Entropy maximization, part II

Let $\mathcal{X} \in \{\alpha_1, \cdots, \alpha_n\}$ be a random variable with finite alphabet, what is the distribution that will achieve maximum entropy given the constraint that $E[f(\mathcal{X})]=\beta$ ?

We define the expectation
$$\sum_i \alpha_i p_i = \beta$$ The optimization problem is given as
$$\max_\mathbf{p} H(\mathbf{p}) \; s.t., \; \sum_i p_i = 1,\; \sum_i \alpha_i p_i = \beta$$ Optimizing using Lagrange multipliers $\lambda$ and $\mu$, we have
$$p_i = exp^{-(1-\lambda)}exp^{\mu \alpha_i}$$ which turns out to be the familiar expression
$$p_i = \frac{1}{Z}exp^{\mu \alpha_i}$$ with $Z=\sum_i exp^{\mu \alpha_i}$ being the partition function.

The above form is called a Gibbs distribution.

Now consider the problem where with $\mathcal{X} \in \mathbb{R}^p$ and we are given the moments $E[f_j(\mathcal{X})]=\beta_j, j=1,\dots,p$.  The optimization problem becomes
$$\max_\mathbf{p} H(\mathbf{p}) \; s.t., \; \sum_i p_i = 1,\; \sum_i \alpha_{ij} p_i = \beta_j$$ The expression have the form
$$p_i = \frac{1}{Z} exp^{\sum_{j=1}^p \mu_j \alpha_{ij}}          =\frac{1}{Z} exp^{\mathbf{\mu}^T \mathbf{\alpha_{i}}}$$ Since the distribution in general has infinite support the constraint on the moment will allow one to reach a unique solution.