Log-Concave Distribution Estimation

Introduction

Let $$n = 1$$ and suppose $$x_i$$ are i.i.d. samples from a log-concave discrete distribution on $$\{0,\ldots,K\}$$ for some $$K \in {\mathbf Z}_+$$. Define $$p_k := {\mathbf {Prob}}(X = k)$$ to be the probability mass function. One method for estimating $$\{p_0,\ldots,p_K\}$$ is to maximize the log-likelihood function subject to a log-concavity constraint , i.e.,

$\begin{array}{ll} \underset{p}{\mbox{maximize}} & \sum_{k=0}^K M_k\log p_k \\ \mbox{subject to} & p \geq 0, \quad \sum_{k=0}^K p_k = 1, \\ & p_k \geq \sqrt{p_{k-1}p_{k+1}}, \quad k = 1,\ldots,K-1, \end{array}$

where $$p \in {\mathbf R}^{K+1}$$ is our variable of interest and $$M_k$$ represents the number of observations equal to $$k$$, so that $$\sum_{k=0}^K M_k = m$$. The problem as posed above is not convex. However, we can transform it into a convex optimization problem by defining new variables $$u_k = \log p_k$$ and relaxing the equality constraint to $$\sum_{k=0}^K p_k \leq 1$$, since the latter always holds tightly at an optimal solution. The result is

$\begin{array}{ll} \underset{u}{\mbox{maximize}} & \sum_{k=0}^K M_k u_k \\ \mbox{subject to} & \sum_{k=0}^K e^{u_k} \leq 1, \\ & u_k - u_{k-1} \geq u_{k+1} - u_k, \quad k = 1,\ldots,K-1. \end{array}$

Example

We draw $$m = 25$$ observations from a log-concave distribution on $$\{0,\ldots,100\}$$. We then estimate the probability mass function using the above method and compare it with the empirical distribution.

set.seed(1)
## Calculate a piecewise linear function
pwl_fun <- function(x, knots) {
n <- nrow(knots)
x0 <- sort(knots$x, decreasing = FALSE) y0 <- knots$y[order(knots$x, decreasing = FALSE)] slope <- diff(y0)/diff(x0) sapply(x, function(xs) { if(xs <= x0) y0 + slope*(xs -x0) else if(xs >= x0[n]) y0[n] + slope[n-1]*(xs - x0[n]) else { idx <- which(xs <= x0) y0[idx-1] + slope[idx-1]*(xs - x0[idx-1]) } }) } ## Problem data m <- 25 xrange <- 0:100 knots <- data.frame(x = c(0, 25, 65, 100), y = c(10, 30, 40, 15)) xprobs <- pwl_fun(xrange, knots)/15 xprobs <- exp(xprobs)/sum(exp(xprobs)) x <- sample(xrange, size = m, replace = TRUE, prob = xprobs) K <- max(xrange) counts <- hist(x, breaks = -1:K, right = TRUE, include.lowest = FALSE, plot = FALSE)$counts
ggplot() +
geom_histogram(mapping = aes(x = x), breaks = -1:K, color = "blue", fill = "orange") We now solve problem with log-concave constraint.

u <- Variable(K+1)
obj <- t(counts) %*% u
constraints <- list(sum(exp(u)) <= 1, diff(u[1:K]) >= diff(u[2:(K+1)]))
prob <- Problem(Maximize(obj), constraints)
result <- solve(prob)
pmf <- result$getValue(exp(u)) The above lines transform the variables $$u_k$$ to $$e^{u_k}$$ before calculating their resulting values. This is possible because exp is a member of the CVXR library of atoms, so it can operate directly on a Variable object such as u. Below are the comparison plots of pmf and cdf. dens <- density(x, bw = "sj") d <- data.frame(x = xrange, True = xprobs, Optimal = pmf, Empirical = approx(x = dens$x, y = dens$y, xout = xrange)$y)
plot.data <- gather(data = d, key = "Type", value = "Estimate", True, Empirical, Optimal,
factor_key = TRUE)
ggplot(plot.data) +
geom_line(mapping = aes(x = x, y = Estimate, color = Type)) +
theme(legend.position = "top") d <- data.frame(x = xrange, True = cumsum(xprobs),
Empirical = cumsum(counts) / sum(counts),
Optimal = cumsum(pmf))
plot.data <- gather(data = d, key = "Type", value = "Estimate", True, Empirical, Optimal,
factor_key = TRUE)
ggplot(plot.data) +
geom_line(mapping = aes(x = x, y = Estimate, color = Type)) +
theme(legend.position = "top") From the figures we see that the estimated curve is much closer to the true distribution, exhibiting a similar shape and number of peaks. In contrast, the empirical probability mass function oscillates, failing to be log-concave on parts of its domain. These differences are reflected in the cumulative distribution functions as well.

Session Info

sessionInfo()
## R version 3.6.0 (2019-04-26)
## Platform: x86_64-apple-darwin18.5.0 (64-bit)
## Running under: macOS Mojave 10.14.5
##
## Matrix products: default
## BLAS/LAPACK: /usr/local/Cellar/openblas/0.3.6_1/lib/libopenblasp-r0.3.6.dylib
##
## locale:
##  en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
##  stats     graphics  grDevices datasets  utils     methods   base
##
## other attached packages:
##  tidyr_0.8.3   ggplot2_3.1.1 CVXR_0.99-6
##
## loaded via a namespace (and not attached):
##   gmp_0.5-13.5      Rcpp_1.0.1        compiler_3.6.0
##   pillar_1.4.1      plyr_1.8.4        R.methodsS3_1.7.1
##   R.utils_2.8.0     tools_3.6.0       digest_0.6.19
##  bit_1.1-14        evaluate_0.14     tibble_2.1.2
##  gtable_0.3.0      lattice_0.20-38   pkgconfig_2.0.2
##  rlang_0.3.4       Matrix_1.2-17     yaml_2.2.0
##  blogdown_0.12.1   xfun_0.7          withr_2.1.2
##  dplyr_0.8.1       Rmpfr_0.7-2       ECOSolveR_0.5.2
##  stringr_1.4.0     knitr_1.23        tidyselect_0.2.5
##  bit64_0.9-7       grid_3.6.0        glue_1.3.1
##  R6_2.4.0          rmarkdown_1.13    bookdown_0.11
##  purrr_0.3.2       magrittr_1.5      scales_1.0.0
##  htmltools_0.3.6   scs_1.2-3         assertthat_0.2.1
##  colorspace_1.4-1  labeling_0.3      stringi_1.4.3
##  lazyeval_0.2.2    munsell_0.5.0     crayon_1.3.4
##  R.oo_1.22.0

R Markdown