# Johnicholas comments on Value of Information: Four Examples - Less Wrong

68 22 November 2011 11:02PM

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Comment author: 20 December 2011 08:47:49PM 0 points [-]

Interesting..

So the original tree T models a tree of "states of affairs", and the original partition or subset T' models each node being either under the decision-maker's control or not. Then the function f would go from elements of T' to successors of those elements - nodes of T, to be sure, but there's a side condition there. Then the probability measure P is a somewhat more powerful way of attaching probabilities to the non-choice nodes - that is, if you have a distribution over the successors of each non-choice node, then you can obtain a probability measure, but a probability measure over paths would allow some additional correlations. The function u can be understood (in a finite tree) as labelling leaves with utilities, because paths of maximum length in a finite tree are isomorphic with leaves - but by describing it the way you did, you leave the door open to applying this formalism to an infinite tree. UP(f) would be the utility of a particular strategy (f), and U(f|\sigma) would be... the utility given a certain initial sequence of events? So \sigma is a finite path segment?

I don't understand the grammar of "Now either .... and ..." - should it be "Now either ... or ..."? Or is it really "Now assume ... and ..."?

When you use U(f) later, I am guessing that's either UP(f) with the P elided, or U(f | the empty path segment) - regardless, we're going to have to fix a P in order to get a utility, right?

Then the phrase "f(sigma)=f(tau) if the set of x such that \sigma(x) is not equal to tau(x) is a subset of S" - If I understand correctly, sigma and tau are finite path segments, which are isomorphic to nodes in the tree - but are they functions from nodes? If they are functions, wouldn't they be from, say, initial sections of the integers 0...n TO nodes? If I understand correctly, they're going to diverge at at most one point - once diverged, since it's a tree, they're not going to be able to rejoin. Or were you saying tree and thinking 'dag'?

I worry about the types of these things; coding it up in something like Haskell or Ocaml might make everything sharper and perhaps suggest simplifications. I'm sure that you can carry through the basic intuition.