So I was having a read of the paper here: Exponential Inequalities for Self-Normalized Martingales. I am particularly interested in Remark 4.2, which states that if $M_n$ is a Gaussian martingale (and is therefore sub-Gaussian), then the following probability inequality holds: $$ (1)~~~~~~~~~\mathbb{P}\left(\frac{M_n}{a + b\langle M\rangle_n} \geq x\right) \leq \inf_{p > 1}\left(\mathbb{E}\left[\exp\left\{-\left(p-1\right) \frac{x^2}{\alpha^2}\left(ab + \frac{b^2}{2}\langle M\rangle_n\right)\right\}\right]\right)^{1/p} $$ This is given in equation 4.6 in the paper above. Here, we require only that $a \geq 0$ and $b > 0$ and $x > 0$. We say that $\left(M\right)_n$ is sub-Gaussian with parameter $\alpha$. For clarity, we also have, $$ \langle M\rangle_n=\sum_{k=1}^n \mathbb{E}\left[\left(M_k-M_{k-1}\right)^2 \bigg| \mathcal{F}_{k-1}\right] $$ which is the predictable quadratic variation.
What I do not understand is why there is an expected value on the right-hand side of (1). It seems to be that all of the parameters on the right-hand side are non-random. So, why is there an expected value inside the infimum?
Any help with understanding this would be appreciated.