Example: Integer-valued data $y = (y_1, ...,y_n):$
$M_1 = Geometric(\theta_1)$ likelihood with $Beta(\alpha_1, \beta_1)$ prior on $\theta_1;$
$M_2=Poisson(\theta_2)$ likelihood with $Gamma(\alpha_2, \beta_2)$ prior on $\theta_2$.
The Bayes factor in favor of $M_1$ over $M_2$ turns out to be
$$\large \frac{\Gamma(\alpha_1 + \beta_1)\Gamma(n +\alpha_1)\Gamma(n \bar{y} + \beta_1)\Gamma(\alpha_2)(n + \beta_2)^{n\bar{y+\alpha_2}}(\prod\limits_{i=1}^n y_i !) }{\Gamma(\alpha_1)\Gamma(\beta_1)\Gamma(n+n\bar{y}+\alpha_1+\beta_1)\Gamma(n\bar{y}+\alpha_2)\beta^{\alpha_2}_2}$$
Diffuse priors: take $(\alpha_1, \beta_1)=(1,1)$ and $(\alpha_2, \beta_2)=(\epsilon,\epsilon)$ for some $\epsilon > 0.$
Bayes factor reduces to
$$\large \frac{\Gamma(n+1)\Gamma(n \bar{y} + 1)\Gamma(\epsilon)(n+\epsilon)^{n\bar{y}+\epsilon}(\prod\limits_{i=1}^n y_i !) }{\Gamma(n + n\bar{y} +2)\Gamma(n\bar{y}+\epsilon)\epsilon^\epsilon}$$
The Bayes factor $\rightarrow +\infty$ depending on the value of $\epsilon$.
I was wondering if someone could explain this example step-by-step, and perhaps break down the calculations for me? Is this a proof already explained somewhere, if so could someone link me to it? Many thanks in advance. :)