My interpretation of the usual Logistic regression is that it maximizes the Likelihood of the data given the parameters. I.e., given responses $Y_j$ and random variables $X_{ij}$, it maximizes the likelihood that the data is generated from
$$Y_j = 1 \ if \ \epsilon + \sum_i \beta_i X_{ij} > 0, \ else \ Y_j = 0 $$
where $\epsilon$ is logistic noise.
When a regularization is added, e.g. of the form
$$ \sum_j (Y_j - \tilde{Y}(X_{j}))^2 + \alpha ||\tilde{Y}(X)||$$
the interpretation I found in the literature is that it corresponds to maximize the posterior of the parameters given the data, $P(\beta|Y)$, by interpreting $\log P(\beta) = \alpha ||\tilde{Y}(X)||$ (regularization is the prior).
Now, in standard textbook material, the interpretation of $\beta_i$ is that the log of the odds ratio increases by $\beta_i$ when $X_i$ increases by 1. My intuition for this is that because in non-regularized logistic regression we are maximizing the Likelihood, the interpretation that Y (i.e. data) increases with $X_i$ seems consistent because the random variable is the data (and the parameters are being conditioned).
However, I do not see how this interpretation holds in a Logistic regression with regularization. Specifically, when we maximize the posterior, we end with a quantity about $\beta$ ($P(\beta|X,Y))$).
This answer claims that the interpretation is still the same, but gives no justification as of why. If yes, why can we still argue that $\exp(\beta_1)$ is the increase of the odds ratio by one unit of $X$?
Hand-wavingly, I would argue something as follows: $\langle \exp(X_1) \rangle$ is the increase of the odds ratio when $\beta_1$ is increased by 1.
