intensity).