Fundamentals of kicking anthropic butt

18 Manfred 26 March 2012 06:43AM


Galactus

Introduction

An anthropic problem is one where the very fact of your existence tells you something. "I woke up this morning, therefore the earth did not get eaten by Galactus while I slumbered." Applying your existence to certainties like that is simple - if an event would have stopped you from existing, your existence tells you that that it hasn't happened. If something would only kill you 99% of the time, though, you have to use probability instead of deductive logic. Usually, it's pretty clear what to do. You simply apply Bayes' rule: the probability of the world getting eaten by Galactus last night is equal to the prior probability of Galactus-consumption, times the probability of me waking up given that the world got eaten by Galactus, divided by the probability that I wake up at all. More exotic situations also show up under the umbrella of "anthropics," such as getting duplicated or forgetting which person you are. Even if you've been duplicated, you can still assign probabilities. If there are a hundred copies of you in a hundred-room hotel and you don't know which one you are, don't bet too much that you're in room number 68.

But this last sort of problem is harder, since it's not just a straightforward application of Bayes' rule. You have to determine the probability just from the information in the problem. Thinking in terms of information and symmetries is a useful problem-solving tool for getting probabilities in anthropic problems, which are simple enough to use it and confusing enough to need it. So first we'll cover what I mean by thinking in terms of information, and then we'll use this to solve a confusing-type anthropic problem.

continue reading »

Information theory and the symmetry of updating beliefs

45 Academian 20 March 2010 12:34AM

Contents:

1.  The beautiful symmetry of Bayesian updating
2.  Odds and log odds: a short comparison
3.  Further discussion of information

Rationality is all about handling this thing called "information".  Fortunately, we live in an era after the rigorous formulation of Information Theory by C.E. Shannon in 1948, a basic understanding of which can actually help you think about your beliefs, in a way similar but complementary to probability theory. Indeed, it has flourished as an area of research exactly because it helps people in many areas of science to describe the world.  We should take advantage of this!

The information theory of events, which I'm about to explain, is about as difficult as high school probability.  It is certainly easier than the information theory of multiple random variables (which right now is explained on Wikipedia), even though the equations look very similar.  If you already know it, this can be a linkable source of explanations to save you writing time :)

So!  To get started, what better way to motivate information theory than to answer a question about Bayesianism?

The beautiful symmetry of Bayesian updating

The factor by which observing A increases the probability of B is the same as the factor by which observing B increases the probability of A.  This factor is P(A and B)/(P(A)·P(B)), which I'll denote by pev(A,B) for reasons to come.  It can vary from 0 to +infinity, and allows us to write Bayes' Theorem succinctly in both directions:

     P(A|B)=P(A)·pev(A,B),   and   P(B|A)=P(B)·pev(A,B)

What does this symmetry mean, and how should it affect the way we think?

A great way to think of pev(A,B) is as a multiplicative measure of mutual evidence, which I'll call mutual probabilistic evidence to be specific.  If pev=1 if they're independent, if pev>1 they make each other more likely, and if pev<1 if they make each other less likely.

But two ways to think are better than one, so I will offer a second explanation, in terms of information, which I often find quite helpful in analyzing my own beliefs:

continue reading »