COU 4: Analyzing a Parameterized Model of Learning

In this COU, you’ll work with a parameterized model of learning inspired by Bueno de Mequita’s analysis of terrorism described in COU 3.

Consider a potential supporter of a militant organization. Suppose this person is unsure about the organization’s competence. She knows, moreover, that the organization’s competence partially determines how successful it will be when it carries out an attack. More specifically, suppose the potential supporter’s prior beliefs about the organization’s competence and whether an attack it stages will be successful are given as follows:

• The group’s competence is either “low” or “high”, and the marginal probability that the group’s competence is low is \frac{2}{3}.
• The next attack staged by the group will either be a “success” or “failure”.
• Conditional on the event that the group’s competence is low, the probability that the next attack will be a success is \frac{1}{2}.
• Conditional on the event that the organization’s competence is high, the probability that the next attack will be a success is \delta, where \delta is a number between \frac{1}{2} and 1.

Here is a marginal-conditional diagram describing the potential supporter’s prior beliefs:

Potential supporter’s prior beliefs

Make sure you note the assumption that \delta lies between \frac{1}{2} and 1. This means that, regardless of the value \delta takes on within its range, the probability that the attack is successful is higher if the group has high competence than if it has low competence.

Prompt 1

Suppose an attack occurs and the potential supporter observes that the attack is a success. Write the expression for the probability that the group’s competence is high according to the potential supporter’s posterior beliefs and mark up the diagram above to show how you derived that expression. You do not need to simplify the expression you write, although you are welcome to. All that matters is whether the expression is correct.

Prompt 2

Suppose an attack occurs and the potential supporter observes that the attack is a failure. Write the expression for the probability that the group’s competence is high according to the potential supporter’s posterior beliefs and mark up the diagram above to show how you derived that expression. You do not need to simplify the expression you write, although you are welcome to. All that matters is whether the expression is correct.

Prompt 3

Draw a graph that shows the value of the parameter \delta on the horizontal axis, ranging from it’s minimum value of \frac{1}{2} on the left to its maximum value of 1 on the right. On the graph, draw approximations of the paths of three expressions as functions of \delta:

1. The probability that the group’s competence is high according to the potential supporter’s prior beliefs.
2. The probability that the group’s competence is high according to the potential supporter’s posterior beliefs when the attack is a success.
3. The probability that the group’s competence is high according to the potential supporter’s posterior beliefs when the attack is a failure.

Make sure to add labels or a legend that makes clear which curve is which.

The shape of each graph does not have to be exactly correct. However, each graph must be drawn in a way that meets the following criteria:

1. It must intersect the y axis at the correct level at \delta = \frac{1}{2}. (Make sure to include labeled tic marks on the y axis so that the grader can tell the levels you intend!)
2. It must be at the correct level along the y axis at \delta = 1.
3. It must slope in the correct direction (either increasing, decreasing or constant) between \delta = \frac{1}{2} and \delta = 1.

Prompt 4

In a few sentences of well-written, double-spaced text, describe how a change in the value of \delta changes how much the potential supporter learns from observing whether the attack is a success or failure.

Rubric

Prompts 1 and 2

For each of these prompts, you get:

• Three points if you write the correct expression and give a marked-up diagram that correctly shows how to derive that expression.
• One point if you write the correct expression but do not give a marked-up diagram that correctly shows how to derive that expression.
• Zero points otherwise.

Prompt 3

• You get four points if you draw a graph with the axes properly labeled and that shows the prior probability and 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. Note that we cannot evaluate whether the levels are correct if you do not put labeled tic marks on the vertical axis! Note also that we cannot evaluate what you’ve written if you do not provide labels or a legend that make clear which of the curves you’ve drawn represent which probability.

• You get three points if you meet all the criteria for four points, except the direction of change in one of the three 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 probabilities is incorrect at either the minimum or maximum value of \delta.

• You get zero points otherwise.

Prompt 4

• You get four points on Prompt 4 only if you got four points on Prompt 3 and if what you write is accurate, coherent and well-written.

• You get three points on Prompt 4 only if you got four points on Prompt 3 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.