How can I convert a rate of behavioral responses into a probability?

[DaveJersey]

B.F. Skinner's law of effect said when a response is followed by a reinforcing stimulus, the rate of response increases. So when a response produces a reinforcing stimulus, responses per unit of time (rate) increases. He said the response strengthens. He said when the rate increases, the probability of response increases. So how can I convert, for example, a rate of five responses per minute into a probability number between 0 and 1? Is that possible?

I anticipate your replies.

Dave in New Jersey

 

[Stephen Tashi]

You'll get better advice if you describe a specific experimental situation.

"The probability of a response" and "the rate of a response" are not precise descriptions. In particlar, the "rate" of a physical process is often a deterministic measurement ( for example, a flow in gallons per minute). A probability "of a response" only meaningful if the "response" is a precisely defined event. For example, is the "response" something that does or does-not happen or is there a strength or degree of the response?

 

[DaveJersey]

A rate of response can be precisely described. A rat pressing a lever is defined by the electric switch the lever activates when the rat presses the lever. A person switching on a light can be defined the same way. Response rates are therefore in this scenario the number of light switches or the number of lever-presses per unit of time. Dimensions such as the force or magnitude of a response can also be measured objectively, as in the force by which a pigeon presses a button, or the decibel level of a classroom of "noisy" children. In an experiment, when an underweight (hungry) rat presses a lever the response results in a mechanical delivery of a pellet of grain. The rate of lever presses is shown to increase on a cumulative recorder that draws a line corresponding with total responses. The slope of the line indicates the rate of response. Behavior strength is said to increase with the increased rate of response after reinforcement. Skinner said the probability of response increased, which was shown by the increased rate of response. He did not, however, convert the rate of response into a number between 0 and 1 to indicate the probability. My question is, can the rate of response be converted into a probability value between 0 and 1. Let's say the maximum number of discrete responses possible in one minute is sixty and the rate of response is one per second, can rate be converted to probability?

 

[Stephen Tashi]

Response rates are therefore in this scenario the number of light switches or the number of lever-presses per unit of time.

The question is still too general to have a definite mathematical answer. To have a well posed mathematical problem, you have to add more detail, which usually means making some specific assumptions.

To speak of probability, you need a probability model for the responses. A simple model would be that after one response is given, there is a random draw to determine the time of the next response. So the random variable is "time to next response". A commonly used distribution for "waiting times" is the exponential distribution. This distribution has a parameter whose size determines the shape of the distribution and hence the statistics of the distribution, such as the mean time to the response. The effect of a reward could be modeled as changing the value of the parameter. You wouldn't be representing the effect of a reward as "a probability", you would represent it as a change in a parameter, which affects the probabilities for the all the events in the distribution (e.g. the probability of responding in l1 second or less, the probability of responding in l.5 seconds or less, etc.)

The fact that the above model is simple and often used doesn't prove it applies to your experiment. It's just the first thing that comes to mind.

To model how a reward strengthens a response, you would still need to define quantitatively how the reward changes the parameter, i.e. give some formula such as $\lambda[n+1] = \lambda[n] + c$ where $\lambda[n+1]$ is the value of the parameter after $n+1$ rewards, $\lambda[n]$ is its value after $n$ rewards and $c$ is some constant. If you try various formulas then there are mathematical ways to compare them to experimental data. If you assume formulas with certain unknown constants then there are ways of estimating the constants from experimental data.