\begin{align*} & \sum_{t=1}^T \frac{\log(p_{t+1}(x)) - \log(p_t(x))}{\eta_t} \\ & = \frac{\log(p_{T+1}(x))}{\eta_{T}} - \frac{\log(p_1(x))}{\eta_1} + \sum_{i=1}^N \log(p_{\tau_i+1}) \left(\frac{1}{\eta}_{\tau_i} - \frac{1}{\eta_{\tau_i+1}} \right) \\ & = \frac{\log(p_{T+1}(x))}{\eta_{T}} - \frac{\log(p_1(x))}{\eta_1} + \frac{\gamma}{1+\gamma} \sum_{i=1}^N \frac{\log(p_{\tau_i+1})}{\eta_{\tau_i}} . \end{align*}