COU 8: Building a Marginal-Conditional Diagram

Most of the models you’ve worked with so far in this lesson are motivated by the idea that the distribution of political power across groups is a major determinant of governments’ policy choices. For instance, the model you built in COU 7 was premised on the idea that when groups that profit from legacy technologies are more powerful, governments might forgoe investments in infrastructure that would induce technological change. In some circumstances, however, the relationship between politics and policy may run in the other direction – i.e. it may be that the policies governments adopt affect the distribution of political power.

Political scientist Andrea Campbell (Campbell 2005), for instance, argues that the U.S. Social Security Program, which provides cash payments to supplement the incomes of adults 65 and older, caused many elderly americans to come think of themselves as members of a distinct group with specific interests in government policy, and to become politically active to that end. As a result, politicians began to think of elderly americans as a powerful group whose political support was necessary for winning and holding power.

Knowing that policies that benefit specific groups of persons can cause members those groups to become more politically active, might politicians use policy to try to “activate” groups that they hope will support them in the future? Or, conversely, might politicians sometimes adopt policies that they hope will disrupt political activism by groups that they see as sources of political opposition? These tactics might, for instance, be at work in the U.S. Democratic party’s consistent advocacy for laws that would facilitate labor organizing by public sector workers, and the U.S. Republican party’s opposition to those same laws.

Imagine a politician attempting to cause the government to enact a new law which, if adopted, will confer benefits on a particular group of persons. Suppose the politician is uncertain about two things:

  1. The extent to which the law, if adopted, will cause the persons in the group to become more politically active.
  2. Whether the persons in the group, however politically active they might be, will tend to support or oppose the politician.

Prompt 1

Start by drawing a grid that depicts a model of joint uncertainty about (a) and (b). Do not assign specific numerical values for the joint probabilities in the model. Instead, add labels to the grid that use the notation for joint probabilities specified in the lesson to denote those probabilities.

Note that it is up to you to specify the exhaustive list of mutually exclusive possible resolutions of each of the two uncertainties you are modeling. As demonstrated in the lesson, make sure to include labels in the grid that make clear which axis of the grid depicts which uncertainty and which rows and columns of the grid correspond to each possible resolution of each uncertainty.

Prompt 2

Now model the lobbyist’s marginal uncertainty about (a). Do so by re-drawing the grid you drew in response to Prompt 1 and then adding specific numerical values for the marginal probabilities to the ends of the rows or columns (depending on which axis you used to depict uncertainty about (a)).

Remember, a model of marginal uncertainty is just like any other probabilistic model of uncertainty in that it consists of an exhaustive list of mutually exclusive possible resolutions, along with probabilities assigned to those resolutions that sum to 1.

Prompt 3

Now build models of uncertainty about (b) conditional on each possible resolution of (a). Specifically, for each possible resolution of uncertainty about (a) specified in your answer to Prompt 1, construct one model of uncertainty about (b) conditional on that resolution of (a). For each of the models of conditional uncertainty you write, assign specific numerical values for the conditional probabilities. Specify each model by re-drawing the grid you drew in response to Prompt 1, crossing out or shading the appropriate columns/rows, and then writing the specific numerical values for the conditional probabilities in the un-shaded cells.

Prompt 4

Draw a marginal-conditional diagram that depicts the model of uncertainty implied by the marginal and conditional uncertainties you specified response to Prompts 2 and 3. Make sure to (1) clearly label the uncertainty that each axis of the diagram depicts, (2) add labels that make clear which portions of each axis correspond to each resolution of each uncertainty, and (3) add labels showing the (marginal or conditional) probability assigned to each resolution.

Rubric

Prompt 1

A completely correct answer meets all the following criteria

  1. It is a grid with two axes.
  2. One axis is labelled in a way that clearly indicates that it represents uncertainty about the extent to which the members of the group will become more politically active if the law is adopted.
  3. The other axis is labelled in a way that clearly indicates that it represents uncertainty about whether Whether the persons in the group, however politically active they might be, will tend to support or oppose the politician..
  4. Each axis is clearly divided into two or more distinct rows (for a vertical axis) or columns (for a horizontal axis). The rows/columns on each axis are labelled in a way that makes absolutely clear what the exhaustive list of mutually exclusive possible resolutions of the uncertainty represented on that axis are.
  5. Each cell includes a label using the correct notation to denote the joint probability of the joint event depicted by that cell.

You can earn up to four points on this prompt:

  • 4 points if what you write fully satisfies all of criteria (a) through (e).
  • 3 points if what you write fully satisfies all of criteria (a) through (d) but does not fully satisfy criterion (e).
  • 2 points if what you write fully satisfies all of criteria (a) through (c) but does not fully satisfy criterion (d).
  • 0 points otherwise.

Prompt 2

A completely correct answer meets all the following criteria

  1. It is a grid with two axes.
  2. One axis is labelled in a way that clearly indicates that it represents uncertainty about the extent to which the members of the group will become more politically active if the law is adopted.
  3. The other axis is labelled in a way that clearly indicates that it represents uncertainty about whether Whether the persons in the group, however politically active they might be, will tend to support or oppose the politician..
  4. Each axis is clearly divided into two or more distinct rows (for a vertical axis) or columns (for a horizontal axis). The rows/columns on each axis are labelled in a way that makes absolutely clear what the exhaustive list of mutually exclusive possible resolutions of the uncertainty represented on that axis are.
  5. Each cell includes a label using the correct notation to denote the joint probability of the joint event depicted by that cell.
  6. Specific numerical values between 0 and 1 that together sum to 1 are at the ends of either the rows or columns (depending on which axis is labeled to depict uncertainty about the extent to which the persons in the group will become more politically active if the law is adopted.)

You can earn up to four points on this prompt:

  • 4 points if what you write fully satisfies all of criteria (a) through (f).
  • 3 points if what you write fully satisfies all of criteria (a) through (d) and (f) but does not fully satisfy criterion (e).
  • 0 points otherwise.

Prompt 3

A response to this prompt can only be evaluated in light of a response to Prompt 1 that earns 3 points or more (i.e. satisfies criteria (a) through (d) for Prompt 1). So, if your response to Prompt 1 earned less than 3 points, you earn 0 points on Prompt 4, regardless of your response.

That said, if your response to Prompt 1 satisfies criteria (a) through (d) for Prompt 1, a completely correct answer to Prompt 4 satisfies all of the following:

  1. It consists of exactly one grid for each resolution of uncertainty about the extent to which the persons in the group will become more politically active if the law is enacted, where the resolutions of that uncertainty are as depicted in the grid submitted in response to Prompt 1.
  2. Each grid is exactly the same as the grid drawn in response to Prompt 1 with two exceptions:
  1. The columns or rows representing the resolutions of the uncertainty about the extent to which the persons in the group will become more politically active other than the resolution represented in that grid are crossed out or shaded.
  2. Exact numerical values for probabilities are given in each unshaded cell.
  3. The numerical values in the unshaded cells of each grid together form a valid probability distribution – i.e. they sum to 1.

If your answer to Prompt 1 earned 3 or more points, you can earn up to 3 points on this prompt. Specifically:

  • 3 points if your answer fully meets criteria (i) and (ii)(A), (ii)(B) and (ii)(C)
  • 2 points if your answer fully meets critera (i) and (ii)(A) and (ii)(B) but does not fully satisfy (ii)(C)
  • 0 points otherwise.

Prompt 4

Your answer to this prompt can only be evaluated if you earned full credit (4 points) on Prompt 2 and full credit (3 points) on Prompt 3. If you earned less than full credit on either Prompt 2 or Prompt 3, you earn 0 points on Prompt 4.

That said, if you earned full credit on Prompt 2 and full credit on Prompt 3, A completely correct diagram drawn for Prompt 4 satisfies all of the following:

  1. It is a rectangle.
  2. The horizontal axis of the rectangle is labeled as depicting uncertainty about the extent to which the members of the group will become more politically active if the law is enacted.
  3. The horizontal axis of the rectangle is divided into sections, one section for each of the resolutions of the uncertainty about the extent to which the members of the group will become more political active exactly as those resolutions are specified in response to Prompt 1.
  4. The sections of the horizontal axis include labels for the marginal probabilities of the relevant sections, and those marginal probabilities exactly the match the relevant marginal probabilities written in response to Prompt 2.
  5. The sections of the horizontal axis are sized relative to one another in a way that approximately resembles the relatively sizes of the relevant marginal probabilities.
  6. Each section along the horizontal axis is divided into sections along the vertical axis, with exactly one section for each resolution of the uncertainty about whether the members of the group will tend to support the politician, as depicted in the response to Prompt 1.
  7. The vertical sections into which each section of the horizontal axis is divided are labeled to specify which resolution each section corresponds to, and those labels clearly match those used in response to Prompt 1.
  8. The vertical sections into which each section of the horizontal axis is divided are labeled with probabilities that match the relevant conditional probabilities written in response to Prompt 3.
  9. The vertical sections into which each section of the horizontal axis is divided are sized relative to one another in a way that approxomates the relevant sizes of the conditional probabilities.

Your response earns…

  • 2 points if it meets all of criteria (i) through (ix)
  • 1 point if it meets all of criteria (i) through (ix) except for (iv) or (viii) or both
  • 0 points otherwise.

References

Campbell, Andrea L. 2005. How Policies Make Citizens: Senior Political Activism and the American Welfare State. New York: Princeton University Press.