R: Bayesian Adaptive Sampling for Bayesian Model Averaging and...

bas.lm {BAS} | R Documentation

## Bayesian Adaptive Sampling for Bayesian Model Averaging and Variable Selection in Linear Models

### Description

Sample without replacement from a posterior distribution on models

### Usage

    bas.lm(
      formula,
      data,
      subset,
      weights,
      contrasts = NULL,
      na.action = "na.omit",
      n.models = NULL,
      prior = "ZS-null",
      alpha = NULL,
      modelprior = beta.binomial(1, 1),
      initprobs = "Uniform",
      include.always = ~1,
      method = "BAS",
      update = NULL,
      bestmodel = NULL,
      prob.local = 0,
      prob.rw = 0.5,
      MCMC.iterations = NULL,
      lambda = NULL,
      delta = 0.025,
      thin = 1,
      renormalize = FALSE,
      force.heredity = FALSE,
      pivot = TRUE,
      tol = 1e-07,
      bigmem = FALSE
    )

### Arguments

`formula` | 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` | a data frame. Factors will be converted to numerical vectors based on the using 'model.matrix'.
`subset` | an optional vector specifying a subset of observations to be used in the fitting process.
`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)).
`contrasts` | an optional list. See the contrasts.arg of 'model.matrix.default()'.
`na.action` | a function which indicates what should happen when the data contain NAs. The default is "na.omit".
`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 the min(n.models, MCMC.iterations) occurs so MCMC.iterations be significantly 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", Zellner's g prior where 'g' is specified using the argument 'alpha' "JZS" Jeffreys-Zellner-Siow prior which uses the Jeffreys prior on sigma and the Zellner-Siow Cauchy prior on the coefficients. The optional parameter 'alpha' can be used to control the squared scale of the prior, where the default is alpha=1. Setting 'alpha' is equal to rscale^2 in the BayesFactor package of Morey. This uses QUADMATH for numerical integration of g. "ZS-null", a Laplace approximation to the 'JZS' prior for integration of g. alpha = 1 only. We recommend using 'JZS' for accuracy and compatibility with the BayesFactor package, although it is slower. "ZS-full" (to be deprecated) "hyper-g", a mixture of g-priors where the prior on g/(1+g) is a Beta(1, alpha/2) as in Liang et al (2008). This uses the Cephes library for evaluation of the marginal likelihoods and may be numerically unstable for large n or R2 close to 1. Default choice of alpha is 3. "hyper-g-laplace", Same as above but using a Laplace approximation to integrate over the prior on g. "hyper-g-n", a mixture of g-priors that where u = g/n and u ~ Beta(1, alpha/2) to provide consistency when the null model is true. "EB-local", use the MLE of g from the marginal likelihood within each model "EB-global" uses an EM algorithm to find a common or global estimate of g, averaged over all models. When it is not possible to enumerate all models, the EM algorithm uses only the models sampled under EB-local.
`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 the default. For the Zellner-Siow prior alpha = 1 by default, but can be used to modify the rate parameter in the gamma prior on g, 1/g ~ G(1/2, n*alpha/2) so that beta ~ C(0, sigma^2 alpha (X'X/n)^-1).
`modelprior` | Family of prior distribution on the models. Choices include `uniform` `Bernoulli` or `beta.binomial`, `tr.beta.binomial`, (with truncation) `tr.poisson` (a truncated Poisson), and `tr.power.prior` (a truncated power family), with the default being a `beta.binomial(1,1)`. Truncated versions are useful for p > n.
`initprobs` | Vector of length p or a character string specifying which method is 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 character 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 approximate the Bayes factor from p-values from the full model to find initial marginal inclusion probabilities; "marg-eplogp" uses`eplogprob.marg` function to approximate 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, to override the 'modelprior' or use 'include.always' to force these variables to always be included in the model.
`include.always` | A formula with terms that should always be included in the model with probability one. By default this is '~ 1' meaning that the intercept is always included. This will also override any of the values in 'initprobs' above by setting them to 1.
`method` | A character variable indicating which sampling method to use: "deterministic" uses the "top k" algorithm described in Ghosh and Clyde (2011) to sample models in order of approximate probability under conditional independence using the "initprobs". This is the most efficient algorithm for enumeration. "BAS" uses Bayesian Adaptive Sampling (without replacement) using the sampling probabilities given in initprobs under a model of conditional independence. These can be updated based on estimates of the marginal inclusion probabilities. "MCMC" samples with replacement via a MCMC algorithm 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 described in Clyde, Ghosh and Littman (2010). "MCMC+BAS" runs an initial MCMC to calculate marginal inclusion probabilities and then samples without replacement as in BAS. For BAS, the sampling probabilities can be updated as more models are sampled. (see update below).
`update` | number of iterations between potential updates of the sampling probabilities for method "BAS" or "MCMC+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 Metropolis proposal; otherwise use a random "flip" move to propose swap a variable that is excluded with a variable in the 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 (deprecated).
`delta` | truncation parameter to prevent sampling probabilities to degenerate to 0 or 1 prior to enumeration for sampling without replacement.
`thin` | For "MCMC", thin the MCMC chain every "thin" iterations; default is no thinning. For large p, thinning can be used to significantly reduce memory requirements as models and associated summaries are saved only every thin iterations. For thin = p, the model and associated output are recorded every p iterations, similar to the Gibbs sampler in SSVS.
`renormalize` | For MCMC sampling, should posterior probabilities be based on renormalizing the marginal likelihoods times prior probabilities (TRUE) or frequencies from MCMC. The latter are unbiased in long runs, while the former may have less variability. May be compared via the diagnostic plot function `diagnostics`. See details in Clyde and Ghosh (2012).
`force.heredity` | Logical variable to force all levels of a factor to be included together and to include higher order interactions only if lower order terms are included. Currently supported with ‘method=’MCMC'' and experimentally with ‘method=’BAS'' on non-Solaris platforms. Default is FALSE.
`pivot` | Logical variable to allow pivoting of columns when obtaining the OLS estimates of a model so that models that are not full rank can be fit. Defaults to TRUE. Currently coefficients that are not estimable are set to zero. Use caution with interpreting BMA estimates of parameters.
`tol` | 1e-7 as
`bigmem` | Logical variable to indicate that there is access to large amounts of memory (physical or virtual) for enumeration with large model spaces, e.g. > 2^25. default; used in determining rank of X^TX in cholesky decomposition with pivoting.

### Details

BAS provides several algorithms to sample from posterior distributions of models for use in Bayesian Model Averaging or Bayesian variable selection. For p less than 20-25, BAS can enumerate all models depending on memory availability. As BAS saves all models, MLEs, standard errors, log marginal likelihoods, prior and posterior and probabilities memory requirements grow linearly with M*p where M is the number of models and p is the number of predictors. For example, enumeration with p=21 with 2,097,152 takes just under 2 Gigabytes on a 64 bit machine to store all summaries that would be needed for model averaging. (A future version will likely include an option to not store all summaries if users do not plan on using model averaging or model selection on Best Predictive models.) 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 probabilities. BAS uses different methods to obtain the `initprobs`, which may impact the results in high-dimensional problems. The deterministic 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. We recommend "MCMC" for problems where enumeration is not feasible (memory or time constrained) or even modest p if the number of models sampled is not close to the number of possible models and/or there are significant correlations among the predictors as the bias in estimates of inclusion probabilities from "BAS" or "MCMC+BAS" may be large relative to the reduced variability from using the normalized model probabilities as shown in Clyde and Ghosh, 2012. Diagnostic plots with MCMC can be used to assess convergence. For large problems we recommend thinning with MCMC to reduce memory requirements. 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 global) g-priors. AIC and BIC are also included, while a range of priors on the model space are available.

### Value

`bas` returns an object of class `bas`

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. This is equivalent to the log Bayes Factor for comparing each model to a base model with intercept only.
`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
`rank` | the rank of the design matrix; if 'pivot = FALSE', this is the same as size as no checking of rank is conducted.
`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. NOTE: 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 coefficient 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`)
`include.always` | indices of variables that are forced into the model

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.bas`. Fitted values and predictions can be obtained using the S3 functions `fitted.bas` and `predict.bas`. BAS objects may be updated to use a different prior (without rerunning the sampler) using the function `update.bas`. For MCMC sampling `diagnostics` can be used to assess whether the MCMC has run long enough so that the posterior probabilities are stable. For more details see the associated demos and vignette.

### Author(s)

Merlise Clyde (clyde@duke.edu) and Michael Littman

### References

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
doi: 10.1198/jcgs.2010.09049

Clyde, M. and Ghosh. J. (2012) Finite population estimators in stochastic search variable selection. Biometrika, 99 (4), 981-988. doi: 10.1093/biomet/ass040

Clyde, M. and George, E. I. (2004) Model Uncertainty. Statist. Sci., 19, 81-94.
doi: 10.1214/088342304000000035

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.

Hoeting, J. A., Madigan, D., Raftery, A. E. and Volinsky, C. T. (1999) Bayesian model averaging: a tutorial (with discussion). Statist. Sci., 14, 382-401.
doi: 10.1214/ss/1009212519

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.
doi: 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.

Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., \& Iverson, G. (2009). Bayesian t-tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 16, 225-237

Rouder, J. N., Morey, R. D., Speckman, P. L., Province, J. M., (2012) Default Bayes Factors for ANOVA Designs. Journal of Mathematical Psychology. 56. p. 356-374.

### See Also

`summary.bas`, `coefficients.bas`, `print.bas`, `predict.bas`, `fitted.bas` `plot.bas`, `image.bas`, `eplogprob`, `update.bas`

Other bas methods: `BAS`, `coef.bas()`, `confint.coef.bas()`, `confint.pred.bas()`, `diagnostics()`, `fitted.bas()`, `force.heredity.bas()`, `image.bas()`, `plot.confint.bas()`, `predict.basglm()`, `predict.bas()`, `summary.bas()`, `update.bas()`, `variable.names.pred.bas()`

### Examples

    library(MASS)
    data(UScrime)
    crime.bic <- bas.lm(log(y) ~ log(M) + So + log(Ed) +
      log(Po1) + log(Po2) +
      log(LF) + log(M.F) + log(Pop) + log(NW) +
      log(U1) + log(U2) + log(GDP) + log(Ineq) +
      log(Prob) + log(Time),
    data = UScrime, n.models = 2^15, prior = "BIC",
    modelprior = beta.binomial(1, 1),
    initprobs = "eplogp", pivot = FALSE
    )

    # use MCMC rather than enumeration
    crime.mcmc <- bas.lm(log(y) ~ log(M) + So + log(Ed) +
      log(Po1) + log(Po2) +
      log(LF) + log(M.F) + log(Pop) + log(NW) +
      log(U1) + log(U2) + log(GDP) + log(Ineq) +
      log(Prob) + log(Time),
    data = UScrime,
    method = "MCMC",
    MCMC.iterations = 20000, prior = "BIC",
    modelprior = beta.binomial(1, 1),
    initprobs = "eplogp", pivot = FALSE
    )

    summary(crime.bic)
    plot(crime.bic)
    image(crime.bic, subset = -1)

    # example with two-way interactions and hierarchical constraints
    data(ToothGrowth)
    ToothGrowth$dose <- factor(ToothGrowth$dose)
    levels(ToothGrowth$dose) <- c("Low", "Medium", "High")
    TG.bas <- bas.lm(len ~ supp * dose,
      data = ToothGrowth,
      modelprior = uniform(), method = "BAS",
      force.heredity = TRUE
    )
    summary(TG.bas)
    image(TG.bas)

    # exmple with non-full rank case

     loc <- system.file("testdata", package = "BAS")
     d <- read.csv(paste(loc, "JASP-testdata.csv", sep = "/"))
    fullModelFormula <- as.formula("contNormal ~  contGamma * contExpon +
                                    contGamma * contcor1 + contExpon * contcor1")

    # should give warning; use pivot = T to fit non-full rank case (fails on i386)
     out = bas.lm(fullModelFormula,
                  data = d,
                  alpha = 0.125316,
                  prior = "JZS",
                  weights = facFifty, force.heredity = FALSE, pivot = FALSE)

    # use pivot = TRUE to fit non-full rank case  (default)
    # This is slower but safer

    out =  bas.lm(fullModelFormula,
                  data = d,
                  alpha = 0.125316,
                  prior = "JZS",
                  weights = facFifty, force.heredity = FALSE, pivot = TRUE)

    # more complete demo's
    demo(BAS.hald)
    ## Not run:
    demo(BAS.USCrime)

    ## End(Not run)

[Package BAS version 1.6.2 Index]