# Bayesian Credible interval in Poisson Regression

Assume that $$\pmb{y} = (y_1,...,y_n)$$ are count data which depends on a dummy predictor $$\pmb{x}$$ (x = 0 if male, x = 1 if female) such that : \begin{align} \pmb{y} \sim Poisson(\lambda) \end{align} With $$\lambda$$ being : $$log(\lambda) = \beta_0 + \beta_1 x_i$$ If after fitting a Bayesian GLM with appropriate priors, I get the following credibles intervals for parameters $$\beta_0$$ and $$\beta_1$$ : $$CI_{\beta_0} = [a,b]$$ $$CI_{\beta_1} = [c,d]$$

Is it then correct to specify the credible interval for $$\lambda$$ for the male and female population are :

1. Male population : $$CI_{\lambda} = \big[exp(a+c\times 0)\ ; \ exp(b + d x 0)] = [exp(a) \ ; \ exp(b) \big]$$
2. Female population : $$CI_{\lambda} = \big[exp(a+c) \ ; \ exp(b + d) \big]$$

## 1 Answer

$$\log(\lambda) = \beta_0$$ for the male population. So you get a credible interval of $$\lambda$$ by taking the exponentials of the bounds of the credible interval of $$\beta_0$$.

For the female population, $$\log(\lambda) = \beta_0 + \beta_1$$. Adding the bounds of the credible intervals of $$\beta_0$$ and $$\beta_1$$ to get a credible interval of $$\beta_0 + \beta_1$$ is not correct. If you have some simulations of the posterior distribution, for $$\beta_0$$ and $$\beta_1$$, you can add them to get simulations of the posterior distribution of $$\beta_0 + \beta_1$$, and take the exponential to get simulations of the posterior distribution of $$\lambda$$. Then you can construct a credible interval of $$\lambda$$ from these simulations, e.g. the equi-tailed credible interval by taking the quantiles, or the HPD interval.