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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 : $$ CI_{\lambda} = \big[exp(a+c) \ ; \ exp(b + d) \big] $$
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$\log(\lambda) = \beta_0$ 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.

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