Bayesian Models from the Eye to the Cosmos

My last post caused me to survey some things related to Bayesian statistics as they relate to mathematics and cognition.  First, I want to say that despite the fact that I have been looking more closely at 19th century developments in mathematics, I didn’t know until today that Laplace, in 1814, described a system of inductive reasoning, based on probabilities, that would today be recognized as Bayesian or that in the 1860’s Hermann Helmholtz modeled the brain’s ability to shape the flux of sensory data into our perceived world, probabilistically.

A survey of the growing applications of Bayesian probabilities leads through vastly different landscapes, from inside of us (in how the nervous system accomplishes perceiving the world) to how we learn and are able to make some very accurate every-day predictions, and finally to how we investigate the enormously far-away fundamental fabric of our universe. Bayesian probabilities are distinguished by the fact that they change with evidence or growing information. Looking at their applications makes it seem like all things human have some mathematical wrapping.

The everyday example was given in the research article: Optimal Predictions in Everyday Cognition. In a recent study, individuals were asked to make interval estimates, like how long a movie might run, or how much money it will gross, how long someone might live.   With respect to life span, participants in the study were asked: how long might the person you just met live?  (given their age when you met them).  A Bayesian predictor computes the probability using Bayes’s rule, that the probability that the person will live to a particular age given their age when you met them is proportional to the product of:

(the likelihood of meeting someone at say age 65 who will live to be say 85)

and

(the likelihood of living to 85).

The second factor of this product is called the prior probability and can be said to reflect our expectations.   A good prediction about the person you just met would be the median of the distribution produced by the Bayesian calculations.  The article describes the study in good detail.  Researchers found that people’s judgments were very close to the optimal predictions calculated by a Bayesian model.  The article concludes:

Assessing the scope and depth of the correspondence between probabilities in the mind and those in the world presents a fundamental challenge for future work.

The same kind of modeling has been applied to the very far questions of cosmology and our innermost puzzles of visual perception.  Variations on these methods have been used to understand how the body maximizes the use of its expectations, or how it minimizes the discrepancy between actual features of the world and representations of those features. And it is from this connecting-information-to-probabilities that we get our current models of the universe.  (An example of the kind of fine tuning that has to be brought to the calculation “when need to argue from a state of maximum ignorance” is described in the physics paper, Getting the Measure of the Flatness Problem).

In the paper I referenced last week, How the Mind Grows, from Joshua Tenenbaum, one of its more striking observations is that we organize the features of our world by building on very early, quickly formed abstractions that seem to be based on fundamental ‘similarity metrics’ or the way we determine the relevant properties that a class of objects will have in common.  In a talk on the same topic, Tenenbaum draws attention to the fact that important scientific insights are often the result of RE-organizing key features, perhaps by reinterpreting the similarity metric.  Mendeleev’s periodic table, for example, was an organizational change.  He created it before the notion of an atomic number was developed or before there was any hint of modern quantum mechanical ideas.  (Both the paper and the talk can be accessed from his web page under the heading Representative readings an talks).

In mathematics, organization, structure, class, and similarity are at the heart of the matter.   It’s no wonder it looks alive to me.