COU 7: Analyzing a Parameterized Model of Learning
In this COU, you’ll work with a version of the model of learning from COU 5 that is structurally somewhat simpler, but uses a parameter to explore how changes in the model affect its behavior.
Consider again a citizen who is considering whether to join an anti-government terrorist or guerilla group. Suppose this citizen is unsure about the group’s competence. She knows, moreover, that the group’s competence partially determines how successful it will be when it carries out an attack. More specifically, suppose the citizen’s prior beliefs about the group’s competence and whether an attack it stages will be successful are given by:
| Group’s Level of Competence | Probability |
|---|---|
| low | \frac{2}{3} |
| high | \frac{1}{3} |
| Attack will be a… | Probability |
|---|---|
| failure | \frac{1}{2} |
| success | \frac{1}{2} |
| Attack will be a… | Probability |
|---|---|
| failure | 1-\delta |
| success | \delta |
where the number \delta lies between \frac{1}{2} and 1.
Notice that this model is in some ways much simpler than the ones you worked with in COUs 3, 4 and 5. In this model, the group has only two levels of possible competence (“low” and “high”), and its attack will have one of only two levels of possible success (“failure” or “success”). On the other hand, this model has a parameter, given by the number \delta. In effect, \delta determines how much more likely an attack is to be successful if the group’s competence is high. Specifically, if the group has low competence, its attack will be successful with probability \frac{1}{2}. If the group has high competence, its attack will be successful with probability \delta. Since \delta is a number between \frac{1}{2} and 1, the attack is more likely to be successful if the group’s competence is high than if the group’s competence is low. But the extent to which a competent group is more likely than an incompetent group to stage an attack that is successful varies with the value of \delta. When \delta is close to the lower bound of its range (i.e. close to \frac{1}{2}), a high-competence group is only slightly more likely than a low-competence group to stage a successful attack. When \delta is close to the upper bound of its range (i.e. close to 1), a high-competence group is much more likely than a low-competence group to stage a successful attack.
Suppose that the group stages an attack and the citizen observes whether the attack is a “success” or a “failure”. Thus the citizen can update her beliefs, learning about the group’s competence. It looks like this:
Analyze this model by responding to the following prompts.
Prompt 1
Draw a conditional probability diagram displaying the citizen’s prior beliefs about the group’s competence and whether the attack will be successful.
Prompt 2
Suppose an attack occurs and the citizen observes that the attack is a success. Write the expressions in the blank spaces in a table like the following describing the citizen’s posterior beliefs about the group’s competence…
| Group’s Level of Competence | Probability |
|---|---|
| low | |
| high |
…and redraw the diagram you drew in response to Prompt 1, marking it up to show how you calculated the values in the table.
Prompt 3
Suppose an attack occurs and the citizen observes that the attack is a failure. Write the expressions in the blank spaces in a table like the following describing the citizen’s posterior beliefs about the group’s competence…
| Group’s Level of Competence | Probability |
|---|---|
| low | |
| high |
…and redraw the diagram you drew in response to Prompt 1, marking it up to show how you calculated the values in the table.
Prompt 4
Draw a graph that shows the value of the parameter \delta on the horizontal axis and the values of two expressions on the vertical axis at each value of \delta:
- P(\text{high competence} | \text{attack is a success}) – i.e. the probability that the group’s competence is high in the posterior belief the citizen will have if the attack is a success.
- P(\text{high competence} | \text{attack is a failure}) – i.e. the probability that the group’s competence is high in the posterior belief the citizen will have if the attack is a failure.
As described in the video in the lesson about analyzing parameterized models of learning, there are three things your graph must correctly depict about each of these expressions: (1) the value of the expression when \delta is at its minimum value of \frac{1}{2}, (2) the value of the expression when \delta is at its maximum value of 1, (3) whether the expression increases, decreases, increases then decreases or decreases then increases as \delta increases from its minimum value to its maximum value.
Prompt 5
In about half a page of well-written, double-spaced text, describe how a change in the value of \delta changes the effect of an attack on the learning process depicted in this model.
Rubric
For Prompt 1
You get four points if the diagram you draw includes all the required information and correctly uses all the relevant graphical conventions.
You get two points if you draw a square with the axes labeled properly and with the space within the square divided up into the appropriate number of boxes, but the relative sizes of any of the boxes are obviously wrong and/or any of the numerical labels for the probabilities are missing or wrong.
You get zero points otherwise.
For Prompt 2
For each of the two blank spaces in the table…
You get two points if you fill in the correct expression and you provide a diagram that correctly shows how you computed that expression.
You get one point if you provide a diagram that correctly shows how you computed the expression, but the expression you write is missing or incorrect.
You get zero points otherwise.
For Prompt 3
For each of the two blank spaces in the table…
You get two points if you fill in the correct expression and you provide a diagram that correctly shows how you computed that expression.
You get one point if you provide a diagram that correctly shows how you computed the expression, but the expression you write is missing or incorrect.
You get zero points otherwise.
For Prompt 4
You get four points if you draw a graph with the axes properly labeled and that shows the two posterior probabilities, each one clearly labeled, and with the level of each posterior probability correct at the minimum and maximum values of \delta, and with the direction of change in each one as \delta changes correct.
You get three points if you meet all the criteria for four points, except the direction of change in one or the both posterior probabilities as \delta changes is incorrect.
You get two points if you meet all the criteria for three points except the level of exactly one of the two posterior probabilities is incorrect at either the minimum or maximum value of \delta is incorrect, but not both.
You get zero points otherwise.
For Prompt 5
You get four points on Prompt 5 only if you got four points on Prompt 4 and if what you write is accurate, coherent and well-written.
You get three points on Prompt 5 only if you got four points on Prompt 4 and if what you write is accurate and coherent and but has errors in grammar, spelling or usage that make it anything less than perfectly easy to understand.
You get zero points otherwise.