# Multinomial Logistic Regression with Heavy-Tailed Priors

#### Longhai Li and Steven Liu

#### 2025-12-13

# 1 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 <- 123

    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] -0.684 0.912 -0.997 -1.262 0.613 ...
    #>   ..- 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] -0.903 -0.632 1.111 1.446 -0.43 ...
    #>   ..- attr(*, "dimnames")=List of 2
    #>   .. ..$ : NULL
    #>   .. ..$ : chr [1:2000] "V1" "V2" "V3" "V4" ...
    #>  $ y.tr: int [1:500] 1 1 2 2 3 3 3 1 2 1 ...
    #>  $ x.te: num [1:10, 1:2000] 1.955 1.188 -0.942 -1.387 0.879 ...
    #>   ..- attr(*, "dimnames")=List of 2
    #>   .. ..$ : NULL
    #>   .. ..$ : chr [1:2000] "V1" "V2" "V3" "V4" ...
    #>  $ y.te: int [1:10] 2 2 3 3 2 3 1 2 2 2

# 2 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
    #>  26.550   2.815  25.480

    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
    #>  43.768   7.882  40.904

    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')

# 3 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 -2.8878762572 -1.1517579945
    #> V1         8.8141309062 -0.4827724459
    #> V2        -4.8441556830  0.2080099540
    #> V3        -0.9843977368  3.4452642420
    #> V4        -0.0032682208 -0.0084350734
    #> V5        -0.0063990263 -0.0004660106
    #> V6        -0.0146128750 -0.0031870033
    #> V7        -0.0068047932 -0.0163097984
    #> V8         0.0032944216 -0.0008527143
    #> V9        -0.0061620707  0.0065641278
    #> V10        0.0109562862 -0.0025272676
    #> V100       0.0007569708 -0.0056009522
    #> V200       0.0084240225  0.0049643559
    #> V1000     -0.0041299339 -0.0118251042
    #> V2000     -0.0144638760 -0.0058143273
    #> 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.

# 4 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,]      3
    #>  [4,]      3
    #>  [5,]      2
    #>  [6,]      3
    #>  [7,]      3
    #>  [8,]      2
    #>  [9,]      2
    #> [10,]      2

    print(paste0("prediction accuracy of model 1 = ",
                 sum(y.class == dat$y.te) / length(y.class)))
    #> [1] "prediction accuracy of model 1 = 0.9"

    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 = 0.8"

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] 0.9993980 0.9996598 0.9960531 0.9146809 0.5601952 0.4681349 0.1604013
    #>  [8] 0.9999976 0.9986511 0.9655200
    #>
    #> $table_eval
    #>    Case ID True Label Pred. Prob 1 Pred. Prob 2 Pred. Prob 3 Wrong?
    #> 1        1          2 2.230163e-04 9.993980e-01 3.789681e-04      0
    #> 2        2          2 3.397451e-04 9.996598e-01 4.055238e-07      0
    #> 3        3          3 3.946910e-03 3.274933e-09 9.960531e-01      0
    #> 4        4          3 8.531913e-02 1.201749e-08 9.146809e-01      0
    #> 5        5          2 4.363766e-01 5.601952e-01 3.428135e-03      0
    #> 6        6          3 1.231557e-01 4.087093e-01 4.681349e-01      0
    #> 7        7          1 1.604013e-01 9.352412e-04 8.386635e-01      1
    #> 8        8          2 2.383022e-06 9.999976e-01 3.584158e-08      0
    #> 9        9          2 1.339940e-03 9.986511e-01 8.967895e-06      0
    #> 10      10          2 3.263762e-02 9.655200e-01 1.842378e-03      0
    #>
    #> $amlp
    #> [1] 0.3299063
    #>
    #> $err_rate
    #> [1] 0.1
    #>
    #> $which.wrong
    #> [1] 7

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.