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[1])
            y0[1] + slope[1]*(xs -x0[1])
        else if(xs >= x0[n])
            y0[n] + slope[n-1]*(xs - x0[n])
        else {
            idx <- which(xs <= x0)[1]
            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 4.0.2 (2020-06-22)
## Platform: x86_64-apple-darwin19.5.0 (64-bit)
## Running under: macOS Catalina 10.15.7
## 
## Matrix products: default
## BLAS/LAPACK: /usr/local/Cellar/openblas/0.3.10_1/lib/libopenblasp-r0.3.10.dylib
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## attached base packages:
## [1] stats     graphics  grDevices datasets  utils     methods   base     
## 
## other attached packages:
## [1] tidyr_1.1.0   ggplot2_3.3.2 CVXR_1.0-9   
## 
## loaded via a namespace (and not attached):
##  [1] gmp_0.6-0        Rcpp_1.0.5       compiler_4.0.2   pillar_1.4.6    
##  [5] tools_4.0.2      digest_0.6.25    bit_1.1-15.2     evaluate_0.14   
##  [9] lifecycle_0.2.0  tibble_3.0.3     gtable_0.3.0     lattice_0.20-41 
## [13] pkgconfig_2.0.3  rlang_0.4.7      Matrix_1.2-18    gurobi_9.0.3.1  
## [17] Rglpk_0.6-4      yaml_2.2.1       blogdown_0.19    xfun_0.15       
## [21] cccp_0.2-4       ECOSolveR_0.5.3  withr_2.2.0      dplyr_1.0.0     
## [25] Rmpfr_0.8-1      stringr_1.4.0    knitr_1.28       generics_0.0.2  
## [29] vctrs_0.3.2      tidyselect_1.1.0 bit64_0.9-7      grid_4.0.2      
## [33] glue_1.4.1       R6_2.4.1         rmarkdown_2.3    bookdown_0.19   
## [37] farver_2.0.3     purrr_0.3.4      magrittr_1.5     codetools_0.2-16
## [41] rcbc_0.1.0.9001  scales_1.1.1     htmltools_0.5.0  ellipsis_0.3.1  
## [45] assertthat_0.2.1 colorspace_1.4-1 labeling_0.3     Rcplex_0.3-3    
## [49] stringi_1.4.6    Rmosek_9.2.3     munsell_0.5.0    slam_0.1-47     
## [53] crayon_1.3.4

Source

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References