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Posts Tagged ‘analysis of behavior’

Here’s a nice argument promoting the use of mathematically modeling in the experimental analysis of behavior.

Described in this way, the predictions of these two theories are difficult to distinguish. Although they make their predictions for different reasons, both theories seem to predict the same general result: punishment will cause a decrease in the punished behavior. Once they are translated into mathematical form, however, the different predictions of the two theories can be seen more easily. Deluty (1976) andde Villiers (19771980) developed two different quantitative models of punishment, which can be viewed as mathematical versions of the avoidance theory of punishment and the negative law of effect, respectively. Both models begin with Herrnstein’s (1961) matching law, but then proceed in different directions.

In its simplest form, the matching law can be written as follows:

A mathematical equation, expression, or formula that is to be displayed as a block (callout) within the narrative flow. The name of referred object is jeab-85-02-02-e01.jpg
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where B1 and B2 are the rates of response on reinforcement schedules 1 and 2, and R1 and R2 are the rates of reinforcement on these two schedules. This equation has often been applied to choice situations in which the two alternatives are variable-interval (VI) schedules of food reinforcement. Imagine that a pigeon responds on two keys, with Key 1 delivering 75 reinforcers per hour and Key 2 delivering 25 reinforcers per hour, so Equation 1 predicts that the pigeon will make 75% of its responses on Key 1. Now suppose that in addition to producing food, responses on both keys begin to deliver punishers (electric shocks) at a rate of 20 shocks per hour for each key. How can Equation 1 be expanded to deal with this situation? 

According to de Villiers (1977), if punishment is the opposite of reinforcement, as the negative law of effect states, then the punishers delivered by each alternative should be subtracted from the reinforcers delivered by that alternative:

A mathematical equation, expression, or formula that is to be displayed as a block (callout) within the narrative flow. The name of referred object is jeab-85-02-02-e02.jpg
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where P1 and P2 are the rates of punishment on the two keys. 

In contrast, Deluty (1976) took the view that punishing one response increases the reinforcement for other responses, as proposed by the avoidance theory of punishment. Therefore, in his equation, the punishers for one alternative are added to the reinforcers for the other alternative:

A mathematical equation, expression, or formula that is to be displayed as a block (callout) within the narrative flow. The name of referred object is jeab-85-02-02-e03.jpg
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To keep this example simple, one shock is given the same weight as one food delivery, but both models could easily give food and shock different weights by multiplying P1 and P2 by some constant other than 1. Using such a constant would not change the general conclusions presented here. In this example, with R1  =  75, R2  =  25, and P1  =  P2  =  20, Equation 2 predicts that the percentage of responses on Key 1 should increase from 75% to 92% when the shocks are added to both keys. Conversely, Equation 3 predicts that the percentage of responses on Key 1 should decrease to 68% when the shocks are added. In an experiment with pigeons, de Villiers (1980) found that preference for the key that delivered more reinforcers increased when shocks were added to both keys with equal frequency. This result therefore favors the predictions of Equation 2 over those of Equation 3

It should be clear that the issue here is more fundamental than simply whether a plus sign or a minus sign should be used in an equation. These two models are based on two very different conceptions of how punishment exerts its effects on behavior. The experimental evidence suggests that punishment exerts its effect by weakening the target behavior, as the negative law of effect stipulates, not by strengthening alternative behaviors, as the avoidance theory proposes. This example illustrates how two psychological theories that seem to make similar predictions when stated verbally actually may make very different predictions when they are presented in mathematical form.

Mathematical Models and the Experimental Analysis of Behavior
James E Mazur

Southern Connecticut State University
Correspondence should be addressed to James E. Mazur, Psychology Department, Southern Connecticut State University, New Haven, Connecticut 06515, e-mail: mazurj1@southernct.edu
Received July 21, 2005; Accepted October 3, 2005.
In the conclusion we get this nice warning.
In a commentary about some competing mathematical models of timing, Killeen (1999) wrote: “If you think models are about the truth, or that there is a best timing model, then you are in trouble. There is no best model, any more than there is a best car model or a best swimsuit model, even though each of us may have our favorites. It all depends on what you want to do with the model” (p. 275). Those who do not enjoy studying mathematical models might take this statement (from a preeminent mathematical modeler) as an excuse to avoid them. Why bother putting in the time and effort to understand current mathematical models of behavior when there is no best model, and when they all have their weaknesses and limitations? Killeen addresses this issue by asserting that “all understanding involves models—reference to systems that exist in a different domain than the thing studied. Loose models make vague reference to ambiguous and ad hoc causes. Tighter models are more careful about definitions and avoid gratuitous entities. Models of phenomena are not causes of phenomena; they are descriptions of hypothetical structures or functions that aid explanation, prediction, and control” (p. 276).
That’s true of mathematical modeling in behavior and pretty much anything else.  The model is not the thing.  However, models help structure thinking, investigation and make it easier to communicate.  
I also want to point out just how useful the Matching Law is in analysis of behavior.  You can use it as a basis for so many investigations and you see it play out in almost every situation you are observing behavior from web metrics to neural studies.

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Have you ever played Civilization, the computer game?

Determined and yet unpredicable

Determined and yet unpredicable

If you haven’t, it’s really fun.  However, that’s not the point of this post nor the title.

What makes Civ or Sim City or any of the other sim games so interesting is that they are a great example of what it means to be completely determined and yet totally unpredictable.

So what, you say?  It’s a computer game.

Ah, I contest this game is far more like the real world than one might suspect.  Yes, I can “freely” control what I do with my civilization – move my citizens around, build different things, arm, disarm, trade, war, research, pray, etc. etc.  However, I’m actually quite forced into almost every move once all the contingencies come home to roost.  Did the computer just attack me?  Do I have the money? Do I have the people? How much power do I have? All of those and more filter the space of all possible moves.  Perhaps there’s a few “free” choices left on each move that become probabilities but even those are held under control based on contingencies (like my preferences for how to move).

So why then are the games (life?) unpredictable?  For one, initial conditions.  The set up of the geography, placement on the map, type of people and so forth change every time you start the game, so the game (under the same predetermined moves) takes a completely different form.  I’m always careful in discussing initial conditions because you can’t really ever make a claim as to what a real “initial” condition is in the real world (or even the game), as something must determine the initial condition you are looking at (even if it is a “random number generator).

Amazingly, it’s not just interesting initial conditions.  Simple rules (even the most basic rules you can imagine) can operate in fully unpredictable ways.  examples: http://mathworld.wolfram.com/Rule110.html and http://www.johnkyrk.com/DNAtranscription.html and http://en.wikipedia.org/wiki/Prime_number.  These three examples all of very simple rules of operation (how you generate them, how they evolve, etc. etc), yet their outcome/their behavior/their product is not predictable.

I digress (and don’t you hate when people say “I digress”?).

Perhaps my silly use of computer games does nothing for my argument about there being things that completely determined and unpredictable.  I could, of course, turn your attention to cellular automata or turing machines or some other simple system -some of these types of systems are unpredictable and they all are completely determined.  Unfortunately, many folks will complain that these aren’t “real world.”  The real world is full of indeterminism and lots of messy things that aren’t like computer games or automata.

It’s true that the world is full of systems that aren’t computer games or automata!  However, it’s not obviously true that these systems are indeterminate! How can we figure that out?

One way, if I can show you determined systems that exhibit the same behavior as the supposed indeterminate systems, that’s some evidence for determinate systems.

If I can drill down on your indetermined system enough I will like find your “rules” that drive it.  That’s my conjecture.  Send me a system, and we can discuss.  (markets, DNA, whatever you want).

Human behavior exists in this world of determined and the unpredictable.  Reinforcers/consequences, conditioned stimuli, unconditioned stimuli, discriminant stimulus — (biology + genetics + nervous system + environment) — control the behavior.  Layers of rules applied to a complex biological system.  It’s all weirdly determined but completely unpredictable.

Falsify my claim.

Suppose this is true.  So what?  The implications are relative to you.  If indeterminism services your world view or scientific approach, perhaps the implications are grave.

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