Hi,
I am a new user of winbugs and I am trying to perform a MCMC analysis of
the model error in GLS Regional Regression. Below I fully describe the
problem and some alternatives I am thinking for implementation in two
cases (model error large and small). My question is how to implement this
in winbugs. I tried but trap messages appear due numerical computation of
inverse matrix during simulation. Does anyone has a similar example that
I could use as basis for my code or just a few hints ...
Thanks
Eduardo
We write the general model
y = X b + D + E
where
D ~ N(0,
g I) are the independent model
errors
E ~ N(0,
S ) are the sampling errors
because y consist of estimators.
Sometimes the model is written
y = X b + N
where
N ~
N(0,
L(g))
with
L(g)
= S +
g I
To make this easy to implement in Winbugs, or any MCMC
code,
we precompute the B matrix so
that
B BT =
S (G)
and
Inv(B) = C
PROBLEM
The data is given as y the skews and X the matrix of covariates,
and
S the specified covariance of the
y estimators. We wish to compute the posterior distribution of b
and
g. Thus we may compute posterior
distribution of:
regression coefficients b
model error variance g
Unobserved innovations of Vi and Ei
In the case wherein
g I is large compared
to
S, the problem is like OLS
regression and E is a small correction. In the opposite case
when
g is small, E is a very small
correction and D will be large. I think this effects how we should set up
the problem.
MCMC #1: Formulation for large
g.
Write the problem as
D = B V
E = y - X b - D
Likelihood function based upon
(V)i = Vi ~ N(0,1)
(E)i = Ei ~ N(0, g)
MCMC generates posterior distribution of V, E, b and g.
Need a prior for g and b.
g ~ Gamma(0.1, 0.1)
b ~ N(0, big)
Because BBT is small compared to gI, the values of Vi can bounce around without causing great harm to the likelihood for the Ei values because D is small. And in a similar way, the data may not really tell us much about V because the overwhelming major source of the error is due to E. Thus the E values will essentially be equal to y - XB, as they would be in OLS regression.
MCMC #2: Formulation for large g.
A compact representation of the model above without explicit Ei values.
Likelihood function based upon
(V)i = Vi ~ N(0,1)
y - X b - B V ~ N(0, g)
MCMC generates posterior distribution of V, b and g. Need prior on b and g.
MCMC #3: Formulation for SMALL g.
Take the fundamental equation y = XB + BV + E and solve for V to obtain:
(E)i ~ N(0, g)
C[y - Xb - E]i ~ N( 0, 1)
Compute posterior for g, b, and Ei. Here Ei is viewed as a small correction to the GLS model with iid errors V, which are defined implicitly. Need prior on b and g.
MCMC #2: Formulation for small g. Equivalent alternative with explicit Vi.
V = C[y - Xb - E ]
Likelihood function based upon
(V)i = Vi ~ N(0,1)
(E)i = Ei ~ N(0, g)
MCMC generates posterior distribution of V, E, b and g. Need prior on b and g.
