**Note: After you read the Woody Allen example, please read the note below it, which describes how I screwed up the analysis!**

One of the most important concepts in science, and sometimes in life, involves something called *Bayesian* Probability or *Bayes* Theorem. Since you are reading a sabermetric blog, you are likely at least somewhat familiar with it. Simply put, it has to do with conditional probability. You have probably read or heard about *Bayes* with respect to the following AIDS testing hypothetical.

Let’s say that you are not in a high risk group for contracting HIV, the virus that causes AIDS, or, alternatively, you are randomly selected from the adult U.S. population at large. And let’s say that in that population, one in 500 persons is HIV positive. You take an initial *ELISA *test, and it turns out positive for HIV. What are the chances that you actually carry the disease?

The first thing you need to know is the false positive rate for that particular test. It is also around one in 500. We’ll ignore the fact that there are better, more accurate tests available or that your blood specimen would be given another test if it had a positive *ELISA*. You might be tempted to think that your chances of carrying the virus is 99.8% or one minus .002, where .002 is the one in 500 false positive rate.

And you would be wrong. Enter *Bayes.* Since you only had a 1 in 500 chance of being HIV+ going in, there is a prior probability which must be added “to the equation.”

To understand how this works, and to avoid any semi-complex *Bayesian* formulas, we can frame the analysis like this:

In a population of 500,000 persons, there would be 1,000 carriers, since we specified that the HIV rate was one in 500. All of them would test positive, assuming a zero false-negative rate. Among the 499,000 non-carriers, there would be 998 false positives (a one in 500 chance).

So in our population of 500,000 persons, there are 1,998 positives and only 1,000 of these truly carry the virus. The other 998 positives are false. If you are selected from this population, and have a positive *ELISA* test, you naturally have a 1,000 in 1,998, or around a 50% chance of having the disease. That is a far cry from 99.8%, and should be somewhat comforting to anyone who fails an initial screening. That is basically how *Bayes* works, although it can get far more complex than that. It also applies to many, many other important things in life, including the guilt or innocence of a defendant in a criminal or civil prosecution, which I will address next.

Another famous, but less well-known, illustration of *Bayes* with respect to the criminal justice system,* *involves a young English woman named Sally Clark who was convicted of killing two of her children in 1999. In 1996, her first-born son died, presumably of Sudden Infant Death Syndrome (SIDS). In 1998, she gave birth to another boy, and he too died at home shortly after birth. She and her husband were soon arrested and charged with the murder of the two boys. The charges against her husband were eventually dropped.

Sally was convicted of both murders and sentenced to life in prison. By the way, she and her husband were affluent attorneys in England. At her trial, the following statistical “evidence” was presented by a pediatrician for the prosecution:

He testified that there was about a 1 in 8,500 chance that a baby in that situation would die of SIDS and therefore the chances that both of her children would perish from natural causes related to that syndrome was 1/8500 times 1/8500, or 1 in 73 million. Sally Clark was convicted largely on the “strength” of the statistical “evidence” that the chance of those babies both dying from SIDS, which was the defense’s assertion, was almost zero.

First of all, the 1 in 73 million might not be accurate. It is possible, in fact likely, according to the medical research, that those two probabilities are not independent. If you want to know the chances of two events occurring, multiplying the chances of one event by the other is only proper when the probability of the two events are independent – Stats 101. In this case, it was estimated by an expert witness for the defense in an appeal, that if one infant in a family dies of SIDS, the chances that another one also dies similarly is 5 to 10 times higher than the initial probability.

So that reduces our probability to between one in 15 million and one in 7 million. In addition, the same expert witness, a Professor of Mathematics who studied the historical SIDS data, argued that the 1 in 8,500 was really closer to 1 in 1,300 due to the gender of the Clark babies and other genetic and environmental characteristics. If that number is accurate, that brings us down to 1 in 227,000 for the chances of her two boys both dying of SIDS. While a far cry from 1 in 73 million, that is still some pretty damning evidence, right?

Wrong! That 1 in 227,000 chance of dying of SIDS, or the inverse, a 99.99955 chance of dying from something other than SIDS, like murder, is like our erroneous 99.8% chance of having HIV when our initial AIDS test is positive. In order to calculate the true odds of Mrs. Clark being guilty of murder based solely on the statistical evidence, we need to know, as with the AIDS test, what the chances are, going in, before we know about the deaths, that a woman like Sally Clark would be a double murderer of her own children. That is exactly the same thing as us needing to know the chances that we are an HIV carrier before we are tested, based upon the population we belong to. Remember, that was 1 in 500, which transformed our odds of having HIV from 99.8% to only 50%.

In this case, it is obviously difficult to estimate that *a priori* probability, the chances that a woman in Sally Clark’s shoes would murder her only two children back to back. The same mathematician estimated that the chances of Sally Clark being a double murderer, knowing nothing about what actually happened, was much rarer than the chances of both of her infants dying of natural causes. In fact, he claimed that it was 4 to 10 times rarer, which means that out of all young, affluent mothers with two new-born sons, maybe 1 in a million or 1 in 2 million would kill both of their children. That does not seem like an unreasonable estimate to me, although I have no way of knowing that off the top of my head.

So, as with the AIDS test, if there were a population of one million similar women with two newly born boys, around 4 of them (1 in 227,000) would suffer the tragedy of back-to-back deaths by SIDS, and only ½ to 1 would commit double infanticide. So the odds, based solely on these statistics, of Sally Clark being guilty as charged was around 10 to 20%, obviously not nearly enough to convict, and just a tad less than the 72,999,999 to 1 that the prosecution implied at her trial.

Anyway, after spending more than 3 years in prison, she won her case on appeal and was released. The successful appeal was based not only on the newly presented *Bayesian* evidence, but on the fact that the prosecution withheld evidence that her second baby had had an infection that may have contributed to his death from natural causes. Unfortunately, Sally Clark, unable to deal with the effects of her children’s deaths, the ensuing trial and incarceration, and public humiliation, died of self-inflicted alcohol poisoning 4 years later.

Which brings us to our final example of how *Bayes* can greatly affect an accused person’s chances of guilt or innocence, and perhaps more importantly, how it can cloud the judgment of the average person who is not statistically *savvy*, such as the judge and jurors, and the public, in the Clark case.

Unless you avoid the internet and the print tabloids like the plague, which is unlikely since you’re reading this blog, you no doubt know that Woody Allen was accused around 20 years ago of molesting his adopted 7-year old daughter, Dylan Farrow. The case was investigated back then, and no charges were ever filed. Recently, Dylan brought up the issue again in a NY Times article, and Allen issued a rebuttal and denial in his own NY Times op-ed. Dylan’s mother Mia, Woody Allen’s ex-partner, is firmly on the side of Dylan, and various family members are allied with one or the other. Dylan is unwavering in her memories and claims of abuse, and Mia is equally adamant about her belief that their daughter was indeed molested by Woody.

I am not going to get into any of the so-called evidence one way or another or comment on whether I think Woody is guilty or not. Clearly I am not in a position to do the latter. However, I do want to bring up how *Bayes* comes into play in this situation, much like with the AIDS and SIDS cases described above, and how, in fact, it comes into play in many “he-said, she-said” claims of sexual and physical abuse, whether the alleged victim is a child or an adult. If you have been following along so far, you probably know where I am going with this.

In cases like this, whether there is corroborating evidence or not, it is often alleged by the prosecution or the plaintiff in civil cases, that there is either no reason for the alleged victim to lie about what happened, or that given the emotional and graphic allegations or testimony of the victim, especially if it is a child, common sense tells us that the chances of the victim lying or being mistaken is extremely low. And that may well be the case. However, as you now know or already knew, according to *Bayes*, that is often not nearly enough to convict a defendant, even in a civil case where the burden on the plaintiff is based on a “preponderance of the evidence.”

Let’s use the Woody Allen case as an example. Again, we are going to ignore any incriminating or exculpatory evidence other than the allegations of Dylan Farrow, the alleged victim, and perhaps the corroborating testimony of her mother. Clearly, Dylan appears to believe that she was molested by Woody when she was seven, and clearly she seems to have been traumatically affected by her recollection of the experience. Please understand that I am not suggesting one way or another whether Dylan or anyone else is telling the truth or not. I have no idea.

Her mother, Mia, although she did not witness the alleged molestation, claims that, shortly after the incident, Dylan told her what happened and that she wholeheartedly believes her. Many people are predicating Allen’s likely guilt on the fact that Dylan seems to clearly remember what happened and that she is a credible person and has no reason to lie, especially at this point in her life and at this point in the timeline of the events. The statute of limitations precludes any criminal charges against Allen, and likely any civil action as well. I would assume however, that hypothetically, if this case were tried in court, the emotional testimony of Dylan would be quite damaging to Woody, as it often is in a sexual abuse case in which the alleged victim testifies.

Now let’s do the same *Bayesian* analysis that we did in the above two situations, the AIDS testing, and the murder case, and see if we can come up with any estimate as to the likely guilt or innocence of Woody Allen and perhaps other people accused of sexual abuse where the case hinges to a large extent on the credibility the alleged victim and his or her testimony. We’ll have to make some very rough assumptions, and again, we are assuming no other evidence, for or against.

First, we’ll assume that the chances of the victim and perhaps other people who were told of the alleged events by the victim, such as Dylan’s mother, Mia Farrow, lying or being delusional are very slim. So we are actually on the hypothetical prosecution or plaintiff’s side. ‘How is it possible that this victim and/or her mother would be lying about something as serious and traumatic as this?’

Now, even common sense tells is that it is *possible*, but not likely. I have no idea what the statistics or the assumptions in the field are, but surely there are many cases of fabrication by victims, false repressed memories by victims who are treated by so-called clinicians who specialize in repressed-memories of physical or sexual abuse, memories that are “implanted” in children by unscrupulous parents, etc. There are many documented cases of all of the above and more. Again, I am *not* saying that this case fits into one of these profiles and that Dylan is lying or mistaken, although clearly that is possible.

Let’s put the number at 1 in a 100 in a case similar to this. I’m not sure that any reasonable person could quarrel too much with that. I could easily make the case that it is higher than that. The population that we are talking about is this: First we have a 7 year-old child. The chances that the recollections of a young child, including the chances that those recollections were planted or at least influenced by an adult, might be faulty, have to be greater than that of an adult. The fact that Woody and Mia were already having severe relationship problems and in a bitter custody dispute also increase the odds that Dylan might have been “coached” or influenced in some manner by her mother. But I’ll leave the odds at 100-1 against. So, Allen is 99% guilty, right? You already know that the answer to that is, “No, not even close.”

So now we have to bring in *Thomas Bayes* as our expert witness. What are the chances that a random affluent and famous father like Woody Allen, again, not assuming anything else about the case or about Woody’s character or past or future behavior, would molest his 7-year old daughter? Again, I have no idea what that number is, but we’ll also say that it’s 100-1 against. I think it is lower than that, but I could be wrong.

So now, in order to compute the chances that Allen, or anyone else in a similar situation, where the alleged victim is a very credible witness – like we believe that there is a 99% chance they are telling the truth – is guilty, we can simply take the ratio of the *prior *probability of guilt, assuming no accusations at all, to the chances of the victim lying or otherwise being mistaken. That gives us the odds that the accused is guilty. In this case, it is .01 divided by .01 or 1, which means that it is “even money” that Woody Allen is guilty as charged, again, not nearly enough to convict in a criminal court. Unfortunately, many, perhaps most, people, including jurors in an actual trial, would assume that if there were a 99% chance that the alleged victim was telling the truth, well, the accused is most likely guilty!

**Edit: As James in the comments section, Tango on the Book blog, and probably others, have noted, I screwed up the Woody Allen analysis. The only way that Bayes would come into play as I describe would be if we assumed that 1 out of 100 random daughters in a similar situation would make a false accusation against a father like Woody. That seems like a rather implausible assumption, but maybe not – I don’t really know. In any case, if that were true, then while my Bayesian analysis would be correct and it would make Allen have around a 50% chance of being guilty, the chances that Dylan was not telling the truth would not be 1% as I indicated. It would be a little less than 50%.**

**So, really, the chances that she is telling the truth is equal to the chances of Allen being guilty, as you might expect. In this case, unlike in the other two examples I gave, the intuitive answer is correct, and Bayes is not really implicated. The only way that Bayes would be implicated in the manner I described would be if a prosecutor or plaintiff’s lawyer pointed out that 99% of all daughters do not make false accusations against a father like Woody, therefore there is a 99% chance that she is telling the truth. That would be wrong, but that was not the point I was making. So, mea culpa, I screwed up, and I thank those people who pointed that out to me, and I apologize to the readers. **

**I should add this:**

**The rate of false accusations is probably not significantly related to the rate of true accusations or the actual rate of abuse in any particular population. In other words, if the overall false accusation rate is 5-10% of all accusations, which is what the research suggests, that percentage will not be nearly the same in a population where the actual incidence of abuse is 20% or 5%. The ratio of true to real accusations is probably not constant. What is likely somewhat constant is the percentage of false accusations as compared to the number of potential accusations, although there are surely factors which would make false accusations more or less likely, such as the relationship between the mother and father.**

**What that means is that the extrinsic (outside of the accusation itself) chance that an accused person is guilty is related to the chances of a false accusation. If in one population the incidence of abuse is 20%, there is probably a much lower chance that a person who makes an accusation is lying, as compared to a population where the incidence of abuse is, say, 5%.**

**So, if an accused person is otherwise not likely to be guilty but for an accusation, a prosecutor would be misleading the jury if he reported that overall only 5% of all accusations were false therefore the chance that this accusation is false, is also 5%.**

**If that is hard to understand, imagine a population of persons where the chance of abuse is zero. There will still be some false accusations in that population, and since there will be no real ones, the chances that someone is telling the truth if they accuse someone is zero. The percentage of false accusations is 100%. If the percentage of abuse in a population is very high, then the ratio of false to true accusations will be much lower than the overall 5-10% number.**

I wonder how Woody Allen’s marriage to Soon-Yi Previn might shift the prior against him.

I think it might shift it in both directions or perhaps only more toward innocence. First, I’m not sure that falling in love with or marrying someone that young or someone who is the daughter of your lover has much to do with whether someone is likely to molest a 7 year old. The other side of the coin is that Mia despised Woody because of that. That makes her more likely to fabricate the accusations against him.

I concede that your point regarding “the other side of the coin” is valid and I would agree it makes the marriage a difficult-to-evaluate prior. However, with regards to the point I was trying to make (and did not articulate well), I was only suggesting that Woody may be a person who struggles to respect traditional boundaries and not that there is a strong correlation between molesting children and a desire for a young wife.

It would be unusual (though obviously not unheard of) for a 57 year old to start molesting children at that age. The fact that there were no allegations before or after this alleged instance suggests that Allen was comparatively unlikely to be a child molester.

Sure, I agree. As I said, there are many things about the case, some we know and some we don’t know, that would implicate a greater or less chance of guilt. I did not want to get into that.

I agree that the “boundaries” thing might make him more likely to have molested his daughter, although not being an expert on that sort of thing, my opinion on that is not worth much.

Good intro, but I’m not buying the Woody Allen part, as the accused is not randomly selected in this example.

You are correct and I screwed up. I will edit the post and make the correction.

MGL, could you help me work through one or two examples.

Take the comment of Jack Clark saying Pujols took PEDs.

Quick assumptions:

1/5 players were taking PEDs at the time. = P(H)

Jack Clark was 50% sure he saw Pujols (could be wrong). = P(D|H)

5% chance a person would wrongly convict another person of PEDs = P(D| notH)

Using:

P(H | D) = P(H) P(D | H) / P(H) P(D | H)+P(notH) P(D | notH)

P(H | D) = (.2 * .5)/(.2*.5+(1-.2)*.05) = 71%

So using the above assumptions, so the chance of Pujols using steroids jumped from 20% to 71% once Jack Clack made his assumption.

I am not 100% clear on your assumptions. Tell me if this is right. I will assume that it is, and work from there:

1/5 players taking PED’s. That’s clear.

Clark 50% sure that he saw Pujols take PED’s. So we are assuming that if he is right, that Pujols definitely took, right? The other 50% of the time that he did not see him, we are making no inference that he took Steroids or not? I ask that because if he did not see him, there is a greater than random (80%) chance he did not take, the assumption being that Clark will occasionally see players take steroids if they do. So that statement is ambiguous. You need to specify exactly what you are implying.

5% chance that a person would wrongly convict another person. I really have no idea what that means. Are you talking about Jack Clark? So, if he did not see him take any steroids, there is a 5% chance that he will say that he did?

Actually, I can’t work through this unless you clear up those statements.

I am working on the problem on my end. I would like to understand the methodology a little better as I think I can use it with injury analysis.

It is just Bayesian math I guess. Your math seems to be correct. I am just not able to translate your words into numbers, because they are not unambiguous enough, as I indicated. At least to me. 😉