Consider a bandit problem in which we want to maximize the probability of click-through based on bid values ($b$ is the value of bid and $\Pr(b)$ shows the probability that a customer clicks on a link given bid $b$). I am wondering what the possible modeling approaches are? If we consider a discrete set of bids ($b\in \{b_1,b_2,\cdots,b_k \}$), it is possible to model the probability as a Bernoulli distribution for each bid $\Pr(b_k)=p_k$ (with Beta or Dirichlet prior, e.g., $p_k \sim Beta(\alpha_k,\beta_k)$ ). Another case is model as a Logistic bandit in which it is possible to extend it to continuous values for bids, $\Pr(b_k)=\frac{1}{1+\exp\left(a_0 + a_1 b_k \right)}$.
I am wondering what other approaches are possible? Can we model it in form $\Pr(b_k)=\beta_0 + \beta_1 b_k$ with Normal prior on $\beta_0$ and $\beta_1$? If yes, what should we do if $\beta$ is estimated such that the probability exceeds 1?
