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We are all Bayesians now

Intent on not being late for an evening session at Tinker.it! last week, I dropped by Bunhill Fields for too short a time, the light beginning to fail and a hurriedly printed off, crumpled map for guide.

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Easy to find the memorials to Blake and his wife and Defoe. But I was on a quest for Thomas Bayes:

Bayes, Thomas (b. 1702, London - d. 1761, Tunbridge Wells, Kent), mathematician who first used probability inductively and established a mathematical basis for probability inference (a means of calculating, from the number of times an event has not occurred, the probability that it will occur in future trials). He set down his findings on probability in "Essay Towards Solving a Problem in the Doctrine of Chances" (1763), published posthumously in the Philosophical Transactions of the Royal Society of London.

It took me too long to find his resting place, railed off and not in a great state of repair, and my rushed photos weren’t worth posting, but here’s one from the ISBA site (taken by Professor Tony O'Hagan of Sheffield University and seemingly not copyright):

Bayes 1 

The famous essay is online (PDF).

I need to spend more time in and around Bunhill Fields, but what prompted me to try to take it in as I sped across London was reading in Chris Frith’s book, Making up the Mind, how important Bayes is to neuroscience:

… is it possible to measure prior beliefs and changes in beliefs? … The importance of Bayes’ theorem is that it provides a very precise measure of how much a new piece of evidence should make us change our ideas about the world. Bayes’ theorem provides a yardstick by which we can judge whether we are using new evidence appropriately. This leads to the concept of the ideal Bayesian observer: a mythical being who always uses evidence in the best possible way. … Our brains are ideal observers when making use of the evidence from our senses. For example, one problem our brain has to solve is how to combine evidence from our different senses. … When combining this evidence, our brain behaves just like an ideal Bayesian observer. Weak evidence is ignored; strong evidence is emphasised. … But there is another aspect of Bayes’ theorem that is even more important for our understanding of how the brain works. … on the basis of its belief about the world, my brain can predict the pattern of activity that should be detected by my eyes, ears and other senses … So what happens if there is an error in this prediction? These errors are very important because my brain can use them to update its belief about the world and create a better belief … Once this update has occurred, my brain has a new belief about the world and it can repeat the process. It makes another prediction about the patterns of activity that should be detected by my senses. Each time my brain goes round this loop the prediction error will get smaller. Once the error is sufficiently small, my brain “knows” what is out there. And this all happens so rapidly that I have no awareness of this complex process. … my brain never rests from this endless round of prediction and updating.

… our brain is a Bayesian machine that discovers what is in the world by making predictions and searching for the causes of sensations.

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