Lower bound on likelihood of binary outcomes

Question

I am wondering about the following: does there exist a stochastic process $(X_n)_{n \ge 1}$ with values in $\{0,1\}$ on a probability space $(\Omega, \mathcal F, \mathbb P)$ such that for all $n \ge 1$ and every non-constant vector of binary outcomes $(x_k)_{1 \le k \le n}$, we have $$ \ln \mathbb P[X_1 = x_1, \dotsc, X_n = x_n] \ge h_n \ln \frac {h_n} {n} + t_n \ln \frac {t_n} {n},$$ where $h_n = \sum_{1 \le k \le n} x_k$ and $t_n=n-x_n$?

In the case of a mixture process using a beta distributed prior on the success probability of a Bernoulli process, using Stirling’s formula it is possible to prove a slightly worse bound (namely, subtract an $O(\ln n)$ positive term to the right-hand side).

I’d be curious to know if it is possible to do better. If the above inequality is not possible, I would also like to know if the $O(\ln n)$ term can be improved asymptotically.

Answer 1

This is impossible already for $n=4$ and $n=5$; e.g. in the latter case, the 30 binary vectors with $t_n h_n \ne 0$ yield $$\sum \exp(h_n \ln (h_n/n) + t_n \ln (t_n /n))=\sum h_n^{h_n} \cdot t_n^{t_n}/n^n=10 \cdot 4^4/5^5+ 20 \cdot 3^3 \cdot 2^2/5^5=1.5104 >1 \,. $$