From some machine learning notes talking about some discriminative classification methods, in particular logistic regression, where y is the class label (0 or 1) and x is the data, it is said that:
if $x|y = 0 \sim \mathrm{Poisson}(λ_0)$, and $x|y = 1 \sim \mathrm{Poisson}(λ_1)$, then $p(y|x)$ will be logistic.
Why is this true?
$Y$ has two possible values for any given value of $X$. According to the assumptions,
$$\Pr(X=x|Y=0) = \exp(-\lambda_0) \frac{\lambda_0^x}{x!}$$
and
$$\Pr(X=x|Y=1) = \exp(-\lambda_1) \frac{\lambda_1^x}{x!}.$$
Therefore (this is a trivial case of Bayes' Theorem) the chance that $Y=1$ conditional on $X=x$ is the relative probability of the latter, namely
$$\Pr(Y=1|X=x) = \frac{\exp(-\lambda_1) \frac{\lambda_1^x}{x!}}{\exp(-\lambda_1) \frac{\lambda_1^x}{x!} + \exp(-\lambda_0) \frac{\lambda_0^x}{x!}}= \frac{1}{1 + \exp(\beta_0 + \beta_1 x)}$$
where
$$\beta_0 = \lambda_1 - \lambda_0$$
and
$$\beta_1 = -\log(\lambda_1/\lambda_0).$$
That indeed is the standard logistic regression model.
