Bayes theorem deals with conditional probabilities. The simple mathematical form is as follows: P(B|A) = (P(A|B)*P(B)/(P(A))

One common application of this theorem is, of course as the equation suggests, to find the probability of B given A has occurred when you already know the probability of A given B. For example, if we know that 80% of all parrots have green feathers and red beaks, what is the probability that the bird that you spotted with green feather and red beaks is actually a parrot.

I would suppose that this computation of Bayes theorem is deeply ingrained in humans to the point of involuntary computation. After all, the P(Dangerous Animal| Sound=Growl) needs to be computed instantaneously once a growl is heard. Humans who could do this better survived and those who did particularly bad calculations either ended up dead or ended up jumping scared for almost any external stimulus.

Fast forward to the present day and this involuntary ability to compute a conditional probability, given a prior, is key and forms part of even the simplest tasks of judgement. P(A is angry | A’s voice is loud) would also vary depending on whether A is a loud person by default, and A’s propensity to shout when angry and importantly on whether A shouts whenever he/she is angry. Or take the case of when kids’ voices (before they crack) are mistaken for a female voice over the phone — The person at the other end of the line calculated P(Gender=F| Voice = Not cracked) and hence concluded that the caller is most likely a female

Almost every response to external stimuli, apart from other complex micro-processes, also includes quick computation(s) of several Bayesian probabilities.

Now, coming to the tricky part — Consider conditional probabilities like P(Nationality = Japanese | Eye = {Small, Narrow})? Or, P(Favourite food = Curd rice | State = Tamilnadu)? Or, P(IQ level = Dumb | Blonde) ? Or P(Driver skill = Bad | Gender = Female)? Or P(Book = Bad | Cover = Bad). All these conditional probabilities fall under conventional notions (/taboos) of stereotyping and fall under one or more -isms. Is it really so wrong to compute these conditional probabilities? If early man did not use his ability to compute such probabilities, he/she would have believed that every growl could actually be any of the million species of animals and not a wild, dangerous animal — And the human race would most certainly have not gotten to 2016 AD.

So is the problem, the computation of such conditional probabilities? Consider I meet 1000 Japanese people and 990 of them have small, narrow eyes and I also meet 1000 small, narrow eyed people and 950 of them are Japanese. When I meet the 1001st small eyed person, am I supposed to (rationally) believe that he is equally likely to be a Japanese or an Australian or Brazilian?

It is just completely stupid to expect the brain to assign equal probabilities for all outcomes basis the above data — It implies that there is no learning at all that happens in our lives and that life is just a series of infinite independent events. And hence, the evaluation of such “racist” or “stereotyping” probabilities is not the problem.

Where I believe most people go wrong is in:

a. Not having a good ‘data enough’ construct— If suppose I have met only 3 Japanese and all of them had small, narrow eyes and actually there are 100 million Japanese, then any conclusion I draw basis 3 observations is obviously useless and wrong. We need to wait for a reasonably sized sample (in comparison to the population) before we start drawing any (strong) conclusions.

b. Equating P(B|A) and P(A|B) — This is the most common error, whereby if, say, 99% of all terrorists are muslims then people also infer 99% of all muslims are terrorists. Whereas in reality, even in a case where 99% of all terrorists are muslims, P(Terrorist|Muslim) is actually very low because the P(Muslim) factor is high (1.8B ~25% of the world) and the P(Terrorist) component is very low (in relation to the world population size)

c. Not updating probabilities basis new observations / Confirmation bias— Assume we have met 100 Chinese people (and no other Asians) and almost all of them had small, narrow eyes. And, assume, basis that we correctly compute a fairly high value for P(Chinese | small, narrow eyes). Assume this probability is 99% and you meet ’n’ people who have small, narrow eyes and are not Chinese. These can be cast off as exceptions to start with (and are acceptable errors). However, as this n starts growing and the error rate increases — say, due to the fact that there is another race ‘Japanese’ with non-negligible population size and high prevalence of small, narrow eyes — we should then update our assumptions and suitably dumb down our values for P(Chinese | small narrow eyes). We may still end up with a very high P(Chinese OR Japanese | small narrow eyes). While of course, there is some utility loss in the second probability, it is far less wrong — Worse still, is to expect new observations to rigidly conform to your probability computations. The P(B|A) computation needs to adapt and change as per real world observations (facts) and not vice versa.

d. Not having an error bound — This is slightly related to point ( c ). We also need to have a bound on errors that our probabilities might have. If we compute P(Likes curd rice | Tamilian) is 0.6, this probability/judgement needs to be discarded because, there is solid 40% chance that the Tamilian in question doesn’t like curd rice. Of course, the error threshold impacts the # of cases in which you can make reasonable judgements.

I don’t believe it is even remotely sensible to stop / shun such stereotyping P(B|A) computation based on non-scientific and absurd considerations like “every individual is different and hence you can’t judge” etc. Moreover, when people take offense to stereotyping and judgements, it is not because the act of stereotyping — Rather, it is because they computed the probability wrong.

There is no outrage when one concludes that P(Gender = Male | Beard= True) as a fairly high number because it is mathematically sound. Concluding that P(Formal Employment = False| Gender = Female) is a wrong conclusion today, because the P(Formal Employment = True | Gender = Female) has increased to a non-insignificant value. It might still have been a valid conclusion in the early 1900s, but not as on date. Similarly, concluding that all muslims are terrorists is wrong and absurd (not because it is wrong to judge people, but) because the math doesn’t support such a conclusion.

Contrary to popular notions, being more Bayesian / ‘judgemental’ (of course accompanied with conscious elimination of biases and errors) would only help us in the long run.