Lesson 5: Models of Learning
Over the next two decades, climate change will rapidly alter the frequency and severity of the environmental hazards the threaten human life and livelihoods in communities all over the globe. Cities close to the equator will face more frequent and severe bouts of life-threateningly high temperatures. Coastal communities will experience rising sea levels and more powerful cyclones and hurricanes. Communities that thrive on agriculture, fishing and forestry will have to adapt to the dying off of plant and animal species on which their economies rely. And cities in arid places will encounter greater volatility in rain and snowfall.
How much human suffering and loss these changes wreak depends to a substantial extent on how much governments do in the coming years to facilitate human resilience to these hazards. The populations of coastal regions, for instance, will suffer less if their governments adopt regulations that prevent the destruction of wetlands that dampen storm surges, while populations of cities at risk of heat waves can be protected by government investments in tree cover and public cooling facilities.
What do you know at this point about the environmental hazards that will threaten the places you call home in the coming decades? What do you know about the investments governments are making in those places to mitigate those hazards? How likely do you think it is that governments will make the investments and other changes required to adapt to our rapidly changing climate?
Many of us are aware that our welfare and that of our loved ones depends on governments adopting policies to mitigate the harms of climate change. But almost none of us are in a position to know the specific details of what mitigation policies our governments are or are not adopting. And even fewer have the expertise to determine exactly what government policies and investments are appropriate in light of the specific hazards facing any given place or region. These details will only be known with certainty by the tiny portion of persons whose full-time jobs involve making, researching or reporting on government policy. The rest of us will be left to judge the performance of our governments based on what we see, hear and read in the media.
Most of us face, then, what political scientists call a problem of political accountability. Although we each rely on governments to act in our interest, we have only a sliver of the information or knowledge we would need to know for sure whether any given government has acted or is acting as such. At best, we can use the partial information we glean from media reports and our day-to-day experiences to make inferences about whether our political leaders have acted responsibly, and use what little political power we have to reward or punish them accordingly.
In PPT, we used models of learning to depict the process through which political actors use limited information to update their beliefs about critical facts that they cannot ascertain with certainty. In this section, you’ll learn how to analyze and construct probabilistic models of learning. We’ll start with the concept of conditional probability.
Conditional Probability
Think once again of the imaginary city sitting on the riverbank that we introduced in Lesson 4. Imagine a resident of that city who, at a given moment in time, is uncertain about two things. First, has the city’s mayor made investments in the city’s flood control infrastructure that are in that resident’s best interest? Second, will the next coastal storm that passes over the city overwhelm the city’s infrastructure, causing the city to flood? Conditional probability is a tool for depicting uncertainties like this one, where a person is uncertain about two or more distinct issues that are related to one another.
We’ll start by modeling the citizen’s uncertainty about the first issue – whether or not the mayor has made investments in flood control that are in the resident’s best interest:
Now we’ll turn to the second issue: Will the next storm that passes near the city overwhelm the city’s flood control infrastructure and cause the city to flood? We want to model uncertainty about this issue in a way that captures the idea that the likelihood that the next storm will overwhelm the city’s infrastructure depends on whether the mayor has made investments in the resident’s best interest. More specifically, we want to depict a situation in which the city’s infrastructure is relatively less likely to be overwhelmed by the next storm if the mayor has made investments in the resident’s best interest, and relatively more likely to be overwhelmed by the next storm if the mayor has not made investments in the resident’s best interest. To do this, we need two distinct probability distributions: One describing the likelihood that the city’s infrastructure will be overwhelmed when the mayor has acted in the resident’s best interest and another describing the likelihood that the city’s infrastructure will be overwhelmed when the mayor has not acted in the resident’s best interest.
This is an instance of the concept of conditional probability because the probability distribution over one set of events – whether or not the city’s infrastructure is overwhelmed by the next storm – depends on which event from another set of events – whether the mayor has acted in the resident’s best interest – occurs. For example:
Make sure to take note of the most important choice we’ve made in depicting the resident’s uncertainty: By assuming that the probability that the next storm will flood the city is \frac{1}{4} when the mayor has acted in the resident’s best interest, but \frac{3}{4} when the mayor has not acted in the resident’s best interest, we’ve assumed that the mayor’s actions condition the relative likelihood that the next storm will flood the city.
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Visualizing Conditional Probability
To understand and analyze a model of conditional probability in which there are two distinct but related uncertainties, we can use a powerful visual tool demonstrated by math educator Grant Sanderson in his fantastic video explaining conditional probability. (And yes, you should watch that video to supplement this material!!!)
To see how the tool works, start with our model of the resident’s uncertainty about the mayor’s investments in flood control infrastructure:
Recall that in any probabilistic model, the probabilities assigned to the full list of mutually exclusive events must add up to 1. Therefore, we can always think about probabilities as equivalent to proportions or shares of a shape. Visually, this means we can represent the probability distribution…
| Event | Probability |
|---|---|
| The mayor has made investments in the resident’s best interest. | \frac{5}{8} |
| The mayor has not made investments in the resident’s best interest. | \frac{3}{8} |
…as a rectangle divided into two segments – one segment for each of the two possible events –, with the size of each segment proportional to the probability of the event corresponding to that segment. It looks like this:
Now, in our model of conditional probability above, there are two issues the resident is uncertain about – whether the mayor has acted in the resident’s interest, and whether the next storm will flood the city. Thus to visualize this model, we need a figure with two dimensions. We start by drawing a square, and label one of the square’s horizontal sides as depicting whether the mayor has acted in the resident’s interest, and one of the square’s vertical sides as depicting whether the next storm will flood the city, like this:
Recall that in our model, we represent the probability that the mayor has made investments that are in the resident’s best interest as unconditional, and the probability that the next storm will flood the city as conditional on whether the mayor has acted in the resident’s interest. Thus, we’ll represent the probabilities of the various events by first dividing the square in a way that represents the probability that the mayor has acted in the resident’s interest. Recall that in our model, the mayor acts in the resident’s interest with probability \frac{5}{8} and does not act in the resident’s interest with probability \frac{3}{8}. We therefore divide the square along the horizontal dimension like so:
Now we’ll add to the diagram to represent the conditional probabilities with which the next storm will cause the city to flood. Lets start with the probability that the next storm will flood the city conditional on the event that the mayor has acted in the resident’s interest. These probabilities are:
| Event | Probability |
|---|---|
| The next storm floods the city | \frac{1}{4} |
| The next storm does not flood the city | \frac{3}{4} |
Since our diagram uses the left-hand portion of the square to capture the event that the mayor has acted in the resident’s interest, we’ll divide up that portion of the square accordingly:
Finally, we’ll divide up the right-hand portion of the square to show the probability that the next storm floods the city conditional on the event that the mayor has not acted in the resident’s best interest – i.e.
| Event | Probability |
|---|---|
| The next storm floods the city | \frac{3}{4} |
| The next storm does not flood the city | \frac{1}{4} |
What you’ve seen above is how to diagram the simplest conditional probability distribution possible. You’ll need to be able to diagram more complex distributions in the following sections. So watch this video to see how:
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Joint, Marginal and Conditional Probabilities
Once you’ve constructed a conditional probability diagram, you can use it to visualize and calculate three kinds of probabilities that are essential for the analysis of conditional models: joint probabilities, marginal probabilities, and conditional probabilities.
Joint Probability
The joint probability of two or more events is the probability that all of those events happen. Given events A, B and C, the joint probability of A, B and C is written as P (\text{$A$ and $B$ and $C$}) For instance, consider the conditional probability model developed in the video above:
In that model, the probability that the seawall is 2 meters tall and the next storm floods the city to a depth of 50 centimeters is a joint probability written as: P(\text{seawall is $2$ m and next storm floods city to $50$ cm})
To use a conditional probability diagram to calculate a joint probability, you just need to apply two rules:
- The probability of any event is equal to the area of the region corresponding to that event in the diagram.
- Given any events A and B, the region corresponding to the event that A and B occur is the intersection of the regions corresponding to events A and B.
For instance, consider the events:
The seawall was built to 2 meters
The next storm floods the city to a depth of 50 centimeters.
The regions corresponding to these events are outlined below in blue and yellow respectively:
The intersection of these two regions is outlined in green below:
By reading the probabilities off of the diagram, one can see that the area outlined in green above amounts to a rectangle with sides of length 0.27 and 0.2. Therefore, the joint probability of the event that the seawall is 2 meters and the next storm floods the city to a depth of 50 centimeters is: P(\text{seawall is $2$ m and next storm floods the city to $50$ cm}) = 0.27 \times 0.2 = 0.054
Importantly, when two or more events occupy regions that do not intersect, their intersection has an area of 0. Thus, the joint probability of all of those events happening is 0.
For instance, consider the two events:
A: The seawall is 1 meters high AND the next storm will flood the city to 1 meter.
B: The seawall is 3 meters high AND the next storm will flood the city to 0 centimeters.
Event ‘A’ is depicted outlined below in blue and event ‘B’ is outlined in yellow:
These events do not overlap on the diagram (they are mutually exclusive!) and thus their joint probability is zero, i.e. P(\text{$A$ and $B$}) = 0
Marginal Probability
The marginal probability of one or more events is the probability that at least one of those events happens. Given events A, B and C, the marginal probability of A, B, C is written as: P(\text{$A$ or $B$ or $C$})
For instance, continue to consider the model developed in the video above and in particular the two events:
The seawall was built to 1 meter tall.
The seawall was built to 3 meters tall.
The marginal probability of these two events can be written as: P(\text{seawall built to $1$ meter tall or seawall built to $3$ meters tall})
To use a conditional probability diagram to calculate the marginal probability of a set of events, identify all parts of the diagram that are included in at least one of those events. The marginal probability is equal to the portion of the square taken up by that entire area.
For instance, consider the three events:
A: The seawall was built to 1 meter tall.
B: The seawall was built to 2 meters tall and the next storm floods the city to a depth of 10 centimeters.
C: The seawall was built to 3 meters tall.
The areas of the square corresponding to these three events are outlined in blue, green, and yellow below:
Reading the probabilities off the diagram, one can see that the areas of these three regions are (A) 1 \times 0.7=0.7, (B) 0.2 \times 0.15 = 0.03, (C) 1 \times 0.1 = 0.1. Since these three regions do not overlap one another, the area including all points that sit within at least one of these areas is the sum of their areas – i.e.: P\left( \begin{array}{c} \text{seawall is $1$m} \\ \text{OR} \\ \text{[seawall is $2$ m AND next storm floods to $10$ cm]} \\ \text{OR} \\ \text{seawall is $3$m} \\ \end{array} \right) = 0.7 + 0.03 + 0.1 = 0.83
When you’re calculating a marginal probability for one or more areas that overlap, make sure not to double-count the area of the overlapping regions. For instance, consider the marginal probability: P \left( \begin{array}{c} \text{the sewall is $2$m tall}\\ \text{OR} \\ \text{the next storm will flood the city to $50$cm} \end{array} \right)
The region consisting of all points that lie in at least one of these events amounts to all points that lie in either the blue or yellow regions in the following diagram:
The yellow and blue areas overlap in the middle of the figure. But that overlapping region’s area counts only once towards the marginal probability. In particular, adding up all the areas included (each one just once!) in the combination of the blue-outlined and yellow-outline regions, we have:
\begin{align*} P \left( \begin{array}{c} \text{the sewall is $2$m tall}\\ \text{OR} \\ \text{the next storm will flood the city to $50$cm}\\ \end{array} \right) = & \\ 0.2\times 0.7 + & \\ 0.5 \times 0.2 + & \\ 0.27 \times 0.2 + & \\ 0.15 \times 0.2 + & \\ 0.08 \times 0.2 + & \\ 0.26 \times 0.1 \end{align*}
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Conditional Probability
Given events A and B, the probability of A conditional on B (sometimes written as “the probability of A given B”), is the probability that A occurs given that B has occurred or is absolutely certain to occur. Given events A and B, the probability of event A conditional on B is written P(A|B)
For instance, continue to imagine the resident of our city who is uncertain about how tall the mayor has built the seawall and the depth to which the next storm will flood the city. Suppose that next storm hits, and the city floods to a depth of 10 centimeters. Suppose further that the resident knows that the city has flooded to 10 centimeters with absolute certainty, but that she remains uncertain about the height of the seawall. In this case, her belief about the probability that the seawall is, say, 1 meter tall would be conditional on the event that the next storm flooded the city to a depth of 10 centimeters. We would write that probability as: P (\text{seawall is $1$ m} | \text{next storm flooded the city to $10$ cm})
There are two steps in using a conditional probability diagram to calculate a conditional probability. The first step, is to shade or cross out all the regions of the diagram representing events that are known to have not occurred, given the event that is known to have occurred. For instance, consider again our model of the resident’s uncertainty:
Suppose that the next storm hits, floods the city to a depth of 10 centimeters, and the resident knows that flood depth with certainty. Because the resident now knows that the next storm flooded the city to a depth of 10 centimeters, all possibilities in which that storm flooded the city to some other depth – i.e. to 0 centimeters, 50 centimeters or 1 meter – are ruled out. Thus, to calculate the probability of any given height of the seawall conditional on the event that the next storm flooded the city to a depth of 10 centimeters, we would first shade all of the areas of the square corresponding to events in which the next storm flooded the city to any depth other than 10 centimeters, like so:
The un-shaded remainder of the square represents the resident’s remaining uncertainty. Further, the relative amounts of space occupied by each of the possibilities within this remaining un-shaded area depict the probabilities of each of those possibilities given the event on which we are conditioning. Thus, the second step in computing a conditional probability is to compute the relative sizes of these areas.
For instance, consider the probability that the seawall is 1 meter tall, conditional on the event that the next storm flooded the city to a depth of 10 centimeters. Continuing to work with our shaded diagram, one can see first that the un-shaded area depicting the possibility that the seawall is 1 meter measures 0.7 units wide and 0.07 units tall, and thus has an absolute area of 0.7 \times 0.07 = 0.049, like so:
Its area relative to the total area occupied by the remaining possibilities, on the other hand, depends on the total area occupied by all those possibilities. Examining the diagram…
…one can see that the total area occupied by all of the remaining possibilities is 0.7 \times 0.07 + 0.2 \times 0.15 + 0.1 \times 0.23 = 0.049 + 0.03 + 0.023 = 0.102
Therefore the area occupied by the possibility that the seawall is 1 meter tall relative to the total area occupied by all the remaining possibilities is: \frac{0.7 \times 0.07}{0.7 \times 0.07 + 0.2 \times 0.15 + 0.1 \times 0.23} = \frac{0.049}{0.102} \approx 0.48 And that is the probability that the seawall is 1 meter tall given that the next storm has flooded the city to a depth of 10 centimeters – i.e., P (\text{seawall is $1$ m} | \text{next storm flooded the city to $10$ cm}) \approx 0.48
The following video walks through two more examples showing how to use a conditional probability diagram to calculate a conditional probability.
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Learning: Priors, Posteriors, and Bayes’ Rule
With an understanding of conditional probability in hand, we can now model learning. To illustrate the key concepts, we’ll use the somewhat more elaborate model of conditional probability demonstrated in the videos above. Specifically, we’ll suppose that the mayor of our imaginary city has built a seawall. We’ll focus on a resident of the city who is uncertain about (a) the height of the seawall the mayor has built and (b) the depth to which the next storm will flood the city. Just as in the video, we’ll use the following model of the resident’s beliefs about these two issues:
Imagine that the next storm hits the city and causes a catastrophic flood in which the water in the city’s streets and homes is 1 meter deep. Suppose this flood depth is reported on the news, and the resident himself measured the depth of the water standing on his street. Thus the resident’s uncertainty has now been partially resolved. The next storm has hit the city and the resident is absolutely sure that it caused a flood of the city 1 meter deep. Thus, using the visual conventions demonstrated in the previous section, the resident’s beliefs have changed like this:
Specifically, prior to the storm the resident was uncertain about both the height of the seawall and the depth of the flood that the next storm would cause. After the storm, the resident knows the depth of the flood was 1 meter, ruling out the possibilities that the flood could have been 0 centimeters, 10 centimeters or 50 centimeters, but remains uncertain about the height of the seawall.
It’s critical to learn some terminology used in PPT to discuss models like this. Notice that the story above depicts two moments in time. First, it depicts a moment before the storm hits the city, when the resident is uncertain about the height of the seawall and how much flooding the next storm will cause. Second, it depicts a moment after the storm in which the resident is no longer uncertain about how much flooding the next storm (which at that moment is the most recent storm!) will cause but continues to be uncertain about the height of the seawall. We use the terms prior belief and posterior belief to refer to beliefs like these, where “prior” and “posterior” are used to clarify which belief is held before an event and which belief is held after, like this:
To calculate posterior beliefs in a model of learning analysts apply Bayes’ Rule. Bayes’ Rule is typically stated as a formula for conditional probability. A common formulation of Bayes’ Rule, for instance, is that the probability P(A|B) of an event A given another event B is P(A|B) = \frac{P(\text{$A$ and $B$})}{P(B)}
However, in these lessons, we will only use models of learning in which the prior belief depicts uncertainty over exactly two issues. Beliefs in such models can always be visualized using conditional probability diagrams like those above. And from the previous sections, you already know how to use these diagrams to compute conditional probabilities. So you have no need at this point to de-code or memorize a formula like Bayes’ Rule! Still, when you see models of learning in published research in political science, you will often see references to Bayes’ Rule, so it is useful to be aware of the term.
To see how to compute posterior beliefs in a model of learning using a conditional probability diagram, consider again our visual representation of the model in which the resident of the city learns about the height of the seawall from observing the depth to which the next storm floods the city:
In the resident’s posterior belief (i.e. the belief depicted on the right-hand side of the figure above) what is the probability that the seawall is 1 meter tall? Using the technique for computing conditional probabilities you learned in the previous section, you can discern that P(\text{seawall is $1$m tall} | \text{storm flooded the city to $1$m}) = \frac{0.7\times 0.7}{0.7\times 0.7 + 0.5 \times 0.2 + 0.3 \times 0.1} which is approximately 0.79.
One more remark before moving on to the next section. We mentioned that the technique used above for visualizing conditional probability was demonstrated in a fantastic video by Grant Sanderson. Now that you have a sense of what Bayes Rule is used for and how it relates to conditional probability, that video is well worth watching!
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Parameterized Models of Learning
Parameterized models help us explore how learning might work under different conditions. To explore parameterized models, we’ll construct a basic model of political accountability, in which a voter learns about the characteristics of an incumbent politician through a natural disaster and then votes for or against the incumbent in next election on the basis of those beliefs. The accountability of governments for prevention of and response to natural disasters is a topic of extensive research in political science – including work on the politics of hurricanes (for instance, Malhotra and Kuo (2008)), river floods (for instance, Bechtel and Hainmueller (2011)), landslides (for instance, Gallego (2018)), wildfires (for instance, Ramos and Sanz (2020)), pandemics (for instance, Gutiérrez, Meriläinen and Rubli (2022)) and even shark attacks (Achen and Bartels 2002).
Continue to imagine a coastal city that is frequently threatened by hurricanes. Suppose the city’s government builds and maintains infrastructure meant to protect residents and their property from the tidal surges and torrential rains that accompany each coastal storm. Suppose that residents are aware of this infrastructure, but lack the expertise and first-hand knowledge required to assess its quality, or to know whether the city’s spending on it is appropriate. Residents know, moreover, that even the best-constructed infrastructure will fail in the face of an unusually strong storm. So when a storm floods the city, residents might suspect that their politicians failed to invest appropriately in the city’s protection, but they cannot be sure.
More specifically, imagine a moment in time at which the city’s residents are uncertain about two things: (a) Whether their incumbent elected officeholders are corrupt and (b) whether the next hurricane will cause the city to flood. Residents know that corrupt politicians make their friends and associates rich through flood-control construction and maintenance spending from which little or no useful flood control improvements are actually accomplished. Honest politicians, on the other hand, make investments in the city’s infrastructure that insure a reasonable level of hurricane protection. Thus residents know that if the incumbents are corrupt, the city is somewhat more likely to by flooded by the next storm than it would be if the incumbents are honest.
Residents’ beliefs, then, are described by the following conditional probability distribution:
The parameter \beta (the Greek letter pronounced “bay-tah”) is a number between 0 and 1 and depicts the extent to which the incumbents’ corruption (or honesty) affects the city’s vulnerability to flooding. There are two things to make sure you understand about how this model works. First, regardless of the value of \beta and regardless of whether the incumbents are corrupt, a storm is relatively unlikely to flood the city. Specifically, if the incumbents are corrupt the probability that the next storm will overwhelm the shoddy infrastructure the corrupt incumbents have built is only \frac{1}{4}. If the incumbents are honest, on the other hand, the probability that the next storm floods the city is \frac{1}{4}(1-\beta). Since \beta is between 0 and 1, 1-\beta is also between 0 and 1 and thus \frac{1}{4}(1-\beta) is less than \frac{1}{4}. Therefore, if the incumbents are honest, the likelihood that the next storm floods the city will be to some extent less than the (already relatively low) likelihood that the next storm floods the city when the incumbents are corrupt.
Second, the value of the parameter \beta determines the extent to which the city’s infrastructure is less likely to fail if its been maintained by honest incumbents than by corrupt incumbents. More specifically, as \beta gets larger, the probability that the next storm floods the city when the incumbents are honest gets smaller, and thus the divergence between the robustness of the city’s infrastructure under honest incumbents and the robustness of the city’s infrastructure under corrupt incumbents gets larger. For instance, when \beta is at its minimum value of 0, the probability that the next storm floods the city when the incumbents are honest is \frac{1}{4}(1-0) = \frac{1}{4}, which is exactly the same as the probability that the next storm floods the city when the incumbents are corrupt. On the other hand, when \beta is at the relatively high value of \frac{9}{10}, the probability that the next storm floods the city when the incumbents are honest is \frac{1}{4}\left(1-\frac{9}{10}\right) = \frac{1}{40}, which is 10 times smaller than the probability that the next storm floods the city when the incumbents are corrupt.
To make sure you can visualize both of these features – i.e. that the probability that the next storm floods the city is relatively low regardless of the corruption or honesty of the incumbents, and that the divergence between the probability that the city floods under corrupt incumbents and honest incumbents gets larger as \beta increases –, use the following slider to vary the value of \beta…
…and observe how this changes the likelihood that the storm will flood the city under the two types of incumbents:
Now suppose that the next hurricane arrives. Based on whether or not the city floods, residents can use what occurs to learn about the relative likelihood that the city’s incumbent politicians are corrupt. Notice that this has exactly the same structure as the model of learning introduced in the previous section. Specifically, the conditional probability distribution diagrammed above is the residents’ prior belief about the city’s incumbents and the likelihood that the next storm floods the city. After the next next storm hits the city and residents observe whether or not the city floods, they learn, thus forming a posterior belief about the likelihood that the city’s incumbents are corrupt.
To calculate these posterior beliefs, start by supposing that the next storm floods the city. Following the procedure introduced in the previous section, we shade in the regions of the diagram in which the city does not flood like so:
Then, we calculate the proportions of the un-shaded parts of the square occupied by the event that the incumbents are corrupt and the event that the incumbents are not corrupt, respectively, like so:
Examining the diagram above, we can see that the proportion of the remaining area corresponding to the event that the incumbents are corrupt is: \frac{\frac{1}{6}}{\frac{1}{6} + \frac{1}{12}(1-\beta)} On the other hand, the proportion of the remaining area corresponding to the event that the incumbents are honest is: \frac{\frac{1}{12}(1-\beta)}{\frac{1}{6} + \frac{1}{12}(1-\beta)} Thus, if the next storm floods the city, the residents’ posterior beliefs about the incumbent politicians are:
| Event | Probability |
|---|---|
| incumbents are corrupt | \frac{\frac{1}{6}}{\frac{1}{6} + \frac{1}{12}(1-\beta)} |
| incumbents are honest | \frac{\frac{1}{12}(1-\beta)}{\frac{1}{6} + \frac{1}{12}(1-\beta)} |
Now suppose the city does not flood when the next hurricane hits the city. Using the same procedure…
…we can see that if the next hurricane does not flood the city, the residents’ posterior beliefs about the incumbent politicians are:
| Event | Probability |
|---|---|
| incumbents are corrupt | \frac{\frac{1}{2}}{\frac{1}{2} + \frac{1}{3}\left[1-\frac{1}{4}(1-\beta)\right]} |
| incumbents are honest | \frac{\frac{1}{3}\left[1-\frac{1}{4}(1-\beta)\right]}{\frac{1}{2} + \frac{1}{3}\left[1-\frac{1}{4}(1-\beta)\right]} |
To summarize, the change in residents’ beliefs about the incumbent depends on whether the next hurricane floods the storm as in the following diagram:
It’s hard to get a sense of what these expressions imply from just looking at them. But if we graph the residents’ posterior beliefs as a function of \beta, some striking patterns appear:
This graph shows the posterior probability that residents assign to the event that the incumbents are corrupt, conditional on the event that there is a flood (the top, green line) and conditional on the event that there is no flood (the bottom, blue line).
Notice first that as \beta gets larger, the magnitude of the effect of the storm on residents beliefs’ about the incumbents goes from negligible to substantial. Recall that \beta ranges from 0 to 1. When \beta is close to the bottom of its range – say around 0.05 –, the residents’ beliefs about the incumbents when the storm floods the city differ very little from their beliefs when the storm does not flood the city. But as \beta approaches its maximum, the effect of the storm becomes completely decisive on residents’ beliefs about the incumbents. When, for instance, \beta is about 0.95, a flood causes resident to be nearly certain that the city’s incumbents are corrupt.
Second, this graph reveals that even when \beta is close to its maximum value residents posterior beliefs about the city’s incumbents differ little from their prior beliefs when a storm does not cause a flood. Why is that? Recall that the model assumes that any given storm is unlikely to cause a flood, even when the city’s infrastructure has been poorly maintained by corrupt incumbents. Thus, even when \beta is close to its maximum so that honest incumbents build much more effective infrastructure than corrupt incumbents, the lack of a flood is by no means definitive evidence that the incumbents are honest.
This model, then, suggests two non-obvious ideas about how natural disasters might affect democratic politics. First of all, we shouldn’t assume that a natural disaster, no matter how terrible, will invariably cause voters to punish incumbent politicians by kicking them out of office. Instead, whether a disaster shifts incumbents’ popularity might depend on what voters’ believe about the extent to which incumbents’ behaviors in office can affect voters’ vulnerability to natural hazards. By allowing the parameter \beta to vary, this model shows how voters’ beliefs about incumbents can be quite un-responsive to natural disasters in the right circumstances. When, for instance, voters understand that even the best efforts by honest incumbents will have only small effects on the city’s vulnerability to flooding (i.e., when \beta is close to 0), they will rightly see a catastrophic infrastructure failure as perfectly consistent with the possibility that incumbent political leaders acted in good faith but faced an impossible situation.
Second, even when politicians’ honesty or corruption can have a decisive effect on the effectiveness of preparations for a storm or other natural hazard (i.e. when \beta is close to its maximum), voters will not necessarily reward incumbent politicians when a storm or other natural hazard fails to cause death and destruction. Voters may well recognize that bad incumbents sometimes produce good outcomes through sheer luck. Thus the absence of a disaster may not persuade them that their politicians have used public funds responsibly.
To see how to analyze another (slightly more elaborate) example of a parameterized model of learning, watch this video:
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References
Achen, Chris H., and Larry M. Bartels. 2002. “Blind Retrospection: Electoral Responses to Drought, Flu and Shark Attacks.”
Bechtel, Michael M., and Jens Hainmueller. 2011. “How Lasting Is Voter Gratitude? An Analysis of the Short- and Long-Term Electoral Returns to Beneficial Policy.” American Journal of Political Science 55 (4).
Gallego, Jorge. 2018. “Natural Disasters and Clientelism: The Case of Floods and Landslides in Columbia.” Electoral Studies 55 (1).
Gutiérrez, Emilio, Jaakko Meriläinen, and Adrian Rubli. 2022. “Electoral Repercussions of a Pandemic: Evidence from the 2009 H1N1 Outbreak.” Journal of Politics 84 (4).
Malhotra, Neil, and Alexander Kuo. 2008. “Attributing Blame: The Public’s Response to Hurricane Katrina.” Journal of Politics 70 (1).
Ramos, Roberto, and Carlos Sanz. 2020. “Backing the Incumbent in Difficult Times: The Electoral Impact of Wildfires.” Comparative Political Studies 53 (3-4).