How can we understand the Kavanaugh hearing and people's reaction to it? The well-established logic of probabilities can help.
The testimony before the Senate Judiciary Committee left a lot of uncertainty and a lot of room for interpretation. Was Christine Blasey Ford really assaulted? Did Brett Kavanaugh do it? Is her testimony a sincere public service or a con job? Is his reaction authentic indignation or the rage of an entitled bully?
We can call on Bayes' Rule to help us understand how we and others can come to different conclusions. Bayes' Rule is a venerated tool in science for analyzing uncertainty, and sometimes, belief states of mind. With appropriate problem formulation, we can disentangle the evidence and hypotheses before us. Not surprisingly, what you conclude depends on your prior assumptions, which largely depend on political persuasion. Bayes' Rule lets us isolate and examine those assumptions.
To apply Bayesian reasoning to this difficult situation, I built a detailed model that calls out key questions and assumptions. The model is implemented in the form of a Bayes' Rule Calculator, with sliders that let you adjust assumptions and see how they change the conclusions. Try it for yourself.
The quick takeaway is this:
Blue observers will conclude that Kavanaugh is probably guilty because they are more likely to believe that: 1. Ford is a credible and honest witness; 2. The alleged assault is not out of line with Kavanaugh's character; 3. Kavanaugh's angry outburst was a smoke screen while an honest person would have remained calm.
Red observers will conclude that Kavanaugh is definitely innocent because they are more likely to believe that: 1. Ford and the Democrats are lying conspirators; 2. Ford suffers from mistaken identity; 3. The alleged assault is inconsistent with Kavanaugh's character.
These arguments surface in the news and opinion media. Bayes' Rule shows how they interact to contribute to a final conclusion.
Let us break down the situation into three core issues.
These issues form a chain of reasoning. Any combination of values is theoretically possible. We cannot know for certain what happened in 1982, and we cannot know for certain the parties' state of mind today, whether they are sincere or lying. The only thing we observe directly is their testimony. We can however reason backwards if we assign some upfront assumptions.
It could have worked out that Dr. Ford never got to testify, due either to lesser motivation, or to political thrashing around. But in the end she did. All agree that Dr. Ford was emotional yet calm. Everyone agrees as well that Judge Kavanaugh was angry and combative. As an aspirant to the Supreme Court, he could have testified in a calm manner, either emotionally or not. That counterfactual bears consideration in our mathematical analysis.
Here are the variables we must consider. Some of them are hypotheticals that could have happened but didn't, but should be considered in comparison to what actually occurred. Each of these forms a True/False pair. As the probability of one variable increases from 0 (no chance) to 1 (absolute certainty), the probability of its complement decreases from 1 to 0.
Each of the links in the causal chain is represented in a table. The rows and columns of the tables are possible truth values of the important questions. The cells can be interpreted in terms of human motivation and behavior to which we can assign realistic assumptions.
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Table 1: Linking Alleged Assault to Ford Belief State
| FBG | FBI | |
|---|---|---|
| KG | Ford correcty recalls that Kavanaugh really assaulted her. p( FBG | KG ) |
Ford forgot the assault or else thinks someone else did it. p( FBI | KG ) |
| KI | Ford believes she was assaulted, but it is either imagined or else it was actually someone else. p( FBG | KI ) |
Ford knows Kavanaugh is innocent. She is lying about the assault. p( FBI | KI ) |
The way to read the table is like this. Consider the upper left cell in Table 1. Suppose just for a moment that the Row Header is true (KG = Kavanaugh is actually Guilty). Then what is the probability that the Column Header is true (FBG = Ford Believes he is Guilty).
Because these are tables in a causal chain, we assign numbers to the cells as conditional probabilities. That's what the notation means. p( B | A ) means the probability of B being true when A is true.
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Table 2: Linking Alleged Assault to Kavanaugh Belief State
| KBG | KBI | |
|---|---|---|
| KG | Kavanaugh knows he committed the assault and is lying about it. p( KBG | KG ) |
Kavanaugh committed the assault but he blacked out or has forgotten about it. p( KBI | KG ) |
| KI | Kavanaugh halluncinated or is brainwashed into falsely thinking he committed assault. p( KBG | KI ) |
Kavanaugh is innocent and he knows it. p( KBI | KI ) |
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Table 3: Linking Ford Belief State to Ford Testimony
| FTC | FNT | |
|---|---|---|
| FBG | Ford submits letter and manages to get a chance to testify. p( FTC | FBG ) |
Ford decides not to submit letter, or her efforts do not lead to her testifying p( FNT | FBG ) |
| FBI | Ford deviously advances her phoney case and the Dems conspire to let her testify. p( FTC | FBI ) |
No issue for Ford to raise, or else Dems conspire but fail to get false accusation to committee. p( FNT | FBI ) |
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Table 4: Linking Kavanaugh Belief State to Kavanaugh Testimony
| KTA | KTC | |
|---|---|---|
| KBG | Kavanaugh knows he has been caught and is lashing out in rage. p( KTA | KBG ) |
Kavanaugh knows he has been caught but cooly holds fast in his lie. p( KTC | KBG ) |
| KBI | Kavanaugh is indignant at believing he is falsely accused. Maybe he cannot contain himself, or perhaps he is putting on a show. p( KTA | KBI ) |
Kavanaugh maintains calm judicial manner and adamantly maintains his innocence while seeking impartial truth about the sources of the accusations against him. p( KTC | KBI ) |
It is well understood that people seldom reason forward in a causal chain such as this. Many times we decide what conclusion we want to reach, then adjust our arguments to fit them. Yet the reasoning chain is still a valid model for how people with different opinions can find common ground and discuss differences when they try to put together logical arguments to support their views.
To complete the chain, we need to fill in one additional factor: What is the prior probability that Brett Kavanaugh assaulted Ford? If there had been no accusation, would you have reason to think this might have happened? Again, people will disagree. The Reds say, Judge Kavanaugh might have been a drinker and a cad, but he was an academic star, an athlete, did public service, has had a distinguished record in government, and has been supporter of women in his official offices. So absent these charges, it is extremely unlikely that someone of his character would have committed this act. Let's say 1% Blues will say that Kavanaugh grew up in a culture of male priviledge and dominance, failed to control himself when drunk, and furthermore, other credible instances of disrespecting women are coming out of the woodwork; so not even knowing about Ford's specific accusation, it is not out of the question that he could have done such a thing while a teenager. Blue might put Kavanaugh and his ilk in the 20% range of committing this sort of act, absent any other information.
The prior probability factor is denoted as p( KG ). The prior probability that Kavanaugh is innocent is of course p( KI ) = 1.0 - p( KG )
When we put the chain together we get this reasoning diagram. We can assign and debate estimates for the prior and conditional probabilities based on our own beliefs and assumptions.
Bayes' Rule Inference
This brings us to the question, "Is Kavanaugh guilty or innocent?," given not only our prior beliefs about the causal chain, but also the actual historic Senate hearing testimony. Some people seem to believe that it doesn't matter whether Kavanaugh actually did try to rape Ford as a teenager, he still belongs on the Supreme court. Others say that Kavanaugh's angry outburst is unbecoming of a Supreme Court justice and is immediately disqualifying anyway. Let's strip away these views from the discussion, and focus on how we can reason about about Kavanaugh's likely innocence or guilt alone.
This is the posterior probability as calculated by Baye's Rule:
p( KG | T ) = p( T | KG ) * p( KG ) / p( T )
Here, the proposition of interest is KG: is Kavanaugh guilty of assault? The proposition T is the testimony. The conditional probability terms expand into combinations of terms from the conditional probability tables and the prior.
For observed variables, we have the testimony of two persons. We consider them first independently.
The calculation based on Ford's testimony is
p( KG | FTC ) = p( FTC | KG ) * p( KG ) / p( FTC )
The calculation based on Kavanaugh's testimony is
p( KG | KTA ) = p( KTA | KG ) * p( KG ) / p( KTA )
Let's plug in some numbers.
The Bayes' Rule Calculator lets you adjust assumptions using sliders. Here are some tables of conditional probabilities that might be held by a Blue-thinking or Red-thinking person.
Table 5: Linking Alleged Assault to Ford Belief State
| Red | Blue | |
|---|---|---|
| p( FBG | KG ) | 1.0 | 1.0 |
| p( FBI | KG ) | 0.0 | 0.0 |
| p( FBG | KI ) | 0.5 | 0.1 |
| p( FBI | KI ) | 0.5 | 0.9 |
The first two rows reflect assumptions that both Red and Blue agree that if Kavanaugh really did assault Ford, then she would know it.
What if he didn't assault her? It is still possible that she mistakenly believes he did. Both Red and Blue agree that Ford is a credible witness. Under most circumstances, she probably produces a reasonable recollection of what she sees. But both sides allow that she could be mis-remembering what happened. Let's say Blue gives Ford an accuracy of 90%, while Red rates her recollections as only 50% accurate.
Table 6: Linking Alleged Assault to Kavanaugh Belief State
| Red | Blue | |
|---|---|---|
| p( KBG | KG ) | 0.5 | 0.8 |
| p( KBI | KG ) | 0.5 | 0.2 |
| p( KBG | KI ) | 0.0 | 0.0 |
| p( KBI | KI ) | 1.0 | 1.0 |
The bottom two rows indicate that if Kavanaugh really did not assault Ford, he would not somehow believe that he actually did. Red and Blue agree on this.
What if he actually did assault her as Ford states? These are the top two rows.
Let's allow that both sides would then give him some benefit of the doubt and allow that he might have blacked out or forgotten the incident. Red is more lenient, and puts the chances at 50% that he was so drunk or oblivious to how terrifying his actions were to Christine Ford, that he honestly does not remember doing it. Blue gives Kavanaugh less slack, and supposes that if he did it, there is 80% chance that he remembers it.
Table 7: Ford Belief State to Ford Testimony
| Red | Blue | |
|---|---|---|
| p( FTC | FBG ) | 0.5 | 0.5 |
| p( FNT | FBG ) | 0.5 | 0.5 |
| p( FTC | FBI ) | 0.5 | 0.01 |
| p( FNT | FBI ) | 0.5 | 0.99 |
Assuming Ford honestly believes she was attacked by Kavanaugh, what were the chances that her testimony would actually be presented? This depends on many factors, including her decision to step forward, and the political negotiations to allow her to testify. It could have gone either way. Red and Blue both put the chances at 50%
But what if Ford secretly knows that Kavanaugh is innocent? What are the chances that she would have conjured up this story about him in 2010, and suffered through harrassment and attacks for the purposes of bringing him down? Red believes this is a credible scenario, and gives it a probability of 50%. Blue thinks such a conspiracy is ridiculous. At worst, they would concede that this could have only a 1% chance of being true.
Table 8: Kavanaugh Belief State to Kavanaugh Testimony
| Red | Blue | |
|---|---|---|
| p( KTA | KBG ) | 0.8 | 0.8 |
| p( KTC | KBG ) | 0.2 | 0.2 |
| p( KTA | KBI ) | 0.5 | 0.3 |
| p( KTC | KBI ) | 0.5 | 0.7 |
Assuming Kavanaugh knows he is guilty of the attack, what would his testimony be like? In such a circumstance, Red and Blue might agree on the same probability that he would erupt in anger, but for different reasons.
Red would say that in the bigger picture, Kavanaugh and the Right believe that his getting onto the Supreme Court far outweighs whatever he might have done way back in high school, and it's an outrage that the Democrats are trying to block him by this late stage tactic, even if the charges are true.
Blue says that Kavanaugh is reacting like a caged animal. If he knew he were guilty but rebutted the charges like a cool cucumber, then he would be more like a psychopath which is even worse than a hothead. Both probabilties come out as 80% erupt, 20% maintain composure.
Now let's assume that Kavanaugh sincerely believes he is innocent. He didn't have to break the demeanor of a federal Judge, he could have responded calmly. Red says, yes, but the affront against him is so outrageous that any innocent person might have reacted this way. They give such a person 50% chance of erupting. Blue says, well, we allow that Kavanaugh is a partisan hothead, but still, a federal Judge who truly believes in his own innocence should be more able to contain himself. A person who truly believes in his own innocence would maintain composure with probability 70%.
Running the Numbers
Plugging in these assumptions and running the calculation, through Bayes' Rule we find:
Red Conclusion - Each Tesimony Alone
Given Ford's testimony alone, Red believes Kavanaugh is innocent with probability 99%.
Given Kavanaugh's testimony alone, Red believes Kavanaugh is innocent with probability 98.7%.
Blue Conclusion - Each Testimony Alone
Given Ford's testimony alone, Blue would believe Kavanaugh is guilty with probability 67%.
Given Kavanaugh's testimony alone, Blue would believe that Kavanaugh is
guilty with probability 36%
The calculation changes when we put both testimonies together. We must consider both Ford's success at getting a chance to testify, and Kavanaugh's angry reaction. Let's assume that these are independent. Ford spoke first and did not know what Kavanaugh's testimony would be like. Kavanaugh said he did not watch Ford's testimony.
p( KG | FTC,KTA ) = p( FTC,KTA | KG ) * p( KG ) / p( FTC,KTA )
p( FTC,KTA ) = p( FTC,KTA | KG ) * p( KG ) + p( FTC,KTA | KI ) * p( KI )
= p( FTC | KG ) * p( KTA | KG ) * p( KG ) + p( FTC | KI) * p( KTA | KI ) * p( KI )
"FTC,KTA" means that Ford's testimony was emotional and calm AND Kavanaugh's testimony was emotional and angry. We consider this in relation to the possibility that Kavanaugh could have been calm in his demeanor.
Surprisingly, from the Blue point of view, the combined testimony is much harder on Kavanaugh than either testimony alone.
Red Conclusion - Both Testimonies
Given Ford's and Kavanaugh's testimony together, Red concludes that Kavanaugh is innocent with probability 98.7%
Blue Conclusion - Both Testimonies
Given Ford's and Kavanaugh's testimony together, Blue concludes that Kavanaugh is guilty with probability 83%
Discussion
Why are the conclusions so different? There are three main reasons.
First, Red gives significant credence to a conspiratorial view of Ford and the Democrats. If it is 50% possible that Ford is lying and the Democrats conspired to get her into the hearing, then that discounts the value of her testimony. Conversely, Blue believes that the only way she could have gone through with this is that she is really deeply sincere. Moreover, she is an intelligent, got-it-together person who is unlikely to have just made up this episode in her head.
The second reason for the difference is Kavanaugh's own reaction. To the Red viewpoint, it is perfectly natural for a falsely accused person to blow up, even if he is a federal judge. To the Blue viewpoint, this smells like a caged animal reacting to getting caught, and if he really believed in his own innocence, he would be cooperative in persuading the Democrats that they are simply wrong. If Kavanaugh had reacted calmly, then under the other assumptions, the Blue conclusion would have been only 47% that he is actually guilty.
Another important difference between Red and Blue are their differing prior probability that Kavanaugh could have committed the assault he is accused of. Red thinks it extremely unlikely, Blue thinks it's unlikely but not out of the question.
As you move the sliders around, you might find surprising interactions. Some factors matter a lot under some assumptions, and very little under others. If we believe the model captures some essential elements of the situation, then the math has something to teach us!
Start with your honest assessments of the conditional probabilities of beliefs and behaviors given assumed preconditions, and see where that lands. Then, like any normal irrational human does, adjust the parameters to reason backward from the conclusion you passionately believe anyway, according to the tribe you are in.