Suppose data $\{z_t\}_{t=1}^T$ are $z_t\substack{{\small i.i.d.} \\{\Large \sim}\\ \color{white}{.}}$ $\text{Bernoulli}(\theta)$ and a four-parameter Beta prior $\theta\sim \text{Beta}_4(\alpha,\beta,a,c)$ (with $\alpha,\beta>0$ and $0\leq a<c\leq 1$). (Details about this distribution are found below for your convenience and at the aforementioned link.)
Questions: What is the posterior after observing $\{z_t\}_{t=1}^T$?
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Details about the Beta$_\boldsymbol{4}$ distribution
If random variable $Y$ follows a four-parameter Beta distribution $\text{Beta}_4(\alpha,\beta,a,c)$ then it has p.d.f. $f_{B4}(\cdot|\alpha,\beta,a,c)$ given by $$f_{B 4}(y| \alpha, \beta, a, c) = \frac{\left(\frac{y-a}{c-a}\right)^{\alpha-1} \left (\frac{c-y}{c-a} \right)^{\beta-1} }{(c-a)B(\alpha, \beta)}=\frac{ (y-a)^{\alpha-1} (c-y)^{\beta-1} }{(c-a)^{\alpha+\beta-1}B(\alpha, \beta)},$$ where $B(\cdot,\cdot)$ is the Beta function; its expected value is $\frac{\alpha c+ \beta a}{\alpha+\beta}$. Relationship to the "usual" (two-parameter) Beta distribution: if $X\sim\text{Beta}(\alpha,\beta)$, its pdf $f_{B2}(\cdot;\alpha,\beta)$ is given by $$ f_{B2}(x;\alpha,\beta)= \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}.$$ Then, $f_{B4}(\cdot|\alpha,\beta,a,c)$ can be re-written in terms of $f_{B2}(\cdot|\alpha,\beta)$ as follows: $$f_{B4}(y|\alpha,\beta,a,c) = \frac{f_{B2}\left(\frac{y-a}{c-a};\alpha,\beta\right)}{c-a}. $$