Bayes and Girls

2533475728_a0dcfd4524_zYou have long hair, never miss the Great British Bake-off and are particularly good at multitasking. You also work in Parliament. So are you more likely to be a man or a woman?

Well, some pretty lazy stereotypes aside, we may believe that the initial description is more likely to be female than male. The cliches certainly push us into thinking that way.

But, we should also consider the under-representation of women at all levels in British politics, a fact which has been highlighted in the news.

So, ignoring the introduction spele, if we were simply asked what is the chance that someone picked randomly from Parliament is a guy or a doll, how could we go about answering this? Well, let’s think in terms of the proportional representation of these two groups. Chances of picking a fella is certainly going to be larger. We can reasonably then go and place our stack of chips with the chaps.

What happens when we also consider the multitasking, baking-lover characteristics? These are things that stereotypically we would associate with the fairer sex so, we may reason, this person is more likely to be female. The result is a conflict between these two guesses.

7204293066_1866f0488a_zHow many men have long hair, like baking shows and can pat their head and rub their stomachs at the same time? Actually, probably quite a few. Maybe not proportionally as many as women but if we then include what we already know about the relative population sizes in Parliament, there may well be, in absolute terms, more men who fit this description. Simply by the prior fact that there are so many blokes in parliament, this may well weigh the answer in their favour.

This example is a different take on a common example of something called decision heuristics which can lead to cognitive bias. That is, rules-of-thumb that you and I use to take mental shortcuts, focusing on one aspect of a problem rather than the whole thing. It involves presupposition and is an example of where our intuition can lead us astray. The description of the person’s characteristics seduces us into imagining a woman but the baseline gender ratio, which is sneaked-in afterwards, is probably a better indicator.

What this little example can show is how we can go about combining two separate bits of information to get an overall, rational answer. In statistics this can be done formally using something called Bayes’ theorem.

In general terms, Bayes’ theorem takes what we already know- called the prior- and combines this with what extra information or data we observe- called the likelihood. As a result, we then have an answer- called a posterior- that is influenced by both of these things. How much it is influenced by each depends on the strength and conviction of the prior belief or data respectively. If they agree, then the answer is more certain than it would have been if we had used only one of them in isolation and if they are conflicting then the answer represents this too.

In the British politics example above, the prior could be what we know about the proportion of men and women and the extra data is the description of the employee. Like a tug-of-war, these two bits of information pull us in different directions. For example, we can not simply go along with the description in isolation and bank on a broad. The conclusion could be that, before hearing the characteristics, we are fairly sure we would get a gent at random and, even after hearing the profile, we may still think this is the most likely outcome, although we’re less sure about it.

5985805174_bd5e2cfe98_zKnowing that the mystery person has long hair moves us towards thinking that it is more likely to be a woman compared to before we knew anything about their hair-do but it’s just not enough to overpower what we already know about the parliamentary gender bias. That said, even though the flowing locks may not actually change our mind, they would introduce more doubt. Bayes’ theorem could help quantify this doubt.

Bayes’ theorem has applications far and wide, including spam filtering, internet search engines and voice recognition software. Originally, its statistical fundamentals were thought a little shaky, so have been extensively discussed and argued but it is fair to say a lot of progress has been made and the theorem has attained acceptance in most fields. That said, It has some way to go before it’s nearly as popular as the Great British Bake-off.

Post by: Nathan Green

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