Bayes' Theorem applied to the FBI findings on Brett Kavanaugh

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Question: What is the probability that Brett Kavanaugh is guilty of the allegation of sexual assault by Dr. Christine Blasey Ford, given the findings of the FBI’s report? How should someone’s estimate of this probability have changed once the report was released?

Conclusion: Unless you are more than about 80% sure that the FBI’s investigation would have found corroborating evidence if Kavanaugh were guilty, your new estimate of the probability of Kavanaugh’s guilt should not have changed much after the investigation (fact). Considering the identities of the interviewees, it seems crazy to put that first part anywhere close to 80% (opinion).

How Bayes' Theorem helps: The question headlining this report is tough to address directly, because it is asking about the probability of a prior event (the allegation) given a more recent event (the report). It's more practical to switch this around, to consider the probability that the later event would follow the former. Bayes' Theorem lets you cast the problem this way. We get to examine what the FBI actually did, and consider how much we should update our estimates of Kavanaugh's guilt based on how thorough we think the investigation was. 

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In this context, Bayes’ Theorem states:

$P(KG|F)=\frac{P(F|KG)}{P(F)}*P(KG)$

$KG$ stands for "Kavanaugh guilty", and $F$ stands for "Findings" (of the report-no corroborating evidence). $P(KG|F)$ is the probability that Kavanaugh is guilty, given the FBI's findings, $P(F|KG)$ is the probability that the FBI report would produce no corroborating evidence, if Kavanaugh were guilty, $P(KG)$ is the probability that Kavanaugh was guilty absent information from the report, and $P(F)$ is the probability that the report would produce no corroborating evidence, regardless of whether Kavanaugh was guilty.

Bayes’ theorem helps us estimate what we are interested in (left-hand side: “Is Kavanaugh guilty, given the findings of the report?”) in terms of things we can estimate more easily. Let’s have at it:

~$P(F|KG)$: If Kavanaugh were guilty, what is the probability that the FBI report would turn up no new corroborating evidence? The FBI interviewed six people with regards to Dr. Ford’s allegations – four of Kavanaugh’s friends (Mark Judge, PJ Smyth, Timothy Gaudette, and Christopher Garrett), one of Ford’s friends (Leland Keyser), and a lawyer for an unnamed one of these people. Let’s just ignore the lawyer. Now, our question boils down to the probability that any of the five individuals interviewed as witnesses would have offered corroborating evidence, if Kavanaugh had been guilty.

In my opinion, this probability is low, though we will still consider it going anywhere from 0 to 1. Kavanaugh’s friends would most likely not have offered corroborating evidence for many different reasons: perhaps they did not actually witness or know about the assault, or maybe they did witness the assault, but would be harming themselves by admitting it, due to either their own participation in it or other assaults, or they may feel uncomfortable from having not reported it long ago. They may also feel allegiance towards Kavanaugh. Keyser is more likely to have reported the assault if she had witnessed it, although Ford only ever claimed that Keyser was present at the party where the allegations occurred, not that she actually witnessed the assault itself. Therefore, it seems most likely that Keyser would just not have actually witnessed the assault, even if it did occur, and therefore not have offered corroborating evidence. After all, a sexual assailant would probably want to be fairly certain that witnesses do not exist.

~$P(KG)$ : What is the probability of Kavanaugh’s guilt, without the findings of the FBI report? We will leave this completely subjective-someone might put this estimate anywhere between 0 and 1.

~$P(F)$: What is the probability that no new corroborating evidence would be acquired in the FBI report, regardless of whether Kavanaugh is guilty? This has to be broken into two possibilities: 1) Kavanaugh is guilty, and 2) Kavanaugh is not guilty:

~$P(F) = P(F|KG)*P(KG)+P(F|KNG)*(1-P(KG))$

We’ve already addressed all the terms in the above, except for $P(F|KNG)$, the probability that the report would produce no new corroborating evidence, given Kavanaugh’s innocence (~$NG$="Not Guilty"). This should be very close to 1-it would only be less certain if we thought one of the five interviewees could be maliciously out to get Kavanaugh by lying. Given that four were Kavanaugh’s friends, and Keyser does not appear closely linked to Ford or Kavanaugh, this is unlikely. We will call it 1 for now. The above simplifies to:

~$P(F)=1+(P(F|KG)-1)*P(KG)$

Now we have what we are interested in, $P(KG|F)$, in terms of things we were able to think through:

~$P(KG|F)=\frac{P(F|KG)}{1+(P(F|KG)-1)*P(KG)}*P(KG)$

This depends on only two things:

1. $P(KG)$, the probability that Kavanaugh is guilty without the FBI report, which we are treating as completely subjective. Peoples’ opinions on this vary all the way between 0 and 1.

2. $P(F|KG)$, the probability that the FBI report would produce no new corroborating evidence, given Kavanaugh’s guilt. I made the argument that this is probably closer to 1, but we will actually consider the whole span anyways. While my argument on the value of $P(F|KG)$ is not completely objective, I still assert that it is more objective than anyone’s estimate of $P(KG)$, since the report is a more recent and transparent event than the alleged assault. More importantly, the estimate of $P(KG|F)$ given one’s subjective estimates is completely objective: in other words, if you violate the equation above, you are contradicting yourself.

The equation above should be simple for you to visualize. Just kidding. Here’s a plot:

Start from the left side. What did you think was the probability that Kavanaugh was guilty before the FBI investigation? Find your line on the y-axis-I put one at every 0.1, or 10%, increment. Now, consider the probability that the FBI report would find corroborating evidence, if Kavanaugh were guilty, knowing who was interviewed. That’s the x-axis, which is actually $1-P(F|KG)$. Follow your line until you reach your estimate of this on the x-axis, and now your place on the y-axis, $P(KG|F)$, tells you your new estimate of the probability that Kavanaugh is guilty, given what you know about the FBI’s report.

Notice that these curves do not drop substantially until $P(F|KG)$ is somewhere in the range of 80%. This means that your estimate of the probability of Kavanaugh’s guilt should not change substantially unless you are at least 80% sure that corroborating evidence would be found by the FBI’s investigation, knowing who the interviewees were, and assuming that Kavanaugh were guilty. I don’t think we’re even close to 80%-it’s probably more like 10%, and down in that region, the new estimate of Kavanaugh’s guilt has hardly changed at all. Even out to 50%, it doesn’t really change much, regardless of where you started.

 Anyways, if you look at this plot for long enough, it starts to seem pretty likely that the FBI’s investigation shouldn’t have actually changed our viewpoints on the situation much. Unfortunately, Republicans in the Senate want you to think otherwise. If they were to read this report and understand it, perhaps they would see that they are logically contradicting themselves.

Am I biased? Of course. Everyone is. As far as Senators go, I most closely align with Bernie Sanders. Feel free to use Bayes’ Theorem to estimate the probability that I’m full of shit.