Using the Bayesian Adaptive Sampling (BAS) Package for Bayesian Model Averaging and Variable Selection

Installing BAS

The stable version can be installed easily in the R console like any other package:

install.packages("BAS")

On the other hand, I welcome everyone to use the most recent version of the package with quick-fixes, new features and probably new bugs. To get the latest development version from GitHub, use the devtools package from CRAN and enter in R:

devtools::install_github("merliseclyde/BAS")

As the package does depend on BLAS and LAPACK, installing from GitHub will require that you have FORTRAN and C compilers on your system.

Demo

We will use the UScrime data to illustrate some of the commands and functionality.

data(UScrime, package = "MASS")

Following other analyses, we will go ahead and log transform all of the variables except column 2, which is the indicator variable of the state being a southern state.

UScrime[, -2] <- log(UScrime[, -2])

To get started, we will use BAS with the Zellner-Siow Cauchy prior on the coefficients.

library(BAS)
crime.ZS <- bas.lm(y ~ .,
  data = UScrime,
  prior = "ZS-null",
  modelprior = uniform(), initprobs = "eplogp",
  force.heredity = FALSE, pivot = TRUE
)

BAS uses a model formula similar to lm to specify the full model with all of the potential predictors. Here we are using the shorthand . to indicate that all remaining variables in the data frame provided by the data argument. Different prior distributions on the regression coefficients may be specified using the prior argument, and include

• “BIC”
• “AIC
• “g-prior”
• “hyper-g”
• “hyper-g-laplace”
• “hyper-g-n”
• “JZS”
• “ZS-null”
• “ZS-full”
• “EB-local”
• “EB-global”

where the default is the Zellner-Siow prior, ZS-null, where all Bayes factors are compared to the null model. The newest prior option, “JZS”, also corresponds to the Zellner-Siow prior on the coefficients, but uses numerical integration rather than a Laplace approximation to obtain the marginal likelihood of models.

By default, BAS will try to enumerate all models if \(p < 19\) using the default method="BAS". The prior distribution over the models is a uniform() distribution which assigns equal probabilities to all models. The last optional argument initprobs = eplogp provides a way to initialize the sampling algorithm and order the variables in the tree structure that represents the model space in BAS. The eplogp option uses the Bayes factor calibration of p-values \(-e p \log(p)\) to provide an approximation to the marginal inclusion probability that the coefficient of each predictor is zero, using the p-values from the full model. Other options for initprobs include

• “marg-eplogp””
• “uniform”
• numeric vector of length p

The option “marg-eplogp” uses p-values from the \(p\) simple linear regressions (useful for large p or highly correlated variables).

Since we are enumerating under all possible models these options are not important and the method="deterministic" may be faster if there are no factors or interactions in the model.

Plots

Some graphical summaries of the output may be obtained by the plot function

plot(crime.ZS, ask = F)

which produces a panel of four plots. The first is a plot of residuals and fitted values under Bayesian Model Averaging. Ideally, if our model assumptions hold, we will not see outliers or non-constant variance. The second plot shows the cumulative probability of the models in the order that they are sampled. This plot indicates that the cumulative probability is leveling off as each additional model adds only a small increment to the cumulative probability, while earlier, there are larger jumps corresponding to discovering a new high probability model. The third plot shows the dimension of each model (the number of regression coefficients including the intercept) versus the log of the marginal likelihood of the model. The last plot shows the marginal posterior inclusion probabilities (pip) for each of the covariates, with marginal pips greater than 0.5 shown in red. The variables with pip > 0.5 correspond to what is known as the median probability model. Variables with high inclusion probabilities are generally important for explaining the data or prediction, but marginal inclusion probabilities may be small if there are predictors that are highly correlated, similar to how p-values may be large in the presence of multicollinearity.

Individual plots may be obtained using the which option.

plot(crime.ZS, which = 4, ask = FALSE, caption = "", sub.caption = "")

BAS has print and summary methods defined for objects of class bas. Typing the objects name

crime.ZS

## 
## Call:
## bas.lm(formula = y ~ ., data = UScrime, prior = "ZS-null", modelprior = uniform(), 
##     initprobs = "eplogp", force.heredity = FALSE, pivot = TRUE)
## 
## 
##  Marginal Posterior Inclusion Probabilities: 
## Intercept          M         So         Ed        Po1        Po2         LF  
##    1.0000     0.8536     0.2737     0.9747     0.6652     0.4490     0.2022  
##       M.F        Pop         NW         U1         U2        GDP       Ineq  
##    0.2050     0.3696     0.6944     0.2526     0.6149     0.3601     0.9965  
##      Prob       Time  
##    0.8992     0.3718

returns a summary of the marginal inclusion probabilities, while the summary function provides

options(width = 80)
summary(crime.ZS)

##           P(B != 0 | Y)  model 1    model 2    model 3   model 4    model 5
## Intercept     1.0000000  1.00000  1.0000000  1.0000000  1.000000  1.0000000
## M             0.8535720  1.00000  1.0000000  1.0000000  1.000000  1.0000000
## So            0.2737083  0.00000  0.0000000  0.0000000  0.000000  0.0000000
## Ed            0.9746605  1.00000  1.0000000  1.0000000  1.000000  1.0000000
## Po1           0.6651553  1.00000  1.0000000  0.0000000  1.000000  1.0000000
## Po2           0.4490097  0.00000  0.0000000  1.0000000  0.000000  0.0000000
## LF            0.2022374  0.00000  0.0000000  0.0000000  0.000000  0.0000000
## M.F           0.2049659  0.00000  0.0000000  0.0000000  0.000000  0.0000000
## Pop           0.3696150  0.00000  0.0000000  0.0000000  1.000000  0.0000000
## NW            0.6944069  1.00000  1.0000000  1.0000000  1.000000  0.0000000
## U1            0.2525834  0.00000  0.0000000  0.0000000  0.000000  0.0000000
## U2            0.6149388  1.00000  1.0000000  1.0000000  1.000000  1.0000000
## GDP           0.3601179  0.00000  0.0000000  0.0000000  0.000000  0.0000000
## Ineq          0.9965359  1.00000  1.0000000  1.0000000  1.000000  1.0000000
## Prob          0.8991841  1.00000  1.0000000  1.0000000  1.000000  1.0000000
## Time          0.3717976  1.00000  0.0000000  0.0000000  0.000000  0.0000000
## BF                   NA  1.00000  0.9416178  0.6369712  0.594453  0.5301269
## PostProbs            NA  0.01820  0.0172000  0.0116000  0.010800  0.0097000
## R2                   NA  0.84200  0.8265000  0.8229000  0.837500  0.8046000
## dim                  NA  9.00000  8.0000000  8.0000000  9.000000  7.0000000
## logmarg              NA 23.65111 23.5909572 23.2000822 23.130999 23.0164741

a list of the top 5 models (in terms of posterior probability) with the zero-one indicators for variable inclusion. The other columns in the summary are the Bayes factor of each model to the highest probability model (hence its Bayes factor is 1), the posterior probabilities of the models, the ordinary \(R^2\) of the models, the dimension of the models (number of coefficients including the intercept) and the log marginal likelihood under the selected prior distribution.

Visualization of the Model Space

To see beyond the first five models, we can represent the collection of the models via an image plot. By default this shows the top 20 models.

image(crime.ZS, rotate = F)

This image has rows that correspond to each of the variables and intercept, with labels for the variables on the y-axis. The x-axis corresponds to the possible models. These are sorted by their posterior probability from best at the left to worst at the right with the rank on the top x-axis.

Each column represents one of the 16 models. The variables that are excluded in a model are shown in black for each column, while the variables that are included are colored, with the color related to the log posterior probability. The color of each column is proportional to the log of the posterior probabilities (the lower x-axis) of that model. The log posterior probabilities are actually scaled so that the 0 corresponds to the lowest probability model in the top 20, so that the values on the axis correspond to log Bayes factors for comparing each model the lowest probability model in the top 20 models. Models that are the same color have similar log Bayes factors which allows us to view models that are clustered together that have Bayes Factors where the differences are not “worth a bare mention”.

This plot indicates that the police expenditure in the two years do not enter the model together, and is an indication of the high correlation between the two variables.

Posterior Distributions of Coefficients

To examine the marginal distributions of the two coefficients for the police expenditures, we can extract the coefficients estimates and standard deviations under BMA.

coef.ZS <- coef(crime.ZS)

an optional argument, n.models to coef will use only the top n.models for BMA and may be more computationally efficient for large problems.

Plots of the posterior distributions averaging over all of the models are obtained using the plot method for the bas coefficient object.

plot(coef.ZS, subset = c(5:6), ask = F)

The vertical bar represents the posterior probability that the coefficient is 0 while the bell shaped curve represents the density of plausible values from all the models where the coefficient is non-zero. This is scaled so that the height of the density for non-zero values is the probability that the coefficient is non-zero. Omitting the subset argument provides all of the marginal distributions.

To obtain credible intervals for coefficients, BAS includes a confint method to create Highest Posterior Density intervals from the summaries from coef.

confint(coef.ZS)

##                    2.5%       97.5%        beta
## Intercept  6.668911e+00 6.782870658  6.72493620
## M          0.000000e+00 2.203416886  1.14359433
## So        -6.512658e-02 0.306207877  0.03547522
## Ed         6.388232e-01 3.232445246  1.85848834
## Po1       -3.964244e-05 1.436220215  0.60067372
## Po2       -1.780610e-01 1.460189320  0.31841766
## LF        -4.898507e-01 1.018124255  0.05933737
## M.F       -1.964603e+00 2.090431998 -0.02702786
## Pop       -1.210907e-01 0.009931915 -0.02248283
## NW         0.000000e+00 0.167252658  0.06668437
## U1        -5.235411e-01 0.370536467 -0.02456854
## U2        -1.214977e-02 0.673390112  0.20702927
## GDP       -2.159482e-03 1.251458882  0.20625063
## Ineq       6.563959e-01 2.110313049  1.39012647
## Prob      -4.134757e-01 0.000000000 -0.21536203
## Time      -5.388119e-01 0.020662784 -0.08433479
## attr(,"Probability")
## [1] 0.95
## attr(,"class")
## [1] "confint.bas"

where the third column is the posterior mean. This uses Monte Carlo sampling to draw from the mixture model over coefficient where models are sampled based on their posterior probabilities.

We can also plot these via

plot(confint(coef.ZS, parm = 2:16))

## NULL

using the parm argument to select which coefficients to plot (the intercept is parm=1).

For estimation under selection, BAS supports additional arguments via estimator. The default is estimator="BMA" which uses all models or n.models. Other options include estimation under the highest probability model

plot(confint(coef(crime.ZS, estimator = "HPM")))

## NULL

or the median probability model

plot(confint(coef(crime.ZS, estimator = "MPM")))

where variables that are excluded have distributions that are point masses at zero under selection.

Prediction

BAS has methods defined to return fitted values, fitted, using the observed design matrix and predictions at either the observed data or potentially new values, predict, as with lm.

muhat.BMA <- fitted(crime.ZS, estimator = "BMA")
BMA <- predict(crime.ZS, estimator = "BMA")

# predict has additional slots for fitted values under BMA, predictions under each model
names(BMA)

##  [1] "fit"         "Ybma"        "Ypred"       "postprobs"   "se.fit"     
##  [6] "se.pred"     "se.bma.fit"  "se.bma.pred" "df"          "best"       
## [11] "bestmodel"   "best.vars"   "estimator"

Plotting the two sets of fitted values,

par(mar = c(9, 9, 3, 3))
plot(muhat.BMA, BMA$fit,
  pch = 16,
  xlab = expression(hat(mu[i])), ylab = expression(hat(Y[i]))
)
abline(0, 1)

we see that they are in perfect agreement. That is always the case as the posterior mean for the regression mean function at a point \(x\) is the expected posterior predictive value for \(Y\) at \(x\). This is true not only for estimators such as BMA, but the expected values under model selection.

HPM <- predict(crime.ZS, estimator = "HPM")

# show the indices of variables in the best model where 0 is the intercept
HPM$bestmodel

## [1]  0  1  3  4  9 11 13 14 15

variable.names(HPM)

## [1] "Intercept" "M"         "Ed"        "Po1"       "NW"        "U2"       
## [7] "Ineq"      "Prob"      "Time"

MPM <- predict(crime.ZS, estimator = "MPM")
variable.names(MPM)

## [1] "Intercept" "M"         "Ed"        "Po1"       "NW"        "U2"       
## [7] "Ineq"      "Prob"

BPM <- predict(crime.ZS, estimator = "BPM")
variable.names(BPM)

##  [1] "Intercept" "M"         "So"        "Ed"        "Po1"       "Po2"      
##  [7] "M.F"       "NW"        "U2"        "Ineq"      "Prob"

GGally::ggpairs(data.frame(
  HPM = as.vector(HPM$fit), # this used predict so we need to extract fitted values
  MPM = as.vector(MPM$fit), # this used fitted
  BPM = as.vector(BPM$fit), # this used fitted
  BMA = as.vector(BMA$fit)
)) # this used predict

BPM <- predict(crime.ZS, estimator = "BPM", se.fit = TRUE)
crime.conf.fit <- confint(BPM, parm = "mean")
crime.conf.pred <- confint(BPM, parm = "pred")
plot(crime.conf.fit)

## NULL

plot(crime.conf.pred)

## NULL

new.pred <- predict(crime.ZS, newdata = UScrime, estimator = "MPM")

Alternative algorithms

BAS has several options for sampling from the model space with or without enumeration. The (current) default method="BAS" samples models without replacement using estimates of the marginal inclusion probabilities using the algorithm described in Clyde et al (2011). The initial sampling probabilities provided by initprobs are updated based on the sampled models, every update iterations. This can be more efficient in some cases if a large fraction of the model space has been sampled, however, in cases of high correlation and a large number of predictors, this can lead to biased estimates Clyde and Ghosh (2012), in which case MCMC is preferred. The method="MCMC" is described below and is better for large \(p\).

A deterministic sampling scheme is also available for enumeration;

system.time(
  for (i in 1:10) {
    crime.ZS <- bas.lm(y ~ .,
      data = UScrime,
      prior = "ZS-null", method = "BAS",
      modelprior = uniform(), initprobs = "eplogp"
    )
  }
)

##    user  system elapsed 
##   0.678   0.022   0.701

system.time(
  for (i in 1:10) {
    crime.ZS <- bas.lm(y ~ .,
      data = UScrime,
      prior = "ZS-null", method = "deterministic",
      modelprior = uniform(), initprobs = "eplogp"
    )
  }
)

##    user  system elapsed 
##   0.493   0.020   0.512

which is faster for enumeration than the default method=“BAS”.

Beyond Enumeration

Many problems are too large to enumerate all possible models. In such cases we may use the method="BAS" to sample without replacement or the method="MCMC" option to sample models using Markov Chain Monte Carlo sampling to sample models based on their posterior probabilities. For spaces where the number of models greatly exceeds the number of models to sample, the MCMC option is recommended as it provides estimates with low bias compared to the sampling without replacement of BAS (Clyde and Ghosh 2011).

crime.ZS <- bas.lm(y ~ .,
  data = UScrime,
  prior = "ZS-null",
  modelprior = uniform(),
  method = "MCMC"
)

This will run the MCMC sampler until the number of unique sampled models exceeds n.models which is \(2^p\) (if \(p < 19\)) by default or until MCMC.iterations has been exceeded, where MCMC.iterations = n.models*2 by default.

diagnostics(crime.ZS, type = "pip", pch = 16)

diagnostics(crime.ZS, type = "model", pch = 16)

crime.ZS <- bas.lm(y ~ .,
  data = UScrime,
  prior = "ZS-null",
  modelprior = uniform(),
  method = "MCMC", MCMC.iterations = 10^6
)

diagnostics(crime.ZS, type="model", pch=16)

Outliers

BAS can also be used for exploring mean shift or variance inflation outliers by adding indicator variables for each case being an outlier (the mean is not given by the regression) or not. This is similar to the MC3.REG function in BMA, although here we are using a g-prior or mixture of g-priors for the coefficients for the outlier means.

Using the Stackloss data, we can add an identify matrix to the original dataframe, where each column is an indicator that the ith variable is an outlier.

data("stackloss")
stackloss <- cbind(stackloss, diag(nrow(stackloss)))
stack.bas <- bas.lm(stack.loss ~ .,
  data = stackloss,
  method = "MCMC", initprobs = "marg-eplogp",
  prior = "ZS-null",
  modelprior = tr.poisson(4, 10),
  MCMC.iterations = 200000
)

The above call introduces using truncated prior distributions on the model space; in this case the distribution of the number of variables to be included has a Poisson distribution, with mean 4 (under no truncation), and the truncation point is at 10, so that all models with more than 10 (one half of cases rounded down) will have probability zero. This avoids exploration of models that are not full rank.

Looking at the summaries

knitr::kable(as.data.frame(summary(stack.bas)))

Factors and Hierarchical Heredity

BAS now includes constraints for factors so that all terms that represent a factor are either included or excluded together. To illustrate, we will use the data set ToothGrowth and convert dose to a factor:

data(ToothGrowth)
ToothGrowth$dose <- factor(ToothGrowth$dose)
levels(ToothGrowth$dose) <- c("Low", "Medium", "High")

and fit the model of main effects and two way interaction without any constraints:

TG.bas <- bas.lm(len ~ supp*dose,
  data = ToothGrowth,
  modelprior = uniform(), method = "BAS"
)

image(TG.bas)

From the image of the model space, we see that levels of a factor enter or drop from the model independently and that interactions may be included without the main effects. This may lead to more parsimonious models, however, the hypotheses that are being tested about the coefficients that represent the factor depend on the choice for the reference group. To force levels of a factor to enter or leave together we can use force.heredity = TRUE.

The force.heredity option also forces interactions to be included only if the main effects are also included, or for models with several factors and higher order interactions, the heredity constraint implies that all lower order interactions must be included before adding higher order interactions.

TG.bas <- bas.lm(len ~ supp * dose,
  data = ToothGrowth,
  modelprior = uniform(), method = "BAS", force.heredity = TRUE
)

image(TG.bas)

The force.heredity is set to FALSE for the sampling methods MCMC+BAS and deterministic. If there are more than 20 predictors and factors, then we recommend using MCMC to enforce the constraints. Alternatively, there is a function, force.heredity.bas, to post-process the output to drop models that violate the hierarchical heredity constraint:

TG.bas <- bas.lm(len ~ supp * dose,
  data = ToothGrowth,
  modelprior = uniform(), method = "BAS", force.heredity = FALSE
)
TG.herid.bas <- force.heredity.bas(TG.bas)

that can be used with those sampling methods.

Weighted Regression

BAS can perform weighted regression by supplying an optional weight vector that is of the same length as the response where the assumption is that the variance of the response is proportional to 1/weights. The g-prior incorporates the weights in the prior covariance, \[ \sigma^2 g (X_\gamma^T W X_\gamma)^{-1} \] where \(X_\gamma\) is the design matrix under model \(\gamma\) and \(W\) is the \(n \times n\) diagonal matrix with the weights on the diagonal.

To illustrate, we will use the climate data, available at the url below

data(climate, package="BAS")
str(climate)

## 'data.frame':    63 obs. of  5 variables:
##  $ deltaT  : num  -2.6 -2.6 -2.9 -2.4 -2.8 -1.2 -2.4 -2.6 -2.4 -2.5 ...
##  $ sdev    : num  0.7 0.8 0.9 0.7 0.7 0.3 1.3 1.3 1.3 0.5 ...
##  $ proxy   : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ T.M     : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ latitude: num  2.5 2.2 0.5 0.3 0.2 -1.1 5.2 11.4 14.6 6.3 ...

summary(climate)

##      deltaT            sdev            proxy            T.M        
##  Min.   :-7.000   Min.   :0.1500   Min.   :1.000   Min.   :0.0000  
##  1st Qu.:-3.900   1st Qu.:0.5000   1st Qu.:2.000   1st Qu.:1.0000  
##  Median :-2.900   Median :0.7000   Median :3.000   Median :1.0000  
##  Mean   :-3.111   Mean   :0.8579   Mean   :3.333   Mean   :0.8254  
##  3rd Qu.:-2.000   3rd Qu.:1.3000   3rd Qu.:5.000   3rd Qu.:1.0000  
##  Max.   : 0.200   Max.   :2.5000   Max.   :8.000   Max.   :1.0000  
##     latitude      
##  Min.   :-22.500  
##  1st Qu.: -3.450  
##  Median :  0.200  
##  Mean   :  2.187  
##  3rd Qu.:  9.700  
##  Max.   : 29.000

which includes measurements of changes in temperature (deltaT) at various latitudes as well as a measure of the accuracy of the measured values sdev for 8 different types proxy of obtaining measurements. We will use this to explore weighted regression and the option to group terms in factors or from poly. For illustration purposes, we will eliminate proxy == 6 which has only one level as the interactions are not estimable, and then convert proxy to a factor.

library(dplyr)
climate <- filter(climate, proxy != 6) %>%
  mutate(proxy = factor(proxy))

We can fit a weighted regression with weights = 1/sdev^2 with the following code

climate.bas <- bas.lm(deltaT ~ proxy * poly(latitude, 2),
  data = climate,
  weights = 1 / sdev^2,
  prior = "hyper-g-n", alpha = 3.0,
  n.models = 2^20, 
  force.heredity=TRUE,
  modelprior = uniform()
)

Examining the image of the top models,

image(climate.bas, rotate = F)

we see that all levels of a factor enter or drop from the model together, as well as the vectors in the design matrix to represent the term poly.
Rerunning without the constraint,

# May take a while to enumerate all 2^20 models
climate.bas <- bas.lm(deltaT ~ proxy * poly(latitude, 2),
  data = climate,
  weights = 1 / sdev^2,
  prior = "hyper-g-n", alpha = 3.0,
  n.models = 2^20,
  modelprior = uniform(),
  force.heredity = FALSE
)
image(climate.bas)

allows one to see which factors levels are different from the reference group.

Summary

BAS includes other prior distributions on coefficients and models, as well as bas.glm for fitting Generalized Linear Models. The syntax for bas.glm and bas.lm are not yet the same, particularly for how some of the priors on coefficients are represented, so please see the documentation for more features and details until this is updated or another vignette is added!

For issues or feature requests please submit via the package’s github page merliseclyde/BAS