La regresión logística puede ser descrito como una combinación lineal
$$ \eta = \beta_0 + \beta_1 X_1 + ... + \beta_k X_k $$
that is passed through the link function $g$:
$$ g(E(Y)) = \eta $$
where the link function is a logit function
$$ E(Y|X,\beta) = p = \text{logit}^{-1}( \eta ) $$
where $Y$ take only values in $\{0,1\}$ and inverse logit functions transforms linear combination $\eta$ to this range. This is where classical logistic regression ends.
However if you recall that $E(Y) = P(Y = 1)$ for variables that take only values in $\{0,1\}$, than $E(Y | X,\beta)$ can be considered as $P(Y = 1 | X,\beta)$. In this case, the logit function output could be thought as conditional probability of "success", i.e. $P(Y=1|X,\beta)$. Bernoulli distribution is a distribution that describes probability of observing binary outcome, with some $p$ parameter, so we can describe $Y$ as
$$ y_i \sim \text{Bernoulli}(p) $$
So with logistic regression we look for some parameters $\beta$ that togeder with independent variables $X$ form a linear combination $\eta$. In classical regression $E(Y|X,\beta) = \eta$ (we assume link function to be identity function), however to model $Y$ that takes values in $\{0,1\}$ we need to transform $\eta$ so to fit in $[0,1]$ range.
Now, to estimate logistic regression in Bayesian way you pick up some priors for $\beta_i$ parameters as with linear regression (see Kruschke et al, 2012), then use logit function to transform the linear combination $\eta$, so to use its output as a $p$ parameter of Bernoulli distribution that describes your $Y$ variable. So, yes, you actually use the equation and the logit link function the same way as in frequentionist case, and the rest works (e.g. choosing priors) like with estimating linear regression the Bayesian way.
The simple approach for choosing priors is to choose Normal distributions (but you can also use other distributions, e.g. $t$- or Laplace distribution for more robust model) for $\beta_i$'s with parameters $\mu_i$ and $\sigma_i^2$ that are preset or taken from hierarchical priors. Now, having the model definition you can use software such as JAGS to perform Markov Chain Monte Carlo simulation for you to estimate the model. Below I post JAGS code for simple logistic model (check here for more examples).
model {
# setting up priors
a ~ dnorm(0, .0001)
b ~ dnorm(0, .0001)
for (i in 1:N) {
# passing the linear combination through logit function
logit(p[i]) <- a + b * x[i]
# likelihood function
y[i] ~ dbern(p[i])
}
}
As you can see, the code directly translates to model definition. What the software does is it draws some values from Normal priors for a and b, then it uses those values to estimate p and finally, uses likelihood function to assess how likely is your data given those parameters (this is when you use Bayes theorem, see here for more detailed description).
The basic logistic regression model can be extended to model the dependency between the predictors using a hierarchical model (including hyperpriors). In this case you can draw $\beta_i$'s from Multivariate Normal distribution that enables us to include information about covariance $\boldsymbol{\Sigma}$ between independent variables
$$ \begin{pmatrix} \beta_0 \\ \beta_1 \\ \vdots \\ \beta_k \end{pmatrix} \sim \mathrm{MVN} \left(
\begin{bmatrix} \mu_0 \\ \mu_1 \\ \vdots \\ \mu_k \end{bmatrix},
\begin{bmatrix} \sigma^2_0 & \sigma_{0,1} & \ldots & \sigma_{0,k} \\
\sigma_{1,0} & \sigma^2_1 & \ldots &\sigma_{1,k} \\
\vdots & \vdots & \ddots & \vdots \\
\sigma_{k,0} & \sigma_{k,1} & \ldots & \sigma^2_k
\end{bmatrix}
\right)$$
...pero esto es entrar en detalles, así que vamos a dejar aquí.
El "Bayesiano" aquí es la elección de los priores, utilizando el teorema de Bayes y la definición del modelo en términos probabilísticos. Consulte aquí la definición de "modelo Bayesiano" y aquí por algún general de la intuición en el enfoque Bayesiano. Lo que también se puede observar es que la definición de los modelos es bastante sencillo y flexible con este enfoque.
Kruschke, J. K., Aguinis, H., & Joo, H. (2012). Ha llegado el momento: Bayesiano métodos para el análisis de datos en la organización de las ciencias. Organizacional Métodos De Investigación, 15(4), 722-752.
Gelman, A., Jakulin, A., Pittau, G. M., y Su, Y. S. (2008). Una débil informativo predeterminado antes de la distribución para la logística y otros modelos de regresión. Los Anales de la Estadística Aplicada, 2(4), 1360-1383.
