are also important from the viewpoint of defensive forecasting ([@defensive-forecasting], [@df-linear]). We again discuss this point in Section \[subsec:monotonicity\].

We now consider the capital process ${{\cal K}}_n^\theta$ of for a fixed $\theta\in \mathbb{R}^d$. Solving for $\hat p_n$ we have $$\label{eq:hatpn}
\hat p_n = \frac{p_n e^{{\theta{'}}c_n}}{1+ p_n (e^{{\theta{'}}c_n}-1)}, \qquad
1-\hat p_n = \frac{1-p_n}{1+ p_n (e^{{\theta{'}}c_n}-1)}.$$ Then $$\hat p_n^{x_n} (1-\hat p_n)^{1-x_n}= p_n^{x_n} (1- p_n)^{1-x_n} \frac{e^{{\theta{'}}c_n x_n}}{1+ p_n (e^{{\theta{'}}c_n}-1)}$$ and the capital process is written as $$\label{eq:capital-theta}
{{\cal K}}_{n}^\theta = \prod_{i=1}^n \frac{\hat p_i^{x_i} (1-\hat p_i)^{1-x_i}}{p_i^{x_i} (1-p_i)^{1-x_i}}=
\frac{e^{{\theta{'}}\sum_{i=1}^n c_i x_i}}{\prod_{i=1}^n (1+p_i (e^{{\theta{'}}c_i}-1))}.$$

Naturally it is better for Skeptic to choose the value of $\theta$ depending on the moves of other players. In this paper we consider a Bayesian strategy, which specifies a prior distribution $\pi(\theta)$ for $\theta$. Bayesian strategies for Binary Probability Forecasting with constant $p_n\equiv p$ was considered in [@ktt-2008saa]. Bayesian strategy is basically the same as the universal portfolio by Cover and his coworkers ([@cover-universal-portfolios], [@cover-ordentlich], [@cover-thomas-2nd]). In the universal portfolio, a prior is put on the betting ratio $\nu$ itself, where as we put a prior on the parameter of Skeptic’s model. Furthermore differently from [@cover-ordentlich] we allow continuous side information.

In the Bayesian logistic strategy with the prior density function 