Bayesian Analysis Flashcards
[Derive the marginal distribution of β from a regression model with an (adjusted) conjugate prior.]
Consider the conjugate linear regression Bayesian framework. Here, the prior of sigma^2 follows an IG-2(delta,v) distribution. We need to derive the marginal distribution of beta. The question is partly how the delta and v will be processed into the posterior.
We use the standard Bayesian approach to solve a linear regression framework for a conjugate prior. v is introduced in the power and delta is introduced as intercept besides the (constructed) w and V. delta is introduced as a constant in sigma^2 tilde and v is introduced as extra degrees of freedom.
[Derive the marginal distribution of sigma^2 from a regression model with an (adjusted) conjugate prior.]
Consider the conjugate linear regression Bayesian framework. Here, the prior of sigma^2 follows an IG-2(delta,v) distribution. We need to derive the marginal distribution of beta. The question is partly how the delta and v will be processed into the posterior.
We use the standard Bayesian approach to solve a linear regression framework for a conjugate prior. The delta enters the scale parameter of the prior marginal distribution IG-2 as a constant and v as an addition to the degrees of freedom. In the derivation, we make use of a few tricks: integral equal to a constant, determinant of a vector times matrix and the decomposition rule.
To avoid the inclusion of prior influence into a Bayesian analysis, researchers often
use a non-informative (flat) prior specification, that is,
p(θ) ∝ 1,
where p(θ) is the prior density for the model parameters θ. A researcher claims that
this flat prior is not completely uninformative as it is informative about nonlinear
functions of the parameters θ. Is he right? Motivate your answer!
He is right. When you are uninformative about θ it does not have to be the case
that you are uninformative on a nonlinear transformation of θ as p(θ) ∝ 1 does not
automatically imply that p(h(θ)) ∝ 1 where h is a nonlinear function of θ. This is
due to the Jacobian of the transformation.
Let the prior be: β∣σ2∼N(b,σ2/γN)) and p(σ2)∝σ−2
Linear regression: y = eβ + ε
The posterior mean of β is a weighted average of b and the mean of y. Derive
the weight!
This prior is a conjugate prior where B = 1/γN. Use the results from the slides that the marginal posterior density β|y follows a t-distribution with a defined location and scale parameter. Take the location parameter. Use that X=e, e’e = N and e’y = Nmean(y). Then, arithmetically solve the formula of the location parameter.
Let the prior be: β∣σ2∼N(b,σ2/γN)) and p(σ2)∝σ−2
It is often stated that the prior does not influence the posterior mean for large
number of observations? Explain why this is not the case for this prior.
The prior variance decreases with N and hence the prior information gets larger
when the sample size increases.
