We now choose small enough $\epsilon>0$ such that $$\label{eq:epsilon-selec}
\epsilon_1 (1-\epsilon_2) - \frac{\epsilon}{2}(1+\epsilon_2)e^{\epsilon L_c}> \frac{\epsilon_1}{2}.$$ Then $${u}(\epsilon \tilde \eta) \ge \frac{\epsilon \epsilon_1}{2} \lambda_{\min,n}   \rightarrow \infty  \qquad (n\rightarrow\infty, n\in {\mathbb N}_2).$$ Note that the convergence is uniform for $\tilde \eta$ in some neighborhood $N(\eta)$ of $\eta$. Since our prior $\pi$ puts a positive weight to $N(\epsilon\eta)$, ${{\cal K}}_n^\pi \rightarrow\infty$ along $n\in {\mathbb N}_2$. This completes our proof of Theorem \[thm:usual-SLLN\].

Weak forcing of a more precise form of the strong law of large numbers {#subsec:main-result}
----------------------------------------------------------------------

As discussed in Section \[sec:formulation\], we can establish a much more precise rate of convergence of the strong law of large numbers based on our Bayesian logistic strategy. Our main theorem of this paper is stated as follows.

\[thm:main\] In BPFSI, by a Bayesian logistic strategy with a prior distribution supporting a neighborhood of the origin, Skeptic can weakly force $$E_1 \ \Rightarrow \ \limsup_n \frac{{{\cal S}}_n{'}{{\cal V}}_n^{-1}{{\cal S}}_n}{\log \det {{\cal V}}_n} \le 1,$$ where $E_1$ is given in .

We give a proof of this theorem in the following three subsections.

### Bounding the maximum likelihood estimate

We now consider the behavior of ${{\cal K}}_n^\theta$ in , when ${{\cal K}}_n^\theta$ is maximized with respect 