While Whoopi Goldberg, Phylicia Rashad and Camille Cosby are still searching for a giant proof, with mass accusers aligning the probability math has swiftly pointed against their favor. On the chart below we show under a range of scenarios, how a Bayesian probabilist would consider the accused's word of innocence, versus that of the accusers. We also show the math is now unfortunately deliberately tilting -not to one real victim- but towards two or more real victims.
We know that in giving benefit to both parties, we must offer some degree of variance against any accusation. Those are fundamental rights we have in the U.S., and it's a bedrock of our criminal justice system. And here we give a sense on what probability is still left over, after splitting this doubt, while simultaneously factoring in the number of accusers on has.
If it were one or two accusers and no recently broadcasted court testimony, then this would all be just uncomfortable. No blog article would've been written. The evidence wouldn't be so compelling from a probabilistic standpoint, to rise above reasonable doubt.
But when there are 25 or 30 or 40 diverse accusers who are all targeting one person at the end, we all must concede -finally- the story is significantly different. We would have to assume that the great majority (>95%) of the accusers are typically liars, and even then the probability of guilt would come to nearly 60%!
The heart-breaking nature of having so many accusers is also that the probabilistic law of numbers begins to severely work against the accused. Even when we reach by assuming that accusers are typically only about 2% honest, or just less than 98% dishonest, we must still begin to sternly consider the likelihood of now not just one -but two or more- real victims.
Now to help explain this illustration immediately above, recall that in the top chart further above that the probability of guilt was shown to be about 60% in the most defendant-friendly scenario given 40 accusers. But with so many accusers, in the bottom illustration immediately above, we decompose this ~60% guilt into the following ranges. For more on Bayes' theory, and multinomial distributions, please use the search feature on the top-right of our blog site.
37% probability of 1 real victim
17% probability of 2 real victims
6% probability of 3 or more real victims
So just less than 40% of the probability of guilt [(17%+6%)/60%], is not for one victim. But meant for two or more victims! It's unfortunate that for opinion to come around on these sorts of cruelties, it first takes so many initial victims to intrepidly come forth, with then what seemed to be non compos mentis accusations. Now their day to see justice played out -and the signal this sends- will come soon enough.
On an aside: this article was enjoyed across multiple networks including influential attorneys, and by both Bloomberg's Ritholtz, and by Abnormal Returns. Also note that after a hundred math articles, this free resource is about to breach one million views on Google+ alone!
We know that in giving benefit to both parties, we must offer some degree of variance against any accusation. Those are fundamental rights we have in the U.S., and it's a bedrock of our criminal justice system. And here we give a sense on what probability is still left over, after splitting this doubt, while simultaneously factoring in the number of accusers on has.
But when there are 25 or 30 or 40 diverse accusers who are all targeting one person at the end, we all must concede -finally- the story is significantly different. We would have to assume that the great majority (>95%) of the accusers are typically liars, and even then the probability of guilt would come to nearly 60%!
The heart-breaking nature of having so many accusers is also that the probabilistic law of numbers begins to severely work against the accused. Even when we reach by assuming that accusers are typically only about 2% honest, or just less than 98% dishonest, we must still begin to sternly consider the likelihood of now not just one -but two or more- real victims.
Now to help explain this illustration immediately above, recall that in the top chart further above that the probability of guilt was shown to be about 60% in the most defendant-friendly scenario given 40 accusers. But with so many accusers, in the bottom illustration immediately above, we decompose this ~60% guilt into the following ranges. For more on Bayes' theory, and multinomial distributions, please use the search feature on the top-right of our blog site.
37% probability of 1 real victim
17% probability of 2 real victims
6% probability of 3 or more real victims
So just less than 40% of the probability of guilt [(17%+6%)/60%], is not for one victim. But meant for two or more victims! It's unfortunate that for opinion to come around on these sorts of cruelties, it first takes so many initial victims to intrepidly come forth, with then what seemed to be non compos mentis accusations. Now their day to see justice played out -and the signal this sends- will come soon enough.
On an aside: this article was enjoyed across multiple networks including influential attorneys, and by both Bloomberg's Ritholtz, and by Abnormal Returns. Also note that after a hundred math articles, this free resource is about to breach one million views on Google+ alone!
You need to attend also to mob behaviour, scapegoating, to the rewards (financial and bragging rights included) available to accusers and to the costlessness of false accusations. You are treating accusations as if they were probabilistically independent but it is entirely possible in these sorts of cases that they are not, indeed, that the chance of a false accusation should increase with every new accusation. It is entirely possible in these sorts of cases, for these sorts of reasons, for 100% of new accusations to be false.
ReplyDeleteThanks much anonymous. This is a point brought up in my Bayesian model through the concave red-line, termed "defendant-friendly dynamic". Each additional accuser can have some degree a mob-crowd impetus, though you put too much emphasis on "sequence". Rightly or wrongly, we have the "overall" truthful RATE come down meaningfully. What's more pertinent here is that with a far greater number of accusers, the "overall" LEVEL of guilt can not suddenly reverse and head back down, towards 0%. That is another incorrect probabilistic analysis, which you are suggesting.
DeleteIncidentally, it now seems as if Ms. Goldberg has finally also come around and agrees with this article:
http://www.people.com/article/whoopi-goldberg-changes-bill-cosby-stance
Don't really have an opinion on the Cosby thing either way - but I wonder - if you ran the same analysis on this case:
ReplyDeletehttps://en.wikipedia.org/wiki/Jovian%E2%80%93Plutonian_gravitational_effect
Would you get the same results? In this case it was *hundreds* of people that claimed they were floating. Surely the statistical analysis would point to this (admitted) hoax being, in fact, absolutely true! Mindblowing.
Thanks much anonymous. Certainly there are some similarities (the Jovian-Plutonian alignment hoax, versus the Bill Cosby accusations) in the mathematical framework, but there are two important differences between these two situations. In the first case of Jovian-Plutonian, 20th century astronomer Patrick Moore led-on listeners by asserting on the radio that gravity reduction would occur at 9:57am. Only after this time did the same audience call in, en masse, to state this gravity reduction they were stimulated to experience. Callers could know one another and hear what other callers before them were saying. In other words, a subset of the captive audience were given specific instructions on what to convey back and immediately did. No one criminally examined these claims over time.
DeleteIn the second case of Bill Cosby, women from all over, and from very different time periods, provided individual testimonies of their private recollections. Their stories weren't all transparently provided, as some sort of playbook for others on the radio. Their traumatic stories also had sensational characteristics that were found to be common them, which perhaps ONLY a real victim and their perpetrator would know. None of these stories, upon investigation were shown to be false. And Bill Cosby didn’t encourage these accusers to feel as if they were assaulted, but instead provided many counter-arguments against them. So there is simply greater independence among the stories in this latter situation versus the one you note above.
Would probability theory have said that the Jovian-Plutonian gravitational phenomenon would have a very low probability of being true? In fact, maybe. It certainly would never have said it was 100% confident to be true. Part of a statistic such as ~60% or some other middling value, is it tells a degree of uncertainty and doubt. It shows in any of these situations what the otherwise, hard-to-define relative odds are about the true disposition of a variable (e.g., someone's truthfulness, or someone's guilt).