# Math Helps Us Decide

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 an online 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. The Bayes’ model shows how they weigh against each other to contribute to a final conclusion.

The detailed model behind the Bayes’ Rule calculator is described here. The full story is too long for this post, but here are some highlights.

Core Issues

• Did Kavanaugh assault Ford or not? (Yes or No)
• What do Kavanaugh and Ford believe today about what happened? (Sincere belief vs. lying?)
• In what manner did they testify in the Senate hearing? (Testify or not, Calm vs. Angry?)

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’ states of mind today, whether they are sincere or lying. The only thing we observe directly is their testimony. Bayes’ Rule lets us make inferences about the unobserved variables 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.

• KG = Kavanagh is Guilty of assaulting Ford in 1982.
• KI = Kavanagh is Innocent.
• FBG = Ford sincerely Believes that Kavanaugh is Guilty of assaulting her.
• FBI = Ford actually Believes Kavanaugh is Innocent.
• KBG = Kavanaugh secretly Believes that he assaulted Ford (Guilty).
• KBI = Kavanaugh sincerely Believes that he did not assault Ford (Innocent), either because he actually did not do it, or else he blacked out, forgot, or put it out of his memory.
• FTC = Dr. Ford Testifies, emotional yet Calm.
• FNT = Dr. Ford does Not Testify, either because she does not come forward, or because the political mechanations do not allow her to testify. Even though this is historically not the case, it is a counterfactual we must consider. It could have turned out this way.
• KTA = Judge Kavanaugh Testifies in an emotional Angry manner.
• KTC = Judge Kavanaugh Testifies in a Calm and collected manner. Even though this is historically not the case, it is another counterfactual we should consider. He could have testified differently.

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.

Here’s one such table.

The way to read the table is like this. Consider the upper left cell. 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 the tables form a causal chain, the numbers we assign to the cells represent conditional probabilities. That’s what the notation means.
p( B | A ) means the probability of B being true when A is true. (B might be true even if A is not true).

Other tables pertain to other parts of the causal reasoning chain.

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 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 ).

Bayes’ Rule Inference

This brings us to the question, “Is Kavanaugh guilty or innocent?,” given not
only our independent assumptions 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.

Plugging in Numbers

The Bayes’ Rule Calculator lets you adjust assumptions using sliders. Here is one slice of the conditional probabilities that a Red-thinking person and a Blue-thinking person might plausibly assign.

Linking Alleged Assault to Ford Belief State

What is the probability that Ford will believe that Kavanaugh assaulted her or not, given the presuppositions that he did or did not commit the act?

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.

The other tables are filled out in a similar way, translating our understanding of how the world works into conditional probabilities for this situation. Expressed in this way, the causal chain allows us to focus on, discuss, and debate particular elements of our arguments and beliefs, and then later calculate how they all unfold in a final conclusion.

Running the Numbers

I took a guess at the assumptions a Red-thinking person and a Blue-thinking person would make. Examine them by clicking the “Red Assumptions” and “Blue Assumptions” buttons on the calculator. Your numbers may vary. Plugging in these assumptions, through Bayes’ Rule we find:

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.

A third 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.

Try it for yourself.

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.

--

--

Eric Saund is a researcher in Cognitive Science and a consultant in AI and Machine Learning.