linear model formula for the full model with all
predictors, Y ~ X. All code assumes that an intercept will be
included in each model and that the X's will be centered.
data
data frame
weights
an optional vector of weights to be used in the fitting
process. Should be NULL or a numeric vector. If non-NULL, Bayes
estimates are obtained assuming that Y ~ N(Xb, sigma^2
diag(1/weights)).
n.models
number of models to sample either without replacement
(method="BAS" or "MCMC+BAS")
or with replacement (method="MCMC"). If NULL, BAS with
method="BAS" will try to enumerate all 2^p models. If enumeration is
not possible (memory or time) then a value should be supplied
which controls the number of sampled models using 'n.models'.
With method="MCMC", sampling will stop once at the min(n.models, MCMC.iterations) so MCMC.iterations be larger than n.models in order to explore the model space. On exit for method=
"MCMC" this is the number of unique models that have been
sampled with counts stored in the output as "freq""
prior
prior distribution for regression coefficients. Choices include
"AIC", "BIC", "g-prior", "ZS-null", "ZS-full", "hyper-g",
"hyper-g-laplace", "hyper-g-n", "EB-local", and "EB-global"
alpha
optional hyperparameter in g-prior or hyper g-prior. For
Zellner's g-prior, alpha = g, for the Liang et al hyper-g or
hyper-g-n method, recommended choice is alpha are between (2 < alpha < 4),
with alpha = 3 recommended.
modelprior
Family of prior distribution on the models. Choices
include uniformBernoulli or beta.binomial with the
default being a beta.binomial(1,1).
initprobs
Vector of length
p or a character string for approach used to create the vector.
This is used to order variables for sampling all methods for
potentially more efficient storage while sampling and provides the
initial inclusion probabilities used for sampling without replacement with
method="BAS". Options for the charactier string giving the method
are: "Uniform" or "uniform" where each predictor variable is
equally likely to be sampled (equivalent to random sampling without
replacement); "eplogp" uses the eplogprob function
to aproximate the Bayes factor from p-values from the full model to
find initial marginal inclusion probabilitites;
"marg-eplogp" useseplogprob.marg function to
aproximate the Bayes factor from p-values from the full model each
simple linear regression. To run a
Markov Chain to provide initial estimates of marginal
inclusion probabilities for "BAS", use method="MCMC+BAS" below. While the
initprobs are not used in sampling for method="MCMC", this
determines the order of the variables in the lookup table and
affects memory allocation in large problems where enumeration is not
feasible. For variables that should always be included set the
corresponding initprobs to 1, i.e. the intercept should be included with
probability one.
method
A character variable indicating which sampling method to
use: method="BAS" uses Bayesian Adaptive Sampling (without
replacement) using the sampling probabilities given in initprobs;
method="MCMC" samples with replacement via a MCMC algorithm based
that combines the birth/death random walk in Hoeting et al (1997) of
MC3 with a random swap move to interchange a variable in the model
with one currently excluded as in Clyde, Ghosh and Littman (2010);
method="MCMC+BAS" runs an initial MCMC to calculate marginal
inclusion probabilities and then samples without replacement as in
BAS. For both BAS and AMCMC, the sampling probabilities can be updated as
more models are sampled. (see update below). We recommend
"MCMC+BAS" or "MCMC" for high dimensional problems where enumeration
is not feasible.
update
number of iterations between potential updates of the
sampling probabilities for method "BAS". If NULL do not update,
otherwise the algorithm will update using the marginal inclusion
probabilities as they change while sampling takes place. For large
model spaces, updating is recommended. If the model space will be
enumerated, leave at the default.
bestmodel
optional binary vector representing a model to
initialize the sampling. If NULL sampling starts with the null
model
prob.local
A future option to allow sampling of models
"near" the median probability model. Not used at
this time.
prob.rw
For any of the MCMC methods, probability of using the
random-walk proposal; otherwise use a random "flip" move to propose
a new model.
MCMC.iterations
Number of iterations for the MCMC sampler; the
default is n.models*10 if not set by the user.
lambda
Parameter in the AMCMC algorithm.
delta
truncation parameter to prevent sampling probabilities
to degenerate to 0 or 1 prior to enumeration for sampling withour
replacent.
thin
For "MCMC", thin the MCMC every "thin"
iterations; default is no thinning.
Details
BAS provides several search
algorithms to find high probability models for use in Bayesian Model
Averaging or Bayesian model selection. For p less than 20-25, BAS can
enumerate all models depending on memory availability, for larger p, BAS
samples without replacement using random or deterministic sampling. The
Bayesian Adaptive Sampling algorithm of Clyde, Ghosh, Littman (2010)
samples models without replacement using the initial sampling
probabilities, and will optionally update the sampling probabilities
every "update" models using the estimated marginal inclusion
probabilties. BAS uses different methods to obtain the initprobs,
which may impact the results in high-dimensional problems.
The deterinistic sampler provides a list of the top models in order of an
approximation of independence using the provided initprobs. This
may be effective after running the other algorithms to identify high
probability models and works well if
the correlations of variables are small to modest. The priors on
coefficients include Zellner's g-prior, the Hyper-g prior (Liang et al
2008, the Zellner-Siow Cauchy prior, Empirical Bayes (local and gobal)
g-priors. AIC and BIC are also included.
Value
bas returns an object of class BMA
An object of class BAS is a list containing at least the following components:
postprob
the posterior probabilities of the models selected
priorprobs
the prior probabilities of the models selected
namesx
the names of the variables
R2
R2 values for the models
logmarg
values of the log of the marginal likelihood for the
models
n.vars
total number of independent variables in the full model,
including the intercept
size
the number of independent variables in each of the models,
includes the intercept
which
a list of lists with one list per model with variables
that are included in the model
probne0
the posterior probability that each variable is
non-zero computed using the renormalized marginal likelihoods of
sampled models. This may be biased if the number of sampled models
is much smaller than the total number of models. Unbiased estimates
may be obtained using method "MCMC".
mle
list of lists with one list per model giving the MLE (OLS)
estimate of each (nonzero) coefficient for each model. The intercept
is the mean of Y as each column of X has been centered by subtracting
its mean.
mle.se
list of lists with one list per model giving the MLE (OLS) standard error of each coefficient for each model
prior
the name of the prior that created the BMA object
alpha
value of hyperparameter in prior used to create the BMA
object.
modelprior
the prior distribution on models that created the BMA object
Y
response
X
matrix of predictors
mean.x
vector of means for each column of X (used in predict.bas)
The function summary.bas, is used to print a summary of
the results. The function plot.bas is used to plot
posterior distributions for the coefficients and
image.bas provides an image of the distribution over models.
Posterior summaries of coefficients can be extracted using
coefficients.bma. Fitted values and predictions can be
obtained using the S3 functions fitted.bas and predict.bas.
BA objects may be updated to use a different prior (without rerunning
the sampler) using the function update.bma.
Clyde, M. Ghosh, J. and Littman, M. (2010) Bayesian Adaptive Sampling
for Variable Selection and Model Averaging. Journal of Computational
Graphics and Statistics. 20:80-101 http://dx.doi.org/10.1198/jcgs.2010.09049
Clyde, M. (1999)
Bayesian Model Averaging and Model Search Strategies (with
discussion). In Bayesian Statistics 6. J.M. Bernardo, A.P. Dawid,
J.O. Berger, and A.F.M. Smith eds. Oxford University Press, pages
157-185.
Liang, F., Paulo, R., Molina, G., Clyde, M. and Berger,
J.O. (2008) Mixtures of g-priors for Bayesian Variable
Selection. Journal of the American Statistical Association.
103:410-423. http://dx.doi.org/10.1198/016214507000001337
Zellner, A. (1986) On assessing prior distributions and Bayesian
regression analysis with g-prior distributions. In Bayesian Inference
and Decision Techniques: Essays in Honor of Bruno de Finetti,
pp. 233-243. North-Holland/Elsevier.
Zellner, A. and Siow, A. (1980) Posterior odds ratios for selected
regression hypotheses. In Bayesian Statistics: Proceedings of the First
International Meeting held in Valencia (Spain), pp. 585-603.
providing 
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