# Is there a unique tilted measure with specified marginals?

Suppose $\mathcal{A},\mathcal{B}$ are finite sets and $\mu_{A,B}(a,b)$ is a probability measure on the product set $\mathcal{A}\times \mathcal{B}$ so that $\mu_{A,B}(a,b)>0$ for each $a\in \mathcal{A}, b\in \mathcal{B}.$ We may tilt the measure $\mu$ by choosing non-negative (not everywhere zero) functions $\phi(a), \psi(b)$ defining a new measure:

$$\nu_{A,B}(a,b):=\frac{\mu_{A,B}(a,b)\phi(a)\psi(b)}{\sum_{a^\prime,b^\prime}\mu_{A,B}(a^\prime,b^\prime)\phi(a^\prime)\psi(b^\prime)}.$$

We restrict ourselves to such tilted measures and ask the questions:

a) Given marginal probability measures $\nu_A(a), \nu_B(b),$ is there always a tilted measure $\nu_{A,B}(a,b)$ that has these marginals?

b) If the answer to a) is yes, is there a unique tilted measure that has these marginals?

The answer to (a) is indeed YES, since we can consider the relative entropy minimization problem:

$$\arg\min_{\tilde{\nu}_A = \nu_A, \tilde{\nu}_B = \nu_B} \sum_{a,b} \tilde{\nu}_{A,B}(a,b)\log\frac{\tilde{\nu}_{A,B}(a,b)}{\mu_{A,B}(a,b)}.$$

This is a convex programming problem with a unique minimizer that can be shown using Lagrange multipliers to be of the tilted measure form.

My questions then are:

(b): Given any pair of marginals, is there a unique tilted measure with these marginals?

(c): Is there a name for tilted measures of the product form as I have considered here?

I suspect the answer to (b) is YES and this suspicion comes from the fact that the space of pairs of marginals has dimension $(|\mathcal{A}|+|\mathcal{B}|-2)$ which is also the dimension of the space of tilted measures.