Monday, November 7, 2011

Sherlock Holmes - Bayesian styles.

Nothing new or overly idiotic from the politicians today so we can look at something fun. This is where large chunks of the people who know me stop reading.
One of the things that I am going to be doing over the next year or so is constructing Bayesian models of gene regulatory networks. Thomas Bayes was a English chap who did everyone a favour and did some very serious thinking about probabilities back in the 1700's. Everyone is, I presume familiar with the idea that if you roll a dice there's a 1 in 6 chance that you'll get a six. If you roll two (non loaded die) then there's a 1 in 36 chance you'll get 2 sixes. You can just multiply the probabilities of the two events together because they're independent. What happens if your two events are conditional though? That is to say, the outcome of the second event is affected by the outcome of the first event? This is what Bayes worked out, how to compute the probability of an event given the probability of second event.

Basically, this says the probability of event A happening, given that event B happens is the probability the B happening provided A happens, times the probability of A happening divided by the probability of B happening. Doesn't sound immediately intuitive, but if we rearrange it slightly you'll see that we get:
Which, if you think about it, describes a pretty basic Venn diagram.
 The probability of A happening is the whole circle. The probability of B happening provided A happens is the intersection of the two circles. It's quite simple in the when you look at it that way. I'll be take the expression profiles of genes and using them as A's and B's. Which, as I'm sure I've mentioned before, will be fun.
So where does Sherlock come in to all this? We (as in humans) are very good at logical fallacies. Not figuring them out, making them I mean. Scientific American had a post a few days ago describing one. Given a description of someone and the a choice of possible occupations, people most often choose statements with two occupations in them, i.e. this person is a musician and an architect rather than just a musician or just an architect. When if you think about it, if being a musician is A and an architect is B, then the chance of being both, is significantly smaller than just A or just B. i.e. not all musicians are architects nor all architects musicians. With Bayes's formula, we can work out those probabilities if we wanted to, which would just show us how wrong we can be. Sherlock didn't make that mistake. And we all know how clever he was.

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