# A Note on Antoniak’s Approximation for Dirichlet Processes

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Antoniak’s 1974 article titled *Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems* (Annals of Statistics 2(6):1152-1174) is a fundamental work for most modern developments in this area. The article gives two expressions for the expected number of distinct values in a sample of size , drawn from a Dirichlet process-distributed probability distribution with precision parameter . The first expression is exact, and the second is approximate. However, I recently discovered (the ‘hard way’) that the approximation is poor for . The following `R` graphic illustrates the disparity for several values of and .

`R` Code

approx <- function(alpha, n) alpha * log((1:n + alpha)/alpha) exact <- function(alpha, n) cumsum(alpha/(alpha + 1:n - 1)) approx_exact_plot <- function(alp, n, ...) { amax <- max(1/alp) appmax <- approx(amax, n) examax <- exact(amax, n) par(mar = c(5, 6, 4, 3) + 0.1) plot(appmax, examax, type = "l", xlim = c(0, max(appmax)), ylim = c(0, max(examax)), main = expression(paste(italic(E)(italic(Z)[italic(n)]), " - Antoniak (1974)")), xlab = expression(paste("Approximate - ", alpha, "log", ((n + alpha)/alpha))), ylab = expression(paste("Exact - ", sum(alpha/(alpha + m - 1), 1, n))), ...) text(max(appmax), max(examax), substitute(paste(alpha, " = ", amax)), pos = 2) abline(a = 0, b = 1, lty = 2) if (length(alp) == 1) return() for (a in alp[-1]) { app <- approx(1/a, n) exa <- exact(1/a, n) points(app, exa, type = "l") text(max(app), max(exa), substitute(paste(alpha, " = ", 1/a)), pos = 2) } } approx_exact_plot(2^(0:4), 10)

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