Boyd & Vandenberghe, example 2.13 — what is the "convex set of joint probabilities" in this example?

Question

I am reading Boyd & Vandenberghe's Convex Optimization. I can't understand the example below:

• what is the "convex set of joint probabilities" in this example?
• what is the convex set $C$ here?

Answer 1

• to make things clear, let's consider the simple example of $p=(p1, p2)$ in a two dimensional real space, $(x1, x2)$. Then, the convex set of joint probabilities, $p=(p1, p2)$ is the line segment $x1+x2=1, 0 \leq x1, x2 \leq 1$, which is convex of course. Here on this example the convex set of joint probabilities in $mn$ dimensional instead of being $2$ dimensional in our example.
• The convex set $C$ is just the set of probability values I explained above, line segment in $2$ dimensional case.

Answer 2

Explanations so far do not do the book justice.

The joint probabilities pij can be seen as n m x n matrix but equivalently as vector of N = m x n nonnegative numbers that sum to one. It helps to picture what is meant if we think of pij as a vector.

As explained in an earlier example 2.5 in the book the set of all possible such probability vectors lie in the probability simplex of affine dimension N-1 embedded in N dimensional space. This simplex is evidently convex, but that is not the point here.

The convex set C in example 2.13 in the book is meant to be any convex subset of this probability simplex.

Each possible vector pij in C then transforms to a vector of conditional probabilities fij of same dimension N by the linear-fractional function given. fij will typically no longer lie in the probability simplex since conditional probabilities need not sum to one.

The claim in the book is that when you sweep out C by varying pij, then fij will sweep out a region f(C) in N-space that is still convex.