Jul 20, 2017

Belief updating under uncertainty

On the state of belief Feynman states:

I can live with doubt and uncertainty and not knowing. I think it's much more interesting to live not knowing than have answers which might be wrong. I have approximate answers and possible beliefs and different degrees of certainty about different things. But I'm not absolutely sure of anything and of many things I don't know anything about... I don't feel frightened by not knowing things.

Karl Popper on subjective knowledge and learning:

The process of learning, of the growth of subjective knowledge, is always fundamentally the same. It is imaginative criticism. This is how we transcend our local and temporal environment by trying to think of circumstances beyond our experience: by criticizing the universality, or the structural necessity, or what may, to us, appear (or what philosophers may describe) as the 'given' or 'habit'; by trying to find, construct, invent, new situations - that is, test situations, critical situations; and by trying to locate, detect and challenge our prejudices and habitual assumptions.

This is how we lift ourselves by our bootstraps out of the morass of our ignorance; how we throw a rope into the air and then swarm up it - if it gets any purchase, however precarious, on any little twig.

Here's how Bayes helps with updating our beliefs or learning new things:

Bayes’ rule specifies what ought be the relation between the probability that we assign to a state of the world after we get a signal (called the posterior probability), the probability that we assign to that state of the world before we get the signal (called the prior probability), the probability of getting that signal given that state of the world (called the likelihood), and the overall probability of getting that signal, regardless of the state of the world (the unconditional probability of the signal, which is sometimes called the marginal likelihood because it is the sum over the likelihoods under all possible states of the world). Bayes’ rule brings together all the relevant uncertainties, specifying the analytic relation between them.


The Bayesian approach to probability begins by noting that our empirically rooted beliefs are rarely if ever categorical; rather, they vary in strength. We doubt that some things are true; we think other things are rather likely to be true; we feel strongly that some things are almost certainly true (the less cautious would omit the “almost”) and that still other things are very unlikely to be true. These beliefs are about distal stimuli, which, as we have already learned, is psychological jargon for states of the world that affect our sensory systems only indirectly. Distal stimuli affect our senses by way of proximal stimuli that bear a complex, noisy, and ambiguous relation to the states of the world about which we entertain beliefs. Given the modern understanding of sensory processes, anything other than a graded “probabilistic” treatment of our empirically rooted beliefs about distal stimuli (states of the world) would be foolish. It would ignore the manifest difficulties in the way of our obtaining true knowledge from sensory experience. These difficulties become ever more apparent as our understanding of sensory mechanisms and the stimuli that act on them deepens.


The Bayesian argues that we must recognize that beliefs are in fact – and, moreover, ought to be – accompanied by a graded internal (mental or brain) quantity that specifies our uncertainty regarding their truth. From a mathematical perspective, graded quantities are represented by real numbers. (Only quantities that get bigger in discrete steps can be represented by integers.) For a Bayesian, a probability is a continuously graded subjective quantity that specifies a subjective uncertainty about states of the world, in a receiver whose knowledge of the world comes from noisy and ambiguous signals. That is, of course, just what Shannon supposed in his analysis of communication. From this perspective (a radical Bayesian perspective), the theory of probability is the theory of how to handle the real numbers with which we represent subjective (receiver) uncertainties in a logically consistent and mathematically sound way (Cox, 1961; Jaynes, 2003; Jeffreys, 1931).


As a formula for the updating of belief, Bayes’ theorem nicely captures our intuitions about what does and does not constitute strong evidence. These intuitions turn on: (1) the prior probability: how probable we think something is a priori, on the basis of logic or extensive prior experience; and (2) relative likelihood: how likely the evidence is if that something is true versus how likely it is otherwise.

Gallistel, C. R., and Adam King. Memory and the Computational Brain: Why Cognitive Science Will Transform Neuroscience., 2010

Among other things, Bayes and PBL have in common activation prior knowledge, identification unknowns variables, and update of beliefs in accordance to the strength of the new evidence. Here's a video illustration by Dr. Peggy Seri├Ęs on how the brain might be using probability to update our beliefs. Evidence and beliefs change according to the context the person is in and how the brain functions may also interfere with the bayesian framework.

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1. You should attempt to re-express your target’s position so clearly, vividly, and fairly that your target says, “Thanks, I wish I’d thought of putting it that way.
2. You should list any points of agreement (especially if they are not matters of general or widespread agreement).
3. You should mention anything you have learned from your target.
4. Only then are you permitted to say so much as a word of rebuttal or criticism.
Daniel Dennett, Intuition pumps and other tools for thinking.

Valid criticism is doing you a favor. - Carl Sagan