## Black Swans, the “Ludic Fallacy”, and Bayesian inference

A few days ago, I finished reading The Black Swan by Nicholas Taleb, which goes in-depth on topics such as judgment under uncertainty and the issues relating to unrealistic models which deliberately ignore unlikely but possibly highly influential phenomena in order to stay simple. Taleb emphatically argues against the use of the Gaussian bell curve, or the GIF (“great intellectual fraud”), as he likes to call it, pointing out that it forecasts events several standard deviations from the norm as extremely unlikely. He points out that an event 4 SD away is twice as likely as 4.15 SD, and that “the precipitous decline in odds of encountering something is what allows you to ignore outliers. Only one curve can deliver this decline, and it is the bell curve (and its nonscalable siblings). Nassim instead expounds scalable “Mandelbrotian” curves, which, like all things Mandelbrotian, are fractal – The speed of the decrease of odds as one moves from the mean is constant, not declining. So the odds of having a net worth of over 8 million pounds is 1 in 4,000, for higher than 16 million pounds it’s one in 16,000, for 32 it’s 1 in 64,000, etc. So not only does the Mandelbrotian curve put more importance on outliers (“Black Swans”, or unpredictable but highly influential events such as stock market crashes that Gaussian models miss), but any small portion of the graph resembles the larger curve in a fractal sort of way.

To explain this better, consider the average human male heights, which are in fact described by the standard bell curve. If you make a curve of the heights of people taller than 7 feet tall, that no longer resembles the bell curve (it just looks like a tapered edge of it). On the other hand, whether you look at the wealth of everyone or only those earning over 1 million pounds a year, the Mandelbrotian curve resembles itself. This is not to say that the Mandelbrot curve should be used everywhere – the bell curve is perfectly adequate to describe human height. However, it falls short when applied ubiquitously to financial forecasting, which is far less predictable and far more susceptible to outliers.

Now onto the Ludic Fallacy – coined by Nick Taleb himself. The Ludic Fallacy pertains mainly to what Taleb deems “nerds” – people who work with models and are only capable of thinking “inside the box,” and mistakes they make when they try to impose the parameters of theoretical games and models onto the real world. Kind of like economists assuming that “everyone is rational” in order to work their models – BIG MISTAKE. Note: If Taleb could have it his way, economists have been out of business long ago. So I don’t want to write out the entire explanation of the Ludic Fallacy but it can be found here. Read it before going on.

This is where Bayesian inference comes in. Bayesian inference gives a probabilistic framework for the scientific method and making decisions about whether or not to accept a hypothesis given data. A fundamental aspect of Bayesian inference is updating your beliefs in light of new evidence. Essentially, you start out with a prior belief and then update it in light of new evidence. An important aspect of this prior belief is your degree of confidence in it.

A similar question came up on an old AP Biology exam, and although it superficially intends to teach the opposite lesson, and seems to have the “wrong” correct answer, all is corrected by Bayesian statistics. The question reads something like, “A woman has given birth to 5 boys, and is expecting a 6th child. What is the probability that the 6th child is also a boy?” Now, unsuspecting “Fat Tonys” are supposedly “tricked” into choosing D) 5/6 whereas the “Dr. Johns” know that the answer is, truly, B) 1/2. The lesson is supposed to be that the next outcome isn’t affected by the previous ones, but in light of the ludic fallacy and Bayesian inference, I think the answer is a bit more nuanced than that (but 1/2 is still the correct answer).

The reason is this – 5 boys in a row isn’t THAT unlikely – 1/2^5, or 1/32. In addition, your prior belief about the likelihood of getting a boy better be about 51% (a little higher than 1/2, to be exact). In addition, as your prior “coin-toss model” you have this outcome played over literally billions of times. No matter how many boys this woman has isn’t going to significantly altered my confidence that roughly half of human babies are boys.

You might object – are some women more likely to have baby boys? Fair point. I think that’s going a little too in-depth for multiple choice question 4 on the 1998 Biology exam or whatever, but just for fun I looked into it – my “research” proved…inconclusive. Moral of the story is, prior outcomes don’t affect future outcomes, only your belief about them, and in this case prior outcomes shouldn’t even affect your belief about future outcomes.