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PTT: Pólya tree type models

PTT provides Bayesian nonparametric density and conditional-density models built from recursive partitions:

• adaptive Pólya trees (APT) and Markov APTs;
• optional Pólya trees (OPT); and
• conditional APT and conditional OPT models.

The implementation supports univariate and multivariate sample spaces, exact forward-backward inference on a finite partition tree, posterior predictive densities, hierarchical MAP partitions, and posterior partition samples.

Installation

Install the development version from GitHub:

install.packages("remotes")
remotes::install_github("MaStatLab/PTT")

The package compiles C++ code and therefore requires the standard R build tools for your platform.

Marginal density estimation

library(PTT)

set.seed(12345)
x <- c(rbeta(150, 3, 10), rbeta(100, 12, 4))
x_grid <- seq(0.01, 0.99, length.out = 200)

fit <- apt(
  X = x,
  Xpred = x_grid,
  max.resol = 7,
  n.s = 5,
  n.grid = 5
)

plot(x_grid, fit$predictive_densities, type = "l",
     xlab = "x", ylab = "Posterior predictive density")
fit$part_points_hmap

Use opt() for an optional Pólya tree with the same interface:

opt_fit <- opt(x, Xpred = x_grid, max.resol = 7)

Conditional density estimation

Prediction locations are paired: row i of Xpred is evaluated with row i of Ypred.

set.seed(12345)
x <- runif(300)
y <- ifelse(x < 0.5, rbeta(300, 3, 9), rbeta(300, 9, 3))

y_grid <- seq(0.02, 0.98, length.out = 100)
x_grid <- rep(c(0.25, 0.75), each = length(y_grid))
y_pred <- rep(y_grid, 2)

conditional_fit <- cond.opt(
  X = x,
  Y = y,
  Xpred = x_grid,
  Ypred = y_pred,
  max.resX = 5,
  max.resY = 7
)

matplot(
  y_grid,
  matrix(conditional_fit$predictive_densities, ncol = 2),
  type = "l", lty = 1,
  xlab = "y", ylab = "Conditional predictive density"
)
legend("top", c("x = 0.25", "x = 0.75"), lty = 1, col = 1:2)

Sample spaces and outputs

The default Omega.type = "unit" (or OmegaX.type/OmegaY.type) requires all corresponding values to lie in [0, 1]. Use "standardized" to construct a padded hyperrectangle from the observed and prediction ranges. The fitted sample-space bounds are returned as Omega, or as OmegaX and OmegaY.

The main returned quantities are:

• logphi: log marginal likelihood;
• logrho: log posterior complete-shrinkage probability at the root;
• predictive_densities: densities in the physical coordinates of the chosen sample space;
• part_points_hmap: hierarchical MAP partition encoded on the integer grid at the requested maximum resolution; and
• part_points_post_samples: posterior partition samples, when requested.

Seven installed demos emphasize visual interpretation and use reproducible simulations. List them with demo(package = "PTT"). Three concise gallery demos are:

demo("marginal_1d_comparison", package = "PTT")
demo("conditional_1d_surface", package = "PTT")
demo("marginal_2d_partition", package = "PTT")

The demos show an APT/OPT comparison with terminal-cell resolution, a conditional-density heatmap with fitted slices, and a bivariate predictive-density map alongside its hMAP terminal partition.

Four additional mixture and conditional-regime workflows are available with the same visual style and consistent naming convention:

demo("marginal_1d_mixture", package = "PTT")
demo("marginal_2d_mixture", package = "PTT")
demo("conditional_1d_regimes", package = "PTT")
demo("conditional_2d_mixture", package = "PTT")

These retain the original models, simulation sizes, and principal fitting settings while adding fitted-versus-true comparisons, coordinated palettes, conditional-density slices, and clearer partition displays.

References

Ma, L. (2017). Adaptive shrinkage in Pólya tree type models. Bayesian Analysis, 12(3), 779-805. doi:10.1214/16-BA1021

Ma, L. (2017). Recursive partitioning and multi-scale modeling on conditional densities. Electronic Journal of Statistics, 11(1), 1297-1325. doi:10.1214/17-EJS1254

Wong, W. H. and Ma, L. (2010). Optional Pólya tree and Bayesian inference. The Annals of Statistics, 38(3), 1433-1459. doi:10.1214/09-AOS755

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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