This question is based on problem 14 from chapter 3 of Gelman et al.'s Bayesian Data Analysis.

We have four data points $y_i$ with covariates $x_i$, $i=1,\cdots,4$. We use the model: $$y_i\sim \mathrm{Bin}(n_i,\theta_i),\quad \mathrm{logit}(\theta_i) = \alpha + \beta x_i.$$ We place an improper uniform prior $(\alpha,\beta)\sim1$. The problem is to show that the resulting posterior distribution of $\alpha,\beta|y,x,n$ is proper over the range $(\alpha,\beta)\in\mathbb R^2$.

What I have tried:
The posterior distribution is (proportional to): $\prod_{i=1}^4 \left[ \left( \frac{\mathrm{exp}(\alpha+\beta x_i)}{1+\mathrm{exp}(\alpha+\beta x_i)} \right)^{y_i} \left( \frac1{1+\mathrm{exp}(\alpha+\beta x_i)}\right)^{n_i-y_i} \right]$. This is less than or equal to $\frac1{1+\mathrm{exp}(\alpha+\beta x_1)}$, and that upper bound is tight, since depending on the values of $y$ and $n$ the (unnormalized) posterior can reach that upper bound. But surely $\int\int_{\mathbb R^2}\frac1{1+\mathrm{exp}(\alpha+\beta x_1)}d\alpha d\beta=\infty$, since for any $\delta$ it's easy to see that the integrand is greater than $1-\delta$ over an infinite region of $\mathbb R^2$.

Where have I gone wrong?


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