to $\theta$. Let $${\hat\theta_n^*}=\operatorname{argmax} {{\cal K}}_n^\theta.$$ We call ${\hat\theta_n^*}$ the maximum likelihood estimate, since ${{\cal K}}_n^\theta$ is of the form of the likelihood function of the logistic regression model. It is easily seen that the maximizer ${\hat\theta_n^*}$ is finite except for a special case that the vectors in $\{ c_i \mid x_i=1\}\cup \{ -c_i \mid x_i=0\}$ lie on a half-space defined by a hyperplane containing the origin. More specifically in Lemma \[lem:small-mle\] we prove that $\Vert {\hat\theta_n^*}\Vert $ is small when $\Vert {{\cal V}}_n^{-1}{{\cal S}}_n\Vert$ is small.

The maximizing ${\hat\theta_n^*}$ can only be computed at the end of day $n$ after seeing all the data $p_1, c_1, x_1, \dots, p_n, c_n, x_n$. Hence we call a strategy using ${\hat\theta_n^*}$ a “hindsight strategy”, which is the same as the best constant rebalanced portfolio (BCRP) in the terminology of the universal portfolio.

We prove the following lemma.

\[lem:small-mle\] Let $L_{c,n} = \max_{1\le i \le n} \Vert c_i \Vert$ and $L_{\lambda,n}=\lambda_{\max,n}/\lambda_{\min,n}$, where we assume $\lambda_{\min,n}>0$. Then $$\Vert {{\cal V}}_n^{-1} {{\cal S}}_n \Vert \le \frac{1}{3L_{c,n} L_{\lambda,n}} \ \ \Rightarrow \ \ \Vert{\hat\theta_n^*}\Vert
\le 3 L_{\lambda,n} \Vert {{\cal V}}_n^{-1} {{\cal S}}_n \Vert.$$

For any fixed $\xi\in E_1$, there exist $L_c, L_\lambda$, such that $L_{c,n}< L_c$ 