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2.11 Monte Carlo Estimates of Model Probabilities\newline
\newline
In this section a Monte Carlo method for estimating posterior model
probabilities based on independent MCMC sampling of two (or more) different
models is presented; this is a modified version of the approach suggested in
Congdon (2006d) and has the advantage of allowing simple iteration specific
model averaging as compared to other approaches. Let $\theta =(\theta
_{1},...,\theta _{K}$) denote the parameter set over all K models, with
dimension (d$_{1}$,..d$_{K}$). Assume a model indicator m $\in $(1,\dots
,K), such that given m=j, $\theta _{j}$ defines the likelihood for y=(y$_{1}$%
,...y$_{n}$) and y is independent of parameters of other models $\theta
_{k},k\neq j$ (Godsill, 2001; Carlin \&\ Chib, 1995). This means that the
marginal likelihood given m=j is

$\qquad \qquad p(y|m=j)=\int p(y|\theta ,m=j)p(\theta |m=j)d\theta $

$\qquad \qquad \qquad \qquad \qquad =\int p(y|\theta _{j},m=j)p(\theta
|m=j)d\theta _{j}$\newline
This framework replicates that of section 2.1 in Carlin \&\ Chib (1995). As
these authors mention, the form of the cross-model priors p($\theta
_{k}|m=j, $ $j\neq k)$ within the product p($\theta |m=j)=\underset{k}{\prod 
}$p($\theta _{k}|m=j)$ is arbitrary, though proper densities are required in
order that p($\theta |m=j)$ integrates to 1. \newline
\newline
Here the assumption is made that p($\theta _{k}|m=j_{1})=$p($\theta
_{k}|m=j_{2})$ for all \{k$\neq j_{1},$ k $\neq j_{2}\},$ so that there will
be K cross-model priors, rather than K(K-1). So in a three model situation p(%
$\theta _{3}|m=1)=p(\theta _{3}|m=2)=g_{3},$ p($\theta _{2}|m=1)=p(\theta
_{2}|m=3)=g_{2},$ and p($\theta _{1}|m=2)=p(\theta _{1}|m=3)=g_{1}.$ \newline
\linebreak With this framework, one may use the output \{$\theta
_{j}^{(t)},t=1,..,T;j=1,..,K\}$ from the K models to estimate iteration
specific model weights and overall posterior model weights. This involves a
sample of the same length (say T iterations) from the posteriors p($\theta
_{j}|y,m=j)$ of all K models under consideration. Such samples might be
obtained by running models in parallel or by running them separately and
then pooling the output. This is conceptually distinct from product space
search algorithms, such as that of Carlin and Chib (1995), when the model j
parameters are only updated when model j is visited. To see how model
weights are estimated from such output, first write

$\qquad p(m=j{|}y)=\int p(m=j,\theta |y)d\theta =\int p(m=j{|}y,\theta
)p(\theta {|}y)d\theta .$\newline
Then a Monte Carlo estimate of P(m=j{\TEXTsymbol{\vert}}y) is obtainable as

$\qquad \overset{-}{w}_{j}=\overset{T}{\underset{t=1}{\text{ }\sum }}p(m=j{|}%
y,\theta ^{(t)})/T\newline
$where \{$\theta $$^{(t)}$= ($\theta _{1}^{(t)}$,\dots ,$\theta _{2}^{(t)}$%
,\dots ,$\theta _{K}^{(t)}$), t=1,T\} are T samples of parameters in all
models.\newline
\newline
For obtaining weights at a a particular iteration let

$w_{j}^{(t)}=p(m=j{|}y,\theta ^{(t)})=\frac{p(m=j,y,\theta ^{(t)})}{%
p(y,\theta ^{(t)})}=\frac{p(y{|}m=j,\theta ^{(t)})p(\theta ^{(t)}{|}%
m=j)p(m=j)}{p(y,\theta ^{(t)})}\qquad (2.15)$\newline
The numerator in (2.15) contains the term

$p(\theta |m=j)$=$p(\theta _{1}|m=j)p(\theta _{2}|m=j)..p(\theta
_{j}|m=j)...p(\theta _{K}|m=j).$ \newline
From above the cross-model prior is arbitrary and the simplifying assumption

$\ p(\theta _{h}{|}m=j)=g_{h}$ \ \ \ \ \ \ ($all$ $j\neq h)$\newline
is made, where g$_{h}$ is a proper density. So\newline
$p(\theta _{h}{|}m=1)=p(\theta _{h}{|}m=2)=..=p(\theta _{h}{|}%
m=h-1)=p(\theta _{h}{|}m=h+1)$\newline
$=..=p(\theta _{h}{|}m=K)=g_{h}$.\newline
and there are K cross model priors \{g$_{1}$,..,g$_{K}$\}.\newline
\newline
As in the model choice procedures of Gelfand and Dey (1994) and Carlin \&\
Chib (1995), one might set g$_{h}$ to be an estimate of p($\theta
_{h}|m=h,y),$ namely the posterior density of $\theta _{h}$ given y and m=h.
This choice of cross-model prior contrasts with the simplification of taking
p($\theta _{k}|m=j,k\neq j)$ uniform, as considered by Congdon (2006d). It
follows that

$p(\theta |m=j)=p(\theta _{j}{|}m=j)\underset{h\neq j}{\overset{K}{\prod }}%
p(\theta _{h}{|}m=j)=p(\theta _{j}{|}%
m=j)[g_{1}g_{2}..g_{j-1}g_{j+1}..g_{K}]. $\linebreak Then

$\qquad w_{j}^{(t)}=\frac{p(y{|}m=j,\theta _{j}^{(t)})p(\theta _{j}^{(t)}{|}%
m=j)[\underset{h\neq j}{\Pi }g\overset{(t)}{_{h}}]p(m=j)}{p(y,\theta ^{(t)})}%
\qquad \qquad \qquad (2.16)$\newline
The denominator in (2.16) can be written

$\qquad p(y,\theta ^{(t)})=\overset{K}{\underset{k=1}{\sum }}p(y,\theta
^{(t)},m=k)$

\qquad \qquad $\qquad =\overset{K}{\underset{k=1}{\sum }}\{p(y{|}\theta
^{(t)},m=k)p(\theta _{k}^{(t)}{|}m=k)[\underset{h\neq k}{\Pi }g\overset{(t)}{%
_{h}}]p(m=k)\}$

$\qquad \qquad \qquad =\overset{K}{\underset{k=1}{\sum }}p(y{|}\theta
_{k}^{(t)},m=k)p(\theta _{k}^{(t)}{|}m=k)[\underset{h\neq k}{\Pi }g\overset{%
(t)}{_{h}}]p(m=k).\newline
$Then

$\qquad w_{j}^{(t)}=\frac{p(y{|}m=j,\theta _{j}^{(t)})p(\theta _{j}^{(t)}{|}%
m=j)[\underset{h\neq j}{\Pi }g\overset{(t)}{_{h}}]p(m=j)}{\overset{K}{%
\underset{k=1}{\sum }\text{ }}p(y{|}\theta _{k}^{(t)},m=k)p(\theta _{k}^{(t)}%
{|}m=k)[\underset{h\neq k}{\Pi }g\overset{(t)}{_{h}}]p(m=k)}$\newline
One may divide through by the product of the K cross-model priors [g$_{1}$,g$%
_{2},$...g$_{K}$] giving

$\qquad w_{j}^{(t)}=\frac{\left\{ \frac{p(y{|}m=j,\theta _{j}^{(t)})p(\theta
_{j}^{(t)}{|}m=j)p(m=j)}{g\overset{(t)}{_{j}}}\right\} }{\overset{K}{%
\underset{k=1}{\sum }\text{ }}\left\{ \frac{p(y{|}\theta
_{k}^{(t)},m=k)p(\theta _{k}^{(t)}{|}m=k)p(m=k)}{g\overset{(t)}{_{k}}}%
\right\} }\qquad \qquad \qquad \qquad (2.17)$\newline
Consider the case K=2. Using the previous notation for the unnormalised
posterior as

$\qquad p^{\ast }(\theta _{j}^{(t)}|y,m=j)=p(y{|}m=j,\theta
_{j}^{(t)})p(\theta _{j}^{(t)}{|}m=j)$\newline
one obtains

$w_{j}^{(t)}=\frac{\left\{ \frac{p^{\ast }(\theta _{j}^{(t)}|y{,}m=j)p(m=j)}{%
g\overset{(t)}{_{j}}}\right\} }{\left\{ \frac{p^{\ast }(\theta _{1}^{(t)}|y{,%
}m=1)p(m=1)}{g\overset{(t)}{_{1}}}]+[\frac{p^{\ast }(\theta _{2}^{(t)}|y{,}%
m=2)p(m=2)}{g\overset{(t)}{_{2}}}\right\} }$\newline
\newline
Incorporating the above assumptions about cross-model priors and the
division through by $\underset{k=1}{\overset{K}{\prod }}$g$_{k}$ as in
(2.17) gives a posterior mean of $p(m=j{|}y)$ over all iterations

$w_{j}=\overset{T}{\underset{t=1}{\text{ }\sum }}\left\{ \frac{\left[ \frac{%
p^{\ast }(\theta _{j}^{(t)}|y{,}m=j)p(m=j)}{g\overset{(t)}{_{j}}}\right] }{%
\left[ \frac{p^{\ast }(\theta _{1}^{(t)}|y{,}m=1)p(m=1)}{g\overset{(t)}{_{1}}%
}+..\frac{p^{\ast }(\theta _{j}^{(t)}|y{,}m=j)p(m=j)}{g\overset{(t)}{_{j}}}%
+..+\frac{p^{\ast }(\theta _{K}^{(t)}|y{,}m=K)p(m=K)}{g\overset{(t)}{_{K}}}%
\right] }\right\} \qquad $(2.18)\newline
The densities of the w$\QATOP{(t)}{k}$may be skewed in which case the
posterior median of the w$\QATOP{(t)}{k}$ will be relevant as a Monte Carlo
summary of location.\newline
\newline
While the above development uses the framework of Carlin \&\ Chib (1995)
there are two main differences. First, the cross-model priors do not
function here as linking densities in a model search algorithm since there
is no switching between models. Instead all models are sampled from at each
iteration. This avoids the problems involved in tuning prior model
probabilities or 'jump' proposal densities in product space algorithms to
ensure that models are visited sufficiently often (Green \&\ O'Hagan, 1998;
Friel \&\ Pettitt, 2006). The second main difference follows from the first.
Switching between models under product space search implies a binary form of
model averaging: if model 1 is selected at several successive iterations
then the averaged parameters at these iterations are in fact the parameter
values sampled from one model only. Under the approach here averaging is on
the basis of continuous quantities, namely the $w_{k}^{(t)}$ obtained for
all models, so some weight in the average at each iteration is given to
inferior models. This may be important when models are closely competing.
Thus at iteration t, one obtains

$\qquad \qquad q_{k}^{(t)}=\log \{p(y{|}\theta _{k}^{(t)},m=k)p(\theta
_{k}^{(t)}{|}m=k)[\underset{a\neq k}{\Pi }g_{a}]p(m=k)\}$, \newline
and deviations $\Delta q_{k}^{(t)}=q_{k}^{(t)}-\underset{k}{max}%
(q_{k}^{(t)}) $, with w$_{k}^{(t)}$ obtained by exponentiating:

\qquad \qquad $w_{k}^{(t)}$=$\frac{exp(\Delta q_{k}^{(t)})}{\underset{k}{%
\sum }exp(\Delta q_{k}^{(t)})}$.\newline
\newline
This approach is illustrated with the nodal involvement dataset considered
in section 2.3 above. The cross-model priors $p(\theta _{1}|m=2)=g_{1}$ and $%
p(\theta _{2}|m=1)=g_{2}$ use the same parameters as the importance
densities used there. A single chain run of 10000 iterations is run (with
burn-in of 1000, giving T=9000) comparing models M8 (model 1) and M9 (model
2) of Chib (1995). The densities of the weights $w\QATOP{(t)}{k}$ are
slightly skewed, so posterior medians are used as summaries. The posterior
medians of $w\QATOP{(t)}{1}$ and $w\QATOP{(t)}{2}$are 0.778 and 0.222 (as
compared to means 0.742 and 0.258). One therefore obtains 3.505 as the ratio
of posterior median weights. This quantity is also the posterior median of
the ratios ($w\QATOP{(t)}{1}$/$w\QATOP{(t)}{2}),$ whereas the mean of this
ratio is considerably higher (at 8.8). It may be noted that the arithmetic
averages of the ratios $\left[ \frac{p^{\ast }(\theta _{j}^{(t)}|y{,}m=j)}{g%
\overset{(t)}{_{j}}}\right] $ in (2.18) are -35.57 (model M8) and -36.87
(model M9), close to the estimates obtained using the bridge sampling method.%
\newline
\newline
One may also use the iteration specific weights (w[k] in the code) to
average over models at each iteration (e.g. predictions of patient specific
nodal involvement probabilities). For example, Figure 2.1 contains the
posterior densities of the involvement probability $\pi _{ij}$ of patient
i=17 under models j; kernel plots are obtained using the matlab code of
Morgan (2000). These posterior densities are for models M8 (model j=1), M9
(model j=2) and the average model (model j=a) with the model averaged
density obtained using the iteration specific averages

$\qquad \qquad \pi \QATOP{(t)}{a}=\pi \QATOP{(t)}{1}w\QATOP{(t)}{1}+\pi 
\QATOP{(t)}{2}w\QATOP{(t)}{2}.$\newline
\newline
The code is as follows:\newline
model \{for (i in 1:N) \{ y1[i] \TEXTsymbol{<}- y[i]; y1[i] \symbol{126}%
dbern(pi[i,1])

y2[i] \TEXTsymbol{<}- y[i]; y2[i] \symbol{126}dbern(pi[i,2])

\# two models

nu[i,1] \TEXTsymbol{<}- b1[1] + b1[2]*log(x1[i])+ b1[3]*x2[i] + b1[4]*x3[i]

nu[i,2] \TEXTsymbol{<}- b2[1] + b2[2]*log(x1[i]) + b2[3]*x2[i] + b2[4]*x3[i]

+ b2[5]*x4[i]

\# averaged probability at each iteration

pi.a[i] \TEXTsymbol{<}- w[1]*pi[i,1]+w[2]*pi[i,2]

for (j in 1:2) \{pi[i,j] \TEXTsymbol{<}- phi(nu[i,j])

\# log-likelihoods

LL[i,j] \TEXTsymbol{<}- y[i]*log(phi(nu[i,j])) +
(1-y[i])*log(1-phi(nu[i,j]))\}\}

\# priors

for (j in 1:4) \{b1[j] \symbol{126}dnorm(M[j],P[j])

PG1[j] \TEXTsymbol{<}- 1/pow(seG1[j],2)

Pr[j,1] \TEXTsymbol{<}- 0.5*log(P[j]/6.28)-0.5*P[j]*pow(b1[j]-M[j],2)

g[j,1] \TEXTsymbol{<}- 0.5*log(PG1[j]/6.28)-0.5*PG1[j]*pow(b1[j]-MG1[j],2)\}

for (j in 1:5) \{b2[j] \symbol{126}dnorm(M[j],P[j])

PG2[j] \TEXTsymbol{<}- 1/pow(seG2[j],2)

Pr[j,2] \TEXTsymbol{<}- 0.5*log(P[j]/6.28)-0.5*P[j]*pow(b2[j]-M[j],2)

g[j,2] \TEXTsymbol{<}- 0.5*log(PG2[j]/6.28)-0.5*PG2[j]*pow(b2[j]-MG2[j],2)\}

\# Ratio of Pstar to importance sample at each iteration

for (k in 1:2) \{SL[k] \TEXTsymbol{<}- max(logR[k]-maxR,-500)

log.Pstar[k] \TEXTsymbol{<}- sum(LL[,k])+sum(Pr[1:p[k],k])+log(PriorMod[k])

logR[k] \TEXTsymbol{<}- log.Pstar[k] - sum(g[1:p[k],k])

logQ[k] \TEXTsymbol{<}- sum(LL[,k])+sum(Pr[1:p[k],k])- sum(g[1:p[k],k])

expSL[k] \TEXTsymbol{<}- exp(SL[k])

\# model weights at iteration t

w[k] \TEXTsymbol{<}- expSL[k]/sum(expSL[]) \}

\# maximum of model likelihoods at iteration t

maxR \TEXTsymbol{<}- ranked(logR[],2)

\# quantities to monitor

mon[1] \TEXTsymbol{<}- w[1]; mon[2] \TEXTsymbol{<}- w[2]; mon[3] \TEXTsymbol{%
<}- w[1]/w[2]

mon[4] \TEXTsymbol{<}- log(w[1]/w[2])\}\newline
\newline
Apart from the case study data, the other inputs in the data file are 
\newline
list(MG1=c(-0.59,1.42,1.06,1.00),MG2=c(-0.68,1.65,1.06,0.86,0.66),seG1=c(0.39,%
\newline
0.67,0.48,0.41),seG2=c(0.41,0.69,0.49,0.44,0.445),p=c(4,5),PriorMod=c(0.5,0.5),%
\newline
N=53,P=c(0.01,0.04,0.04,0.04,0.04),M=c(0,0.75,0.75,0.75,0.75).\newline
\newline
For models with subject level random effects b$_{i}$, let fixed effects
parameters be denoted $\eta $ and hyperparameters for the random effects be
denoted $\Phi $, with $\theta =(\beta ,\phi ).$ Define the integrated
likelihood

\qquad \qquad \qquad p(y\TEXTsymbol{\vert}m=k,$\theta )=$ $\int \overset{}{%
\underset{}{\overset{N}{\underset{i=1}{\Pi }}p(\text{y}_{\text{i}}|\beta
,b_{i})}}p(b_{i}|\Phi )db_{i}.$\newline
Then the posterior mean model weight is $\overset{-}{w}_{j}$= $\underset{t=1}%
{\overset{T}{\Sigma }}w\QATOP{(t)}{j}/T$ , where

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $w_{j}^{(t)}=\frac{p(y{|}m=j,\theta
_{j}^{(t)})p(\theta _{j}^{(t)}{|}m=j)[\underset{h\neq j}{\Pi }g\overset{(t)}{%
_{h}}]p(m=j)}{p(y,\theta ^{(t)})}$\newline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \newline
and the cross-model priors g$_{h}$ =p($\theta _{j}|m=h)$, h$\neq j,$ involve
(proper density) approximations for p$(\theta _{j}|y,m=j)=$ p$(\eta
_{j},\Phi _{j}|y,m=j).$ In conjugate models (e.g. Poisson models with gamma
random effects) the integrated likelihoods p(y\TEXTsymbol{\vert}m=k,$\theta
) $ are available analytically, whereas in general linear mixed models
devices such as numerical integration by Simpson's rule or quadrature
(Aitkin and Alf\`{o}, 2003), or importance sampling (Geweke, 1989) are
required, and are applied at each iteration to obtain p(y\TEXTsymbol{\vert}%
m=k,$\theta ^{(t)})$. For example consider a scalar random effect in a
Poisson lognormal model with y$_{i}$ \symbol{126}Po(E$_{i}\mu _{i})$ where E$%
_{i}$ are expected events, and log($\mu _{i})$=X$_{i}\beta +b_{i},$ where b$%
_{i}\backsim $ N(0,1/$\tau _{b}).$ Then\ an appropriate range for Simpson's
integration over b$_{i}$ may be obtained from an earlier MCMC\ run.\newline
\newline
2.12 References\newline
\newline
Aitkin, M (1991) Posterior Bayes factors. J. Roy. Stat. Soc. B 53, 111-142%
\newline
Aitkin, M (1997) The calibration of P-values, posterior Bayes factors and
the AIC from the posterior distribution of the likelihood, Statistics and
Computing, 7, 253-261\newline
Alqallaf, F, Gustafson, P (2001) On cross-validation of Bayesian models.
Canadian Journal of Statistics, 29, 333-340.\newline
Azevedo-Filho, A, Shachter, R (1994) Laplace's method approximations for
probabilistic inference in belief networks with continuous variables. In
Proc. 10th Conf. on Uncertainty in Artificial Intelligence, Seattle, de
Mantaras, R, Poole, D (eds). Morgan Kaufmann, San Mateo, 28-36\newline
Barbieri, M, Berger, J (2004) Optimal predictive model selection. Ann.
Statist. 32 , 870--897\newline
Besag J, Green P, Higdon D, Mengersen, K (1995) Bayesian computation and
stochastic systems, Statistical Science, 10, 3-41\newline
Bos, C (2002) A comparison of marginal likelihood computation methods. In
COMPSTAT 2002: Proceedings in Computational Statistics, H\"{a}rdle, W, Ronz,
B (eds) 111-117.\newline
Brooks, S (1998) Markov Chain Monte Carlo Method and its application The
Statistician, 47, 69-100.\newline
Brooks, S (2002) Discussion to Spiegelhalter et al. Journal of the Royal
Statistical Society, Series B, 64, 616-618\newline
Brooks, S, Gelman, A. (1998) General Methods for Monitoring Convergence of
Iterative Simulations. J. Comp. Graph. Stat., 7, 434-455. \newline
Brooks, S P and Roberts, G O (1998) Assessing convergence of Markov Chain
Monte Carlo algorithms. Statistics and Computing. 8, 319-335.\newline
Burnham K, Anderson, D (2002) Model Selection and Multimodel Inference: A
Practical Information-theoretic Approach, 2nd Edition. Springer-Verlag: New
York \newline
Carlin, B, Chib, S (1995) Bayesian model choice via Markov chain Monte Carlo
methods. J. Roy. Stat. Soc B 57, 473-484\newline
Carlin, B and Louis, T (1997) Bayes and Empirical Bayes Methods for Data
Analysis, Chapman and Hall, London\newline
Chib, S (1995) Marginal likelihood from the Gibbs output. J. Amer. Stat.
Assoc, 90, 1313-1321\newline
Chib, S, Jeliazkov, I (2001) Marginal likelihood from the
Metropolis-Hastings output. J. Amer. Stat. Assoc., 96, 270-281.\newline
Chipman H, George E, McCulloch R (2001) The practical implementation of
Bayesian model selection. In Model Selection, IMS Monograph 38, Lahiri, P
(ed.). Beachwood, OH: Institute of Mathematical Statistics, 67-116. \newline
Chu, P, Zhao, X ( 2004) Bayesian change-point analysis of tropical cyclone
activity: the Central North Pacific case. J. Climate, 17, 4893-4901\newline
Congdon, P (2000) A Bayesian approach to prediction using the gravity model,
with an application to patient flow modelling. Geographical Analysis, 32,
205-224\newline
Congdon, P (2005) Bayesian predictive model comparison via parallel
sampling. Computational Statistics and Data Analysis, 48, 735-753\newline
Congdon, P (2006a) A model for nonparametric spatially varying regression
effects. Comp. Stat. \& Data Ana., 50, 422-445\newline
Congdon, P (2006b) A model framework for mortality and health data
classified by age, area and time. Biometrics, 61 (forthcoming)\newline
Congdon, P (2006c) Bayesian model comparison via parallel model output. J.
Statist. Computation \& Simulation, 76, 149-165\newline
Congdon, P (2006d) Bayesian model choice based on Monte Carlo estimates of
posterior model probabilities. Computational Statistics \& Data Analysis,
50, 346-357\newline
Czado, C Raftery, A (2005) Choosing the link function and accounting for
link uncertainty in generalized linear models using Bayes Factors.
Statistical Papers, to appear.\newline
Dellaportas, P, Roberts, G (2003) Introduction to MCMC. In: Spatial
Statistics and Computational Methods (M\o ller, J, ed). New York: Springer,
1--42.\newline
Dempster, A (1997) The direct use of the likelihood ratio for significance
testing, Statistics and Computing, 7, 247-252\newline
Dey, D, Peng, F, Gelfand, A (1997) Overdispersed generalized linear models.
Journal of Statistical Planning and Inference 64, 93-107\newline
Elsalloukh, H, Young, D, Guardiola, J (2005) The epsilon-skew exponential
power distribution family. Far East Journal of Theoretical Statistics, 16,
97-112.\newline
Fernandez, C, Ley, E, Steel, M (2001) Benchmark priors for Bayesian model
averaging. J. Econometrics, 100, 381-427.\newline
Friel, N, Pettitt, A (2006) Marginal likelihood estimation via power
posteriors. Working Paper, Department of Statistics, University of Glasgow.%
\newline
Fruhwirth-Schnatter, S (1995) Bayesian model discrimination and Bayes
factors for linear Gaussian state space models. J. Roy. Stat. Soc B 57,
237--46.\newline
Fruhwirth-Schnatter, S (2004) Estimating marginal likelihoods for mixture
and Markov switching models using bridge-sampling techniques, Econometrics
J., 7, 143-167\newline
Gamerman, D (1998) Markov Chain Monte Carlo, Chapman and Hall, London\newline
Gelfand, A (1996) Model determination using sampling based methods, Chapter
9 in Markov Chain Monte Carlo in Practice, Gilks, W, Richardson, S,
Spiegelhalter, D (Eds), Chapman \& Hall/CRC, Boca Raton\newline
Gelfand, A, Dey, D, Chang, H (1992) Model determination using predictive
distributions with implementations via sampling-based methods, in Bayesian
Statistics 4 (eds J. Bernardo et al) Oxford Univ Press, 147-168\newline
Gelfand, A, Dey, D (1994) Bayesian model choice: asymptotics and exact
calculations. J. Roy. Stat. Soc. B 56, No.3, 501-514\newline
Geisser, S, Eddy, W (1979) A predictive approach to model selection. J. Am.
Stat. Assoc. 74, 153-160\newline
Gelman, A, Rubin, D (1995) Avoiding model selection in Bayesian social
research. Discussion of \textquotedblleft Bayesian model selection in social
research,\textquotedblright\ by A. Raftery. Sociological Methodology 1995,
165--173.\newline
Gelman, A, Meng, X, Stern, H (1996) Posterior predictive assessment of model
fitness via realized discrepancies. Stat. Sin. 6, 733-807\newline
George, E, McCulloch, R (1993) Variable selection via Gibbs sampling. J. Am.
Stat. Assn., 88, 881-89.\newline
George, E (1999) Bayesian model selection. In Encyclopedia of Statistical
Sciences Update (Vol. 3), Kotz S, Read C, Banks D (eds). Wiley: New York,
39--46.\newline
Geyer, C (1992) Practical Markov Chain Monte Carlo, Stat. Sci. 7, 473-511%
\newline
Godsill, S (2001) On the relationship between MCMC model uncertainty
methods. J. Comput. Graph. Stat.,10, 230-48\newline
Green, P (1995) Reversible jump Markov chain Monte Carlo computation and
Bayesian model determination. Biometrika 82, 711--732\newline
Green, P, O'Hagan, A (1998) Model choice with MCMC on product spaces without
using pseudo-priors. Tech. Report 98-13, Univ. Nottingham\newline
Han, C, Carlin, B (2001) Markov chain Monte Carlo methods for computing
Bayes factors: a comparative review. J. Am. Stat. Assn, 96, 1122--1132%
\newline
Ibrahim, J, Chen, M.-H, Sinha, D (2001) Bayesian Survival Analysis.
Springer-Verlag, New York.\newline
Jeffreys, H (1961) Theory of Probability (3rd ed.). Oxford Univ. Press%
\newline
Kass, R, Raftery, A (1995) Bayes factors. J. Am. Stat. Assoc. 90, 773-79%
\newline
Kuo, L, Mallick, B (1998) Variable selection for regression models. Sankhya
B, 60, 65--81.\newline
Kuo, L, Peng, F (2000) A mixture-model approach to the analysis of survival
data. In: Generalized Linear Models: A Bayesian Perspective. Dey, D, Ghosh,
S, Mallick, B (eds) Marcel Dekker, 255-270\newline
Lawson, A, Clark, A (2002) Spatial mixture relative risk models applied to
disease mapping. Statist. Med., 21, 359-370\newline
Lewis, S, Raftery, A (1997) Estimating Bayes factors via posterior
simulation with the Laplace-Metropolis estimator. J. Amer. Statist. Assoc.
92, 648-655.\newline
Liang, F, Pauloy, R, Molinaz, G, Clyde, M, Berger, J (2005) Mixtures of
g-priors for Bayesian variable selection. Duke University, ISDS Discussion
Paper 05-12.\newline
Lopes, H, West, M (2004) Bayesian model assessment in factor analysis.
Statistica Sinica, 14, 41-67\newline
McCullagh, P, Nelder, J (1989) Generalized Linear Models. Chapman and Hall:
London\newline
McCulloch, R, Tsay, R (1994) Statistical analysis of economic time series
via Markov switching models. J. Time Series Ana., 15, 523-539.\newline
McNeil, A, Wendin, J (2005) Bayesian Inference for Generalized Linear Mixed
Models of Portfolio Credit Risk. Manuscript, Departement Mathematik, ETH
Zurich.\newline
Meng, X,Wong, W (1996) Simulating ratios of normalizing constants via a
simple identity: a theoretical exploration. Statistica Sinica, 6, 831-860 
\newline
Meng, X, Schilling, S (2002) Warp bridge sampling. Journal of Computational
and Graphical Statistics, 552-586\newline
Meyer, M, Laud, P (2002) Predictive variable selection in generalized linear
models. J. Am. Stat. Assn., 97, 859-871\newline
Mira, A, Nicholls, G (2004) Bridge estimation of the probability density at
a point. Statistica Sinica, 14, 603-612\newline
Morgan, B (2000) Applied Stochastic Modelling. Arnold: London\newline
Mudholkar, G, Hutson, A (2000) The epsilon-skew-normal distribution for
analyzing near-normal data. Journal of Statistical Planning and Inference,
83 291-309\newline
Naylor, J., Smith, A. (1982) Applications of a method for the efficient
computation of posterior distributions. J. R. Stat. Soc., Ser. C 31, 214-22%
\newline
Newton, M, Raftery, A (1994) Approximate Bayesian inference with the
weighted likelihood bootstrap. J. R. Stat. Soc. B 56, 3-48\newline
Ohlssen, D, Sharples, L, Spiegelhalter, D (2006) Tutorial in Biostatistics:
Flexible random-effects models using Bayesian semi-parametric models:
applications to institutional comparisons. Statist Med (forthcoming)\newline
Pourahmadi, M, Daniels, M (2002) Dynamic conditionally linear mixed models
for longitudinal data. Biometrics, 58, 225-231\newline
Raftery, A (1995) Bayesian model selection in social research, in
Sociological Methodology 1995, ed. Marsden,P, Oxford, Blackwell, 111-163%
\newline
Raftery, A (1996) Hypothesis testing and model selection. In Markov Chain
Monte Carlo in Practice, Gilks, W, Richardson, S, Spiegelhalter, D (eds).
Chapman \& Hall: London, 163-187\newline
Raftery, A, Richardson, S (1996) Model selection for generalized linear
models via GLIB, with application to epidemiology. In Bayesian
Biostatistics, Berry, D, Stangl, D (eds), Marcel-Dekker: New York, 321-354%
\newline
Robert, C. (1995) Convergence control techniques for MCMC algorithms.
Statistical Science 10, 231-253.\newline
Rubin, D, Stern, H (1994). Testing in latent class models using a posterior
predictive check distribution. In: Latent Variables Analysis: Applications
for Developmental Research, von Eye, A, Clogg, C (eds.), Sage Publications:
Thousand Oaks, CA., 420--438.\newline
Sahu, S (2004) Applications of formal model choice to archaeological
chronology building. In: Tools for Constructing Chronologies: Crossing
Disciplinary Boundaries, Buck, C, Millard, A (eds). Springer-Verlag, 111-127%
\newline
Sahu, S. and Roberts, G. (1999). On convergence of the EM algorithm and the
Gibbs sampler, Statistics Computing, 9, pp. 55-64\newline
Schwarz, G (1978) Estimating the dimension of a model. Annals of Statistics,
6, 461-64.\newline
Self S, Liang K (1987) Large sample properties of the maximum likelihood
estimator and the likelihood ratio test on the boundary of parameter space.
J Am Stat Assoc, 82, 605--10.\newline
Shively, T, Kohn, R, Wood, S (1999) Variable selection and function
estimation in additive nonparametric regression using a data-based prior. J.
Amer. Stat. Assn, 94, 777-794.\newline
Sinharay S, Stern H (2002) On the sensitivity of Bayes factors to the prior
distributions. The American Statistician, 56,196-201\newline
Sinharay S, Stern H (2005) An empirical comparison of methods for computing
Bayes factors in generalized linear mixed models. J. Comp. Graph. Stat., 14,
415-435.\newline
Song, X, Lee, S (2004) A Bayesian model selection method with applications.
Computational Statistics \& Data Analysis, 40, 539 -- 557\newline
Spiegelhalter, D, Best, N, Carlin, B, van der Linde, A (2002) Bayesian
measures of model complexity and fit. J. Royal Stat. Soc. B, 64, 583-639%
\newline
Spiegelhalter, D (2006) Two brief topics on modelling with WinBUGS. ICEBUGS
presentation, Hanko, Finland. http://www.math.helsinki.fi/openbugs/IceBUGS/%
\newline
IceBUGSAbstracts.html\newline
Tierney, L, Kadane, J (1986) Accurate approximations for posterior moments
and marginal densities. J. Amer. Stat. Assoc. 81, 82-86\newline
Zijlstra, B, van Duijn, M, Snijders, T (2005) Model selection in random
effects models for directed graphs using approximated Bayes factors.
Statistica Neerlandica, 59, 107 -118 \newline
Vehtari, A, Lampinen, J (2002) Bayesian model assessment and comparison
using cross-validation predictive densities. Neural Computation, 14,
2439-2468\newline
Weakliem, D (1999) A critique of the Bayesian information criterion for
model selection. Soc. Meth. Res., 27, 359-397.\newline
Yang, X, Belin, T, Boscardin, W (2005) Imputation and variable selection in
linear regression models with missing covariates. Biometrics 61, 498--506%
\newline
Yau, P, Kohn, R,Wood, S (2003) Bayesian variable selection and model
averaging in high dimensional multinomial nonparametric regression. J. Comp.
Graph. Stat., 12,23-54\newline
Yi, N, George, V, Allison, D (2003) Stochastic search variable selection for
identifying multiple quantitative trait loci. Genetics, 164, 1129-38\newline

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