Multinomial Logistic Regression with Heavy-Tailed Priors

Data Generation

Load the necessary libraries:

library(HTLR)
library(bayesplot)
#> This is bayesplot version 1.11.1
#> - Online documentation and vignettes at mc-stan.org/bayesplot
#> - bayesplot theme set to bayesplot::theme_default()
#>    * Does _not_ affect other ggplot2 plots
#>    * See ?bayesplot_theme_set for details on theme setting

The description of the dataset generating scheme is found from Li and Yao (2018).

There are 4 groups of features:

  • feature #1: marginally related feature

  • feature #2: marginally unrelated feature, but feature #2 is correlated with feature #1

  • feature #3 - #10: marginally related features and also internally correlated

  • feature #11 - #2000: noise features without relationship with the y

SEED <- 1234

n <- 510
p <- 2000

means <- rbind(
  c(0, 1, 0),
  c(0, 0, 0),
  c(0, 0, 1),
  c(0, 0, 1),
  c(0, 0, 1),
  c(0, 0, 1),
  c(0, 0, 1),
  c(0, 0, 1),
  c(0, 0, 1),
  c(0, 0, 1)
) * 2

means <- rbind(means, matrix(0, p - 10, 3))

A <- diag(1, p)

A[1:10, 1:3] <-
  rbind(
    c(1, 0, 0),
    c(2, 1, 0),
    c(0, 0, 1),
    c(0, 0, 1),
    c(0, 0, 1),
    c(0, 0, 1),
    c(0, 0, 1),
    c(0, 0, 1),
    c(0, 0, 1),
    c(0, 0, 1)
  )

set.seed(SEED)
dat <- gendata_FAM(n, means, A, sd_g = 0.5, stdx = TRUE)
str(dat)
#> List of 4
#>  $ X  : num [1:510, 1:2000] -1.423 -0.358 -1.204 -0.556 0.83 ...
#>   ..- attr(*, "dimnames")=List of 2
#>   .. ..$ : NULL
#>   .. ..$ : chr [1:2000] "V1" "V2" "V3" "V4" ...
#>  $ muj: num [1:2000, 1:3] -0.456 0 -0.456 -0.376 -0.376 ...
#>  $ SGM: num [1:2000, 1:2000] 0.584 0.597 0 0 0 ...
#>  $ y  : int [1:510] 1 2 3 1 2 3 1 2 3 1 ...

Look at the correlation between features:

# require(corrplot)
cor(dat$X[ , 1:11]) %>% corrplot::corrplot(tl.pos = "n")

Split the data into training and testing sets:

set.seed(SEED)
dat <- split_data(dat$X, dat$y, n.train = 500)
str(dat)
#> List of 4
#>  $ x.tr: num [1:500, 1:2000] 0.889 -0.329 1.58 0.213 0.214 ...
#>   ..- attr(*, "dimnames")=List of 2
#>   .. ..$ : NULL
#>   .. ..$ : chr [1:2000] "V1" "V2" "V3" "V4" ...
#>  $ y.tr: int [1:500] 2 3 2 1 2 3 3 3 1 2 ...
#>  $ x.te: num [1:10, 1:2000] 0.83 -0.555 1.041 -1.267 1.15 ...
#>   ..- attr(*, "dimnames")=List of 2
#>   .. ..$ : NULL
#>   .. ..$ : chr [1:2000] "V1" "V2" "V3" "V4" ...
#>  $ y.te: int [1:10] 2 3 2 1 2 2 2 1 2 3

Model Fitting

Fit a HTLR model with all default settings:

set.seed(SEED)
system.time(
  fit.t <- htlr(dat$x.tr, dat$y.tr)
)
#>    user  system elapsed 
#> 228.881   0.171  58.000
print(fit.t)
#> Fitted HTLR model 
#> 
#>  Data:
#> 
#>   response:  3-class
#>   observations:  500
#>   predictors:    2001 (w/ intercept)
#>   standardised:  TRUE 
#> 
#>  Model:
#> 
#>   prior dist:    t (df = 1, log(w) = -10.0)
#>   init state:    lasso 
#>   burn-in:   1000
#>   sample:    1000 (posterior sample size) 
#> 
#>  Estimates:
#> 
#>   model size:    4 (w/ intercept)
#>   coefficients: see help('summary.htlr.fit')

With another configuration:

set.seed(SEED)
system.time(
  fit.t2 <- htlr(X = dat$x.tr, y = dat$y.tr, 
                 prior = htlr_prior("t", df = 1, logw = -20, sigmab0 = 1500), 
                 iter = 4000, init = "bcbc", keep.warmup.hist = T)
)
#>    user  system elapsed 
#> 367.710   0.548  93.659
print(fit.t2)
#> Fitted HTLR model 
#> 
#>  Data:
#> 
#>   response:  3-class
#>   observations:  500
#>   predictors:    2001 (w/ intercept)
#>   standardised:  TRUE 
#> 
#>  Model:
#> 
#>   prior dist:    t (df = 1, log(w) = -20.0)
#>   init state:    bcbc 
#>   burn-in:   2000
#>   sample:    2000 (posterior sample size) 
#> 
#>  Estimates:
#> 
#>   model size:    4 (w/ intercept)
#>   coefficients: see help('summary.htlr.fit')

Model Inspection

Look at the point summaries of posterior of selected parameters:

summary(fit.t2, features = c(1:10, 100, 200, 1000, 2000), method = median)
#>                 class 2       class 3
#> Intercept -3.4152441685 -0.9029461257
#> V1        10.9550978781  0.1139902897
#> V2        -6.8077131737 -0.0722711733
#> V3         0.0378395213  3.4469321695
#> V4         0.0132365292 -0.0003944103
#> V5        -0.0059859587  0.0089446064
#> V6         0.0004462757 -0.0007938343
#> V7        -0.0012941611  0.4338033125
#> V8         0.0025413224  0.0003075648
#> V9        -0.0092135141 -0.0108003250
#> V10        0.0045097384  0.0092975612
#> V100       0.0033002759 -0.0017995756
#> V200       0.0060097424 -0.0036481106
#> V1000      0.0035715341  0.0121408681
#> V2000     -0.0056809289 -0.0004195035
#> attr(,"stats")
#> [1] "median"

Plot interval estimates from posterior draws using bayesplot:

post.t <- as.matrix(fit.t2, k = 2)
## signal parameters
mcmc_intervals(post.t, pars = c("Intercept", "V1", "V2", "V3", "V1000"))

Trace plot of MCMC draws:

as.matrix(fit.t2, k = 2, include.warmup = T) %>%
  mcmc_trace(c("V1", "V1000"), facet_args = list("nrow" = 2), n_warmup = 2000)

The coefficient of unrelated features (noise) are not updated during some iterations due to restricted Gibbs sampling Li and Yao (2018), hence the computational cost is greatly reduced.

Make Predictions

A glance at the prediction accuracy:

y.class <- predict(fit.t, dat$x.te, type = "class")
y.class
#>       y.pred
#>  [1,]      2
#>  [2,]      2
#>  [3,]      2
#>  [4,]      3
#>  [5,]      2
#>  [6,]      2
#>  [7,]      2
#>  [8,]      3
#>  [9,]      2
#> [10,]      3
print(paste0("prediction accuracy of model 1 = ", 
             sum(y.class == dat$y.te) / length(y.class)))
#> [1] "prediction accuracy of model 1 = 0.7"

y.class2 <- predict(fit.t2, dat$x.te, type = "class")
print(paste0("prediction accuracy of model 2 = ", 
             sum(y.class2 == dat$y.te) / length(y.class)))
#> [1] "prediction accuracy of model 2 = 0.7"

More details about the prediction result:

predict(fit.t, dat$x.te, type = "response") %>%
  evaluate_pred(y.true = dat$y.te)

#> $prob_at_truelabels
#>  [1] 0.98881252 0.32727524 0.98210498 0.03068972 0.99976125 0.70578256
#>  [7] 0.99982799 0.07120373 0.98702519 0.97246844
#> 
#> $table_eval
#>    Case ID True Label Pred. Prob 1 Pred. Prob 2 Pred. Prob 3 Wrong?
#> 1        1          2 1.116066e-02 9.888125e-01 2.682052e-05      0
#> 2        2          3 1.342592e-01 5.384656e-01 3.272752e-01      1
#> 3        3          2 1.789154e-02 9.821050e-01 3.476804e-06      0
#> 4        4          1 3.068972e-02 2.751081e-10 9.693103e-01      1
#> 5        5          2 3.283030e-05 9.997613e-01 2.059150e-04      0
#> 6        6          2 2.309747e-01 7.057826e-01 6.324276e-02      0
#> 7        7          2 1.953297e-05 9.998280e-01 1.524749e-04      0
#> 8        8          1 7.120373e-02 2.890835e-04 9.285072e-01      1
#> 9        9          2 1.286960e-02 9.870252e-01 1.052145e-04      0
#> 10      10          3 2.752929e-02 2.271286e-06 9.724684e-01      0
#> 
#> $amlp
#> [1] 0.7662135
#> 
#> $err_rate
#> [1] 0.3
#> 
#> $which.wrong
#> [1] 2 4 8
Li, Longhai, and Weixin Yao. 2018. “Fully Bayesian Logistic Regression with Hyper-LASSO Priors for High-Dimensional Feature Selection.” Journal of Statistical Computation and Simulation 88 (14): 2827–51.