However, I am now convinced that this idea is a dead end, for the following reason: Just because the point you start with has all coordinates between 0 and 1, does not mean that the projection to the subspace containing coherent assignments still has all coordinates between 0 and 1. (Imagine a 3d unit cube, and imagine that a theory is coherent if x+y+z=1. If you project 1,1,0 onto this subspace, you get 2⁄3,2/3,-1/3, which is not a valid probability assignment) I am now convinced that this idea will not be fruitful.
Why is this an issue? Just project onto the simplex (sum x_1… x_n =1, x_1 >=0 … x_n >= 0) instead of the affine subspace. This is perfectly possible in O(n) time Duchi ’08. You can add many more constraints and still make efficient projections.
But I have to admit I’m confused about what use this is. What is the application supposed to be, and why is simply dividing by the sum of probabilities insufficient?
There are 2 issues. First, the actual situation is an infinite dimension vector space, so it is not clear that that result applies. Second, the idea was to take advantage of properties of projection in order to converge to the projection as a series of local moves, and I don’t see a way to do that with a simplex.
The goal was to use this projection and apply it to the point that assigns 1⁄2 to all statements, in order to get a coherent probability distribution on sentences and a simple procedure that converges to it, so that we can look at the properties of this distribution.
It also could have given an answer to the question: How should I correct if I observe that my probability assignment is incoherent.
The basis vectors do not in general correspond to a set of propositions exactly one of which is true, but merely a set of propositions with some set of logical constraints (e.g. if x, y, and z are all logically equivalent, then the assignment is coherent iff x=y=z. The situation where the assignment is coherent iff x+y+z=1 was just an example).
This is correct. My response was thinking he was saying project onto the polytope, (which is not always a simplex), which is a good suggestion, but doesn’t work in terms of local moves.
This modified simplex, where some coordinates are constrained to be equal, shouldn’t be impossible to project on, but it does seem like you have larger problems. Ah well.
But I figure there should be some approach that works spiritually with whatever you’re trying to do (which is not 100% clear to me) just because there are so many ways to project vectors, or sampled versions of those vectors.
What specifically do you hope to gain from your probability distribution over 1⁄2 probability on all sentences?
Why is this an issue? Just project onto the simplex (sum x_1… x_n =1, x_1 >=0 … x_n >= 0) instead of the affine subspace. This is perfectly possible in O(n) time Duchi ’08. You can add many more constraints and still make efficient projections.
But I have to admit I’m confused about what use this is. What is the application supposed to be, and why is simply dividing by the sum of probabilities insufficient?
There are 2 issues. First, the actual situation is an infinite dimension vector space, so it is not clear that that result applies. Second, the idea was to take advantage of properties of projection in order to converge to the projection as a series of local moves, and I don’t see a way to do that with a simplex.
The goal was to use this projection and apply it to the point that assigns 1⁄2 to all statements, in order to get a coherent probability distribution on sentences and a simple procedure that converges to it, so that we can look at the properties of this distribution.
It also could have given an answer to the question: How should I correct if I observe that my probability assignment is incoherent.
The basis vectors do not in general correspond to a set of propositions exactly one of which is true, but merely a set of propositions with some set of logical constraints (e.g. if x, y, and z are all logically equivalent, then the assignment is coherent iff x=y=z. The situation where the assignment is coherent iff x+y+z=1 was just an example).
This is correct. My response was thinking he was saying project onto the polytope, (which is not always a simplex), which is a good suggestion, but doesn’t work in terms of local moves.
This modified simplex, where some coordinates are constrained to be equal, shouldn’t be impossible to project on, but it does seem like you have larger problems. Ah well.
But I figure there should be some approach that works spiritually with whatever you’re trying to do (which is not 100% clear to me) just because there are so many ways to project vectors, or sampled versions of those vectors.
What specifically do you hope to gain from your probability distribution over 1⁄2 probability on all sentences?