How to show this distribution is proper?
$begingroup$
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: $$y_isim mathrm{Bin}(n_i,theta_i),quad mathrm{logit}(theta_i) = alpha + beta x_i.$$ 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 $(alpha,beta)inmathbb R^2$.
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 $intint_{mathbb R^2}frac1{1+mathrm{exp}(alpha+beta x_1)}dalpha dbeta=infty$, 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?
probability-distributions improper-integrals bayesian density-function
asked Dec 7 '18 at 21:28
CephCeph
874415
$endgroup$
add a comment |
$begingroup$
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: $$y_isim mathrm{Bin}(n_i,theta_i),quad mathrm{logit}(theta_i) = alpha + beta x_i.$$ 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 $(alpha,beta)inmathbb R^2$.
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 $intint_{mathbb R^2}frac1{1+mathrm{exp}(alpha+beta x_1)}dalpha dbeta=infty$, 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?
probability-distributions improper-integrals bayesian density-function
asked Dec 7 '18 at 21:28
CephCeph
874415
$endgroup$
add a comment |
$begingroup$
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: $$y_isim mathrm{Bin}(n_i,theta_i),quad mathrm{logit}(theta_i) = alpha + beta x_i.$$ 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 $(alpha,beta)inmathbb R^2$.
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 $intint_{mathbb R^2}frac1{1+mathrm{exp}(alpha+beta x_1)}dalpha dbeta=infty$, 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?
probability-distributions improper-integrals bayesian density-function
asked Dec 7 '18 at 21:28
CephCeph
874415
$endgroup$
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: $$y_isim mathrm{Bin}(n_i,theta_i),quad mathrm{logit}(theta_i) = alpha + beta x_i.$$ 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 $(alpha,beta)inmathbb R^2$.
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 $intint_{mathbb R^2}frac1{1+mathrm{exp}(alpha+beta x_1)}dalpha dbeta=infty$, 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?
probability-distributions improper-integrals bayesian density-function
probability-distributions improper-integrals bayesian density-function
asked Dec 7 '18 at 21:28
CephCeph
874415
asked Dec 7 '18 at 21:28
CephCeph
874415
edited Dec 7 '18 at 21:38
Ceph
asked Dec 7 '18 at 21:28
CephCeph
874415
asked Dec 7 '18 at 21:28
CephCeph
874415
asked Dec 7 '18 at 21:28
CephCeph
874415
874415
add a comment |
add a comment |
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