The Maximum Entropy Principle

In Theorem 3.6, it was shown that for a non-empty finite result set Ω, the probability measure on \({\mathcal{P}}(\Omega)\) with maximum entropy is given by \(\mathbb{P}(\{\omega\})=\frac{1}{|\Omega|},\quad\omega\in\Omega\quad(|\Omega|\text{ number of elements of $\Omega$}).\) Now we investigate the same question under constraints. Let \(n\in\mathbb{N}\) and \(f_{\mathbb{S}}:[0,1]^{n}\to\mathbb{R}^{+}_{0},\quad\mathbf{x}=(x_{1},\ldots,x_{n})\mapsto-\sum_{j\in J_{\mathbf{x}}}x_{j}\mathop{\mathrm{ld}}(x_{j})\) with \(J_{\mathbf{x}}=\{k\in\{1,\ldots,n\};\;x_{k}> 0\},\) then \(f_{\mathbb{S}}\) is continuous and strictly concave on \([0,1]^{n}\) due to \(\lim_{x\to 0}x\mathop{\mathrm{ld}}(x)=0\) (see Theorem 2.1).