ex08. Conditional DPMGPD (SB Backend)

Website workflow note. This page reflects the current exported API and recommended wrapper-first usage. Last updated: 2026-02-19.

For the full package narrative, see the main package vignettes (basic, unconditional, conditional, and causal).

Conditional CausalMixGPD: Stick-Breaking Backend

Purpose: Apply fixed-component stick-breaking truncation to covariate-dependent mixtures while keeping the GPD tail. This vignette mirrors ex10 but with the SB backend.

What you’ll learn

How to combine conditional modeling (y X) with SB truncation and GPD tail augmentation.
How to keep the workflow wrapper-first: bundle() → dpmgpd() → predict()/plot().
How to compare bulk kernels under the same SB+GPD structure to diagnose bulk vs tail sensitivity.

When to use this template

You need tail-aware conditional predictions but want fixed-complexity mixtures for iteration and tuning.
You prefer choosing an explicit upper bound (components) over CRP-style partition flexibility.

Next steps

Try the CRP conditional tail version (ex07) to compare conditional tail uncertainty under a different backend.

Data Setup

Code

data("nc_posX100_p5_k4")
y <- nc_posX100_p5_k4$y
X <- as.matrix(nc_posX100_p5_k4$X)
if (is.null(colnames(X))) {
  colnames(X) <- paste0("x", seq_len(ncol(X)))
}

summary_tbl <- tibble(
  statistic = c("N", "Mean", "SD", "Min", "Max"),
  value = c(length(y), mean(y), sd(y), min(y), max(y))
)

ggplot(data.frame(y = y, x1 = X[, 1]), aes(x = x1, y = y)) +
  geom_point(alpha = 0.6, color = "purple") +
  geom_smooth(method = "loess", color = "orange", fill = NA) +
  labs(title = "Tail Outcome vs x1 (SB)", x = "x1", y = "y") +
  theme_minimal()

Code

summary_tbl

# A tibble: 5 × 2
  statistic   value
  <chr>       <dbl>
1 N         100    
2 Mean        1.94 
3 SD          1.15 
4 Min         0.488
5 Max         5.28

Threshold

Code

u_threshold <- quantile(y, 0.85)

ggplot(data.frame(y = y), aes(x = y)) +
  geom_histogram(aes(y = after_stat(density)), bins = 40, fill = "lightgreen", alpha = 0.6, color = "black") +
  geom_vline(xintercept = u_threshold, linetype = "dashed", color = "black") +
  labs(title = "Threshold for SB tail (85%)", x = "y", y = "Density") +
  theme_minimal()

Model Specification

Code

bundle_sb_cond_gpd_gamma <- bundle(
  y = y,
  X = X,
  kernel = "gamma",
  backend = "sb",
  GPD = TRUE,
  components = 5,
  param_specs = list(
    gpd = list(
      threshold = list(mode = "link", link = "exp")
    )
  ),
  mcmc = mcmc
)

bundle_sb_cond_gpd_laplace <- bundle(
  y = y,
  X = X,
  kernel = "laplace",
  backend = "sb",
  GPD = TRUE,
  components = 5,
  param_specs = list(
    gpd = list(
      threshold = list(mode = "link", link = "exp")
    )
  ),
  mcmc = mcmc
)

MCMC Execution

Code

fit_sb_cond_gpd_gamma <- load_or_fit("ex08-conditional-dpmgpd-sb-fit_sb_cond_gpd_gamma", dpmgpd(bundle_sb_cond_gpd_gamma))
fit_sb_cond_gpd_laplace <- load_or_fit("ex08-conditional-dpmgpd-sb-fit_sb_cond_gpd_laplace", dpmgpd(bundle_sb_cond_gpd_laplace))
summary(fit_sb_cond_gpd_gamma)

MixGPD summary | backend: Stick-Breaking Process | kernel: Gamma Distribution | GPD tail: TRUE | epsilon: 0.025
n = 100 | components = 5
Summary
Initial components: 5 | Components after truncation: 2

Summary table
          parameter   mean    sd q0.025 q0.500 q0.975     ess
         weights[1]  0.601 0.145  0.423  0.553  0.861   3.424
         weights[2]  0.257 0.085  0.138  0.231  0.464   7.234
              alpha  0.857 0.624  0.151  0.609  2.268   2.048
   beta_scale[1, 1] -0.224 0.261 -0.772 -0.238  0.245   25.65
   beta_scale[2, 1]   0.22 0.818 -1.449  0.193  1.989  32.749
   beta_scale[3, 1] -0.205 1.698 -3.295 -0.162  4.516  93.175
   beta_scale[4, 1]  0.161 1.785 -3.108  0.126  3.773  80.657
   beta_scale[5, 1]  0.083 1.951 -3.706  0.236  3.405 110.762
   beta_scale[1, 2]  -0.47  0.57 -1.735 -0.417  0.568  28.862
   beta_scale[2, 2] -0.044 1.147 -1.867 -0.029  2.635  14.745
   beta_scale[3, 2] -0.005 1.693 -2.995 -0.182  3.654  45.849
   beta_scale[4, 2]  0.142 1.755 -3.189  0.019  4.323 105.242
   beta_scale[5, 2]  0.087 1.755 -3.362  0.022  3.388  57.353
   beta_scale[1, 3]  0.208 0.295 -0.304  0.181  0.854  62.894
   beta_scale[2, 3] -0.131 0.708 -1.764 -0.095  1.272  56.406
   beta_scale[3, 3] -0.103 1.785 -3.561  0.041  2.934 137.278
   beta_scale[4, 3] -0.152 1.699 -3.538 -0.181  3.282  74.715
   beta_scale[5, 3] -0.232 1.823 -4.179  0.007  3.244     150
   beta_scale[1, 4]  0.296 1.131 -1.155 -0.032  2.993  15.652
   beta_scale[2, 4]  0.684 1.605 -1.598  0.449  4.042  25.515
   beta_scale[3, 4]  0.828 1.935 -3.011  0.952  4.157  71.315
   beta_scale[4, 4]  0.491 2.094 -3.392  0.672  3.956  48.517
   beta_scale[5, 4]  0.257 1.827 -3.322  0.203  3.814 107.576
   beta_scale[1, 5]  0.052 0.225 -0.277  0.033  0.458  43.619
   beta_scale[2, 5]  0.449 0.775 -0.951  0.338  1.995   39.47
   beta_scale[3, 5]  0.055 1.565 -3.013  0.023  3.103  31.064
   beta_scale[4, 5]  0.463 1.734 -3.063  0.192  3.629  77.519
   beta_scale[5, 5]  0.522 1.862 -3.639  0.501   3.76     150
 beta_tail_scale[1]  0.224  0.13 -0.023  0.224  0.493     150
 beta_tail_scale[2]   0.01 0.203 -0.361  0.001  0.386 112.961
 beta_tail_scale[3] -0.031 0.093 -0.237 -0.028  0.112     150
 beta_tail_scale[4]  0.489  0.24 -0.025  0.517  0.896  53.679
 beta_tail_scale[5] -0.049 0.094 -0.245 -0.053  0.129 116.138
  beta_threshold[1]      0     0      0      0      0       0
  beta_threshold[2]      0     0      0      0      0       0
  beta_threshold[3]      0     0      0      0      0       0
  beta_threshold[4]      0     0      0      0      0       0
  beta_threshold[5]      0     0      0      0      0       0
         tail_shape -0.024 0.115 -0.217 -0.039  0.255  27.235
           shape[1]  2.481 0.475  1.363  2.454  3.352  14.946
           shape[2]  2.989 1.189  1.273  2.714  6.179  13.375

Code

summary(fit_sb_cond_gpd_laplace)

MixGPD summary | backend: Stick-Breaking Process | kernel: Laplace Distribution | GPD tail: TRUE | epsilon: 0.025
n = 100 | components = 5
Summary
Initial components: 5 | Components after truncation: 1

Summary table
           parameter   mean    sd q0.025 q0.500 q0.975    ess
          weights[1]  0.834 0.115  0.588  0.852      1  3.525
               alpha  0.775 0.368   0.24  0.718  1.648 31.211
 beta_location[1, 1]      0     0      0      0      0      0
 beta_location[2, 1]      0     0      0      0      0      0
 beta_location[3, 1]      0     0      0      0      0      0
 beta_location[4, 1]      0     0      0      0      0      0
 beta_location[5, 1]      0     0      0      0      0      0
 beta_location[1, 2]      0     0      0      0      0      0
 beta_location[2, 2]      0     0      0      0      0      0
 beta_location[3, 2]      0     0      0      0      0      0
 beta_location[4, 2]      0     0      0      0      0      0
 beta_location[5, 2]      0     0      0      0      0      0
 beta_location[1, 3]      0     0      0      0      0      0
 beta_location[2, 3]      0     0      0      0      0      0
 beta_location[3, 3]      0     0      0      0      0      0
 beta_location[4, 3]      0     0      0      0      0      0
 beta_location[5, 3]      0     0      0      0      0      0
 beta_location[1, 4]      0     0      0      0      0      0
 beta_location[2, 4]      0     0      0      0      0      0
 beta_location[3, 4]      0     0      0      0      0      0
 beta_location[4, 4]      0     0      0      0      0      0
 beta_location[5, 4]      0     0      0      0      0      0
 beta_location[1, 5]      0     0      0      0      0      0
 beta_location[2, 5]      0     0      0      0      0      0
 beta_location[3, 5]      0     0      0      0      0      0
 beta_location[4, 5]      0     0      0      0      0      0
 beta_location[5, 5]      0     0      0      0      0      0
  beta_tail_scale[1]  0.197  0.12  -0.07    0.2   0.39    150
  beta_tail_scale[2] -0.027   0.2 -0.379 -0.038  0.385    150
  beta_tail_scale[3] -0.018 0.098 -0.221 -0.024  0.161    150
  beta_tail_scale[4]  0.541 0.236  0.131  0.552  0.979 80.822
  beta_tail_scale[5] -0.038 0.091 -0.206  -0.04  0.142    150
   beta_threshold[1]      0     0      0      0      0      0
   beta_threshold[2]      0     0      0      0      0      0
   beta_threshold[3]      0     0      0      0      0      0
   beta_threshold[4]      0     0      0      0      0      0
   beta_threshold[5]      0     0      0      0      0      0
          tail_shape -0.064 0.083 -0.158 -0.101  0.119 57.819
            scale[1]  3.765 0.834  2.403   3.67  5.639    150

Code

params_sb_cond <- params(fit_sb_cond_gpd_gamma)
params_sb_cond

Posterior mean parameters

$alpha
[1] 0.8566

$w
[1] 0.6009 0.2565

$shape
[1] 2.481 2.989

$beta_scale
            x1        x2      x3     x4      x5
comp1 -0.22420 -0.470200  0.2079 0.2959 0.05209
comp2  0.22010 -0.044060 -0.1314 0.6836 0.44910
comp3 -0.20500 -0.005438 -0.1027 0.8280 0.05459
comp4  0.16110  0.142200 -0.1518 0.4909 0.46330
comp5  0.08309  0.087220 -0.2323 0.2569 0.52210

$beta_threshold
[1] 0 0 0 0 0

$beta_tail_scale
[1]  0.223800  0.009858 -0.031200  0.488500 -0.049220

$tail_shape
[1] -0.02435

Conditional Predictions

Code

X_new <- rbind(
  c(-1, 0, 0, 0, 0),
  c(0, 0, 0, 0, 0),
  c(1, 1, 0, 0, 0)
)
colnames(X_new) <- colnames(X)
y_grid <- seq(0, max(y) * 1.2, length.out = 200)

df_pred_gamma <- lapply(seq_len(nrow(X_new)), function(i) {
  pred <- predict(fit_sb_cond_gpd_gamma, newdata =as.matrix(X_new[i, , drop = FALSE]), y = y_grid, type = "density")
  data.frame(
    y = pred$fit$y,
    density = pred$fit$density,
    label = paste("x1=", X_new[i, 1], ", x2=", X_new[i, 2], sep = ""),
    model = "Gamma"
  )
})

df_pred_laplace <- lapply(seq_len(nrow(X_new)), function(i) {
  pred <- predict(fit_sb_cond_gpd_laplace, newdata =as.matrix(X_new[i, , drop = FALSE]), y = y_grid, type = "density")
  data.frame(
    y = pred$fit$y,
    density = pred$fit$density,
    label = paste("x1=", X_new[i, 1], ", x2=", X_new[i, 2], sep = ""),
    model = "Laplace"
  )
})

bind_rows(df_pred_gamma, df_pred_laplace) %>%
  ggplot(aes(x = y, y = density, color = label)) +
  geom_line(linewidth = 1) +
  facet_wrap(~ model) +
  labs(title = "Conditional Density (SB + GPD)", x = "y", y = "Density") +
  theme_minimal() +
  theme(legend.position = "bottom")

Tail Quantiles

Code

X_grid <- cbind(x1 = seq(-1, 1, length.out = 5), x2 = 0, x3 = 0, x4 = 0, x5 = 0)
colnames(X_grid) <- colnames(X)
quant_probs <- c(0.90, 0.95)

pred_q_gamma <- predict(fit_sb_cond_gpd_gamma, newdata =as.matrix(X_grid), type = "quantile", p = quant_probs)
pred_q_laplace <- predict(fit_sb_cond_gpd_laplace, newdata =as.matrix(X_grid), type = "quantile", p = quant_probs)

quant_df_gamma <- pred_q_gamma$fit
quant_df_gamma$x1 <- X_grid[quant_df_gamma$id, "x1"]
quant_df_gamma$model <- "Gamma"

quant_df_laplace <- pred_q_laplace$fit
quant_df_laplace$x1 <- X_grid[quant_df_laplace$id, "x1"]
quant_df_laplace$model <- "Laplace"

bind_rows(quant_df_gamma, quant_df_laplace) %>%
  ggplot(aes(x = x1, y = estimate, color = factor(index), group = index)) +
  geom_line(linewidth = 1) +
  geom_point(size = 2) +
  facet_wrap(~ model) +
  labs(title = "Tail Quantiles vs x1 (SB)", x = "x1", y = "Quantile", color = "Probability") +
  theme_minimal()

Residuals & Diagnostics

Code

plot(fitted(fit_sb_cond_gpd_gamma))

Code

plot(fit_sb_cond_gpd_gamma, family = c("traceplot", "autocorrelation", "geweke"))


=== traceplot ===


=== autocorrelation ===


=== geweke ===

Code

plot(fit_sb_cond_gpd_laplace, family = c("density", "running", "caterpillar"))


=== density ===


=== running ===


=== caterpillar ===

Takeaways

Conditional stick-breaking mixtures capture covariate-dependent bulk structure while the GPD handles extremes.
predict() and plot() remain consistent for densities, posterior-mean quantiles, and residuals.
Expect threshold-selected posterior-mean tail quantiles to shift with x1 even when components is fixed.
Next: move into causal models starting with no-X CRP (ex09).

Prereqs

Required packages and data for this page are listed in the setup chunks above.

Outputs

This page renders model fits, diagnostics, and summary artifacts generated by package APIs.

Interpretation

Canonical concept page: Model Umbrella
Treat this page as an application/example view and use the canonical page for core definitions.

Continue to the linked canonical concept page, then return for implementation-specific details.

Conditional CausalMixGPD: Stick-Breaking Backend

What you’ll learn

When to use this template

Next steps

Data Setup

Threshold

Model Specification

MCMC Execution

Conditional Predictions

Tail Quantiles

Residuals & Diagnostics

Takeaways

Workflow Navigation

Prereqs

Outputs

Interpretation

Next