samedi 17 août 2013

What are the odds that a person is a rapist, given that they are accused of rape?

The argument has been made elsewhere on these forums recently that since most rape accusations are true, that most people accused of rape are therefore rapists. It sounds trivially obvious, but educated skeptics know that this is a kind of maths which humans are very bad at doing in their heads.



Please note that this thread is for a scientific and mathematical approach to this question. Ideological approaches are off-topic. Start your own thread if you wish to discuss the matter from an ideological point of view.



Also this is all a bit rough and ready. I'm aiming for a very loose ballpark figure not something terribly rigorous.



Formally, the question is what posterior probability we should allocate to the hypothesis "X is a rapist", after they have been accused. We can't answer this question unless we have values for three other probabilities: The probability that a person will be accused given they are a rapist, the prior probability they are a rapist, and the prior probability they will be accused of rape.



A figure of 6% of men being rapists seems reasonably uncontroversial, so I'll go with that for now. P(rapist)=0.06.



The probability that a rapist will be accused of rape is much more problematic, and has a major impact on the posterior probability. This is where I'm having trouble settling on a figure to plug in. Lots of rapes go unreported, but then again lots of rapists are serial offenders. As best I can puzzle things out from this source it looks like about one third of sexual assaults are reported in Australia, but this wikipage says a 2007 report said 75% to 95% go unreported. As a very rough estimate if we assume each sexual assault has a 25% chance of being reported and the average offender commits four such crimes, the chance they will be accused at some point is close enough to 70%. So P(accused|rapist) we could call 0.7.



The probability that a person will be accused of rape regardless of whether they are a rapist is again tough to estimate.



Studies give widely varying rates of false rape reporting, but the lower figures of 2% for Australia and 8% for the FBI seem better supported than the higher values. For now I'll split the difference and call it 5%, so using US figures of 90,000 incidents per year (source) that would mean in a typical modern year there will be 90,000 reports of which maybe 4500 (5%) are false. Let's make a (false) simplifying assumption that we're only looking at men in the USA, and a (false) simplifying assumption that false rape reports are distributed at random, and say that means that of the 150 million men in the USA, each year the probability they will be falsely accused of rape is p=0.00003. Let's say they're at risk of being accused for fifty years of their life, so the lifetime risk of being accused unjustly is p=0.0015 or so, compared to p=0.7 for actual sex offenders. (I'm being lazy and ignoring the chance that a real rapist will by chance be unfairly accused).



So the lifetime risk that you will be accused of rape is 70% for the 6% who are rapists, and 0.15% for the 94% who are not, or 0.042 plus so for the general population it's 4.31% or 0.0431.



So p(rapist|accused)=(p(accused|rapist)*p(rapist))/p(accused), or (0.7*0.06)/0.0431= 0.9745.



So unless I've screwed up a decimal place or something, as a very rough bit of Bayesian updating it seems reasonable to update p(rapist) for any given person accused of rape from a prior probability of about 0.06 to a posterior probability of 0.9745, or about a 97.5% chance they did it.



This figure seems likely to be fairly robust - all of my figures are questionable to a greater or lesser extent, but the probability shift is so large they would have to be very far off to meaningfully change the outcome.



Again, assuming a random distribution of accusations it seems incredibly likely that a person who gets accused twice independently did it. False rape accusations are rare enough, and distributed amongst such a large non-offending population, that getting unfairly accused twice is very highly improbable.



So the accusation that, for example, one individual person had been the target of six separate rape accusations would be incredibly powerful evidence they were a sex offender. Of course this assumes that it turned out that they really had been accused six separate times, as opposed to someone with an axe to grind claiming six such accusations had been made.



Granted if we start assuming that all accused people are guilty that would lead to all sorts of problems, including an increase in the false accusation rate. However that hasn't happened yet, so as of now it seems like false skepticism to not believe that a person credibly accused is probably a rapist, and that someone accused multiple times is very highly likely to be guilty.



I'm open to the possibility I've screwed something up, since this is just a rough estimate I dashed off this afternoon. If so, hopefully someone will point it out.





via JREF Forum http://forums.randi.org/showthread.php?t=263893&goto=newpost

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