My question is: What is the mathematical relationship between the Beta distribution and the coefficients of the logistic regression model?
To illustrate: the logistic (sigmoid) function is given by
$$f(x) = \frac{1}{1+\exp(-x)}$$
and it is used to model probabilities in the logistic regression model. Let $A$ be a dichotomous $(0,1)$ scored outcome and $X$ a design matrix. The logistic regression model is given by
$$P(A=1|X) = f(X \beta).$$
Note $X$ has a first column of constant $1$ (intercept) and $\beta$ is a column vector of regression coefficients. For example, when we have one (standard-normal) regressor $x$ and choose $\beta_0=1$ (intercept) and $\beta_1=1$, we can simulate the resulting 'distribution of probabilities'.
This plot reminds of the Beta distribution (as do plots for other choices of $\beta$) whose density is given by
$$g(y;p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)} y^{(p-1)} (1-y)^{(q-1)}.$$
Using maximum likelihood or methods of moments it is possible to estimate $p$ and $q$ from the distribution of $P(A=1|X)$. Thus, my question comes down to: what is the relationship between choices of $\beta$ and $p$ and $q$? This, to begin with, adresses the bivariate case given above.