|02-17-2007, 04:53 AM||#1|
Join Date: Jul 2001
Location: New Brunswick, NJ, USA
Are our brains optimized Bayesian computers?
The word "Bayesian" seems to be cropping up everywhere. It crops up now even in clinical trial design where it means a flexible clinical trial that changes according to the results that it is getting. What? A clinical trial that changes its design based on the results? Yup. If it is quite clear that you are getting results that indicate that you will not get a statistically significant result from your clinical trial, why waste the time doing the clinical trial as planned. Change it so that you can get significant results? But Bayesian theory is much more than that.
A recent article on brain function suggests that our brain operate as Bayesian computers (Source). This controversial theory by Alex Pouget of the University of Rochester explains several puzzling observations concerning our brains. It seems that our brains are full of "noise" or seemingly random activity. How can this be? Pouget found that our brain noise actually fits a frequency distribution called "Poisson Distribution" which suggests that it is not noise. Studies have shown that Bayesian computation is most efficiently done when the data have a Poisson distribution format. Wow, what does that mean?
The word Bayesian refers to the work of Thomas Bayes (1702-1761) who proved a special case of what is now called Bayes' theorem. The term Bayesian was first used in 1950 and it is not clear that Bayes himself would approve of the broad interpretation of probability that is associated with his name. Laplace solved a more general version of Bayes' theorem. The theorem states that the concept of probability can be defined as the degree to which a person believes a proposition. Bayesian theory also suggested that Bayes' theorem can be used as a rule to infer or update the degree of belief when there is new information.
In classical probability, the odds of success is simply expressed as a frequency compared to chance frequency. The Bayesian approach suggests that there are other factors that influence the degree of belief in a given proposition, including "subjective probability", "personal probability", "epistemic probability", and "logical probability". The sum of these represent the "degree of belief" that an individual has in a given proposition. The degree of belief may be subjective although there are some Bayesians that do not accept the subjectivity claim. Some Bayesians hold that the degree of belief refers to the extent to which a person is willing to bet on the proposition at hand.
The frequency probability and the Bayesian interpretation of probability have important ramifications for statistical practice. Classical probability theory, as developed by Fisher, Perason, and others during the first half of the 20th Century, rigidly tests hypotheses based on frequency probability and errors. The main frequentist hypothesis is the null hypothesis, that the results of a clinical trial does not differ from that due to chance alone. A clinical trial is regarded to be statistically significant if the results deviates markedly from chance results. An alternative is that the observed data are a misleading set.
Bayesian theory assigns probabilities to multiple hypotheses of a model, updates the probabilities from the incoming data, and adjusts the "degree of belief" accordingly. An example is Bayesian email spam filtering. Suppose that one has identified a body of email messages that are spam. Bayes' theorem says that the probability that a received email is spam, given that it has certain words in it, is equal to the probability of finding those words in the body of spam email, divided by the probability of finding those words in any email. Each word of a received email contributes to the email's spam probability. If that summed probabilty exceeds a certain value, say 95%, the spam detector would mark the email as spam.
Note that in the Bayesian approach, each word has a probability function and hence is a hypothesis. However, as data accumulates, the probability functions may change. So, for example, every day, I go through the exercise of deciding which new registrant to the CareCure web site is a spammer. There are registrants who are definitely spammers, because they have posted spam on our site. There are registrants who have not posted anything on our site and therefore we cannot know for certain that they are or are not spammers. Finally, there are registrants who post legitimate material and are not spammers.As collections of spammers and non-spammer registrations accumulate, Bayesian filters continually hone probabilities of spammers from specific words contained in answers during registration.
Now, imagine this approach being applied to clinical trials. Let us assume that we have a body of people whom we have categorized as recovered or not recovering from spinal cord injury. We have collect a whole bunch of data on these patients, i.e. their injury level, their injury classification (A, B, C, D, E), the presence of neuropathic pain, spasticity, etc. etc. Can we build a probability profile that would predict whether any single person would be able to walk? The answer to this question is yes. Can we build a probability profile that would predict the probability that a person would walk given a particular treatment? The answer is again yes. A third and very interesting Bayesian approach is to automatically adjust combination treatment doses as data accumulates showing that certain doses work better. So, this represents a dramatic new approach to clinical trial design.
One reason why Bayesian theory is intuitively attractive is because it is close to the way people think. In fact, this is how we utilize our experience and make judgements about the probability of success. We categorize experiences as success (reward) or failures (punishment or lack of reward). Then we evaluate each incoming experience by comparing various features of the the experience with those stored as successes or failures in our minds. It is the way we decide whether an email is a spam. It is how we decided whether we jump or not jump in a fire. Bayesian thinking is how we make decisions.
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