How to reduce credible sets in over-specified linear regression while maintaining global coverage...
$begingroup$
Vectors $A, B$ and covariance matrix $C$ are fixed and known. I have a vector of measurements, $Yinmathbb{R}^n$, sampled from
$$
M_1: Y sim N(Aalpha_* + Bbeta_*, C)
$$
My goal, roughly speaking, is to find a small 50% Bayesian credible set, $S$, for $alpha$ that is robust: $S$ should exhibit at least 50% coverage probability for all value of $alpha$, meaning that if $Ysim N(Aalpha_* + Bbeta_*, C)$ for fixed $(alpha_*, beta_*)$, then $alpha_* in S$ with probability at least 50%.
The classical approach of jointly estimating $(alpha, beta)$ yields unacceptably large $S$ when $beta=0$, since in this case we are estimating more degrees of freedom than are necessary. If we knew a priori that $beta=0$, we would estimate compute $S$ by classical regression on $Ysim N(Aalpha, C)$ and be done. Of course when $beta neq 0$, this approach results in degraded coverage probability for $S$.
I'm looking for a way to get smaller credible sets than classical methods afford when $beta=0$ (or near zero) while maintaining coverage probability for all $alpha_*, beta_*$.
Here is one idea. Consider the Bayesian posterior distribution of $alpha$ assuming flat (improper) prior distributions on $alpha, beta$:
$$
P(alpha | Y) propto int P(Y|alpha, beta), dbeta
$$
where the likelihood function $P(Y|alpha, beta)$ is given by the normal distribution above. The posterior $P(alpha|Y)$ is Gaussian with mean and variance determined by classical linear regression. Instead, introduce an informative prior $pi(beta)propto P(Y|beta)$, where
$$
P(Y|beta):=int P(Y|alpha, beta), dalpha
$$
so that
$$
P(alpha|Y) propto int P(Y|alpha, beta) P(Y|beta) , dbeta
$$
In cases were $beta$ is near 0, the integrand should be more concentrated near the mode of the likelihood function, thus reducing the size of $S$. When $beta neq 0$ and is 'observable', $P(Y|beta)$ should be concentrated near the correct value.
Using the same data in the likelihood and prior in this way feels abusive, but it's all I've come up with so far.
bayesian linear-regression
asked Jan 10 at 2:04
MathManMMathManM
427310
$endgroup$
add a comment |
$begingroup$
Vectors $A, B$ and covariance matrix $C$ are fixed and known. I have a vector of measurements, $Yinmathbb{R}^n$, sampled from
$$
M_1: Y sim N(Aalpha_* + Bbeta_*, C)
$$
My goal, roughly speaking, is to find a small 50% Bayesian credible set, $S$, for $alpha$ that is robust: $S$ should exhibit at least 50% coverage probability for all value of $alpha$, meaning that if $Ysim N(Aalpha_* + Bbeta_*, C)$ for fixed $(alpha_*, beta_*)$, then $alpha_* in S$ with probability at least 50%.
The classical approach of jointly estimating $(alpha, beta)$ yields unacceptably large $S$ when $beta=0$, since in this case we are estimating more degrees of freedom than are necessary. If we knew a priori that $beta=0$, we would estimate compute $S$ by classical regression on $Ysim N(Aalpha, C)$ and be done. Of course when $beta neq 0$, this approach results in degraded coverage probability for $S$.
I'm looking for a way to get smaller credible sets than classical methods afford when $beta=0$ (or near zero) while maintaining coverage probability for all $alpha_*, beta_*$.
Here is one idea. Consider the Bayesian posterior distribution of $alpha$ assuming flat (improper) prior distributions on $alpha, beta$:
$$
P(alpha | Y) propto int P(Y|alpha, beta), dbeta
$$
where the likelihood function $P(Y|alpha, beta)$ is given by the normal distribution above. The posterior $P(alpha|Y)$ is Gaussian with mean and variance determined by classical linear regression. Instead, introduce an informative prior $pi(beta)propto P(Y|beta)$, where
$$
P(Y|beta):=int P(Y|alpha, beta), dalpha
$$
so that
$$
P(alpha|Y) propto int P(Y|alpha, beta) P(Y|beta) , dbeta
$$
In cases were $beta$ is near 0, the integrand should be more concentrated near the mode of the likelihood function, thus reducing the size of $S$. When $beta neq 0$ and is 'observable', $P(Y|beta)$ should be concentrated near the correct value.
Using the same data in the likelihood and prior in this way feels abusive, but it's all I've come up with so far.
bayesian linear-regression
asked Jan 10 at 2:04
MathManMMathManM
427310
$endgroup$
add a comment |
$begingroup$
Vectors $A, B$ and covariance matrix $C$ are fixed and known. I have a vector of measurements, $Yinmathbb{R}^n$, sampled from
$$
M_1: Y sim N(Aalpha_* + Bbeta_*, C)
$$
My goal, roughly speaking, is to find a small 50% Bayesian credible set, $S$, for $alpha$ that is robust: $S$ should exhibit at least 50% coverage probability for all value of $alpha$, meaning that if $Ysim N(Aalpha_* + Bbeta_*, C)$ for fixed $(alpha_*, beta_*)$, then $alpha_* in S$ with probability at least 50%.
The classical approach of jointly estimating $(alpha, beta)$ yields unacceptably large $S$ when $beta=0$, since in this case we are estimating more degrees of freedom than are necessary. If we knew a priori that $beta=0$, we would estimate compute $S$ by classical regression on $Ysim N(Aalpha, C)$ and be done. Of course when $beta neq 0$, this approach results in degraded coverage probability for $S$.
I'm looking for a way to get smaller credible sets than classical methods afford when $beta=0$ (or near zero) while maintaining coverage probability for all $alpha_*, beta_*$.
Here is one idea. Consider the Bayesian posterior distribution of $alpha$ assuming flat (improper) prior distributions on $alpha, beta$:
$$
P(alpha | Y) propto int P(Y|alpha, beta), dbeta
$$
where the likelihood function $P(Y|alpha, beta)$ is given by the normal distribution above. The posterior $P(alpha|Y)$ is Gaussian with mean and variance determined by classical linear regression. Instead, introduce an informative prior $pi(beta)propto P(Y|beta)$, where
$$
P(Y|beta):=int P(Y|alpha, beta), dalpha
$$
so that
$$
P(alpha|Y) propto int P(Y|alpha, beta) P(Y|beta) , dbeta
$$
In cases were $beta$ is near 0, the integrand should be more concentrated near the mode of the likelihood function, thus reducing the size of $S$. When $beta neq 0$ and is 'observable', $P(Y|beta)$ should be concentrated near the correct value.
Using the same data in the likelihood and prior in this way feels abusive, but it's all I've come up with so far.
bayesian linear-regression
asked Jan 10 at 2:04
MathManMMathManM
427310
$endgroup$
Vectors $A, B$ and covariance matrix $C$ are fixed and known. I have a vector of measurements, $Yinmathbb{R}^n$, sampled from
$$
M_1: Y sim N(Aalpha_* + Bbeta_*, C)
$$
My goal, roughly speaking, is to find a small 50% Bayesian credible set, $S$, for $alpha$ that is robust: $S$ should exhibit at least 50% coverage probability for all value of $alpha$, meaning that if $Ysim N(Aalpha_* + Bbeta_*, C)$ for fixed $(alpha_*, beta_*)$, then $alpha_* in S$ with probability at least 50%.
The classical approach of jointly estimating $(alpha, beta)$ yields unacceptably large $S$ when $beta=0$, since in this case we are estimating more degrees of freedom than are necessary. If we knew a priori that $beta=0$, we would estimate compute $S$ by classical regression on $Ysim N(Aalpha, C)$ and be done. Of course when $beta neq 0$, this approach results in degraded coverage probability for $S$.
I'm looking for a way to get smaller credible sets than classical methods afford when $beta=0$ (or near zero) while maintaining coverage probability for all $alpha_*, beta_*$.
Here is one idea. Consider the Bayesian posterior distribution of $alpha$ assuming flat (improper) prior distributions on $alpha, beta$:
$$
P(alpha | Y) propto int P(Y|alpha, beta), dbeta
$$
where the likelihood function $P(Y|alpha, beta)$ is given by the normal distribution above. The posterior $P(alpha|Y)$ is Gaussian with mean and variance determined by classical linear regression. Instead, introduce an informative prior $pi(beta)propto P(Y|beta)$, where
$$
P(Y|beta):=int P(Y|alpha, beta), dalpha
$$
so that
$$
P(alpha|Y) propto int P(Y|alpha, beta) P(Y|beta) , dbeta
$$
In cases were $beta$ is near 0, the integrand should be more concentrated near the mode of the likelihood function, thus reducing the size of $S$. When $beta neq 0$ and is 'observable', $P(Y|beta)$ should be concentrated near the correct value.
Using the same data in the likelihood and prior in this way feels abusive, but it's all I've come up with so far.
bayesian linear-regression
bayesian linear-regression
asked Jan 10 at 2:04
MathManMMathManM
427310
asked Jan 10 at 2:04
MathManMMathManM
427310
asked Jan 10 at 2:04
MathManMMathManM
427310
asked Jan 10 at 2:04
MathManMMathManM
427310
asked Jan 10 at 2:04
MathManMMathManM
427310
427310
add a comment |
add a comment |
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