# Simulate Cluster Labels from Unnormalised Log-Probabilities using the Gumbel-Max Trick Samples cluster labels for N observations from G clusters efficiently using log-probabilities and the so-called Gumbel-Max trick, without requiring that the log-probabilities be normalised; thus redundant computation can be avoided. ```r gumbel_max(probs, slice = FALSE) ``` ## Arguments - `probs`: An N x G matrix of unnormalised probabilities on the log scale, where N is he number of observations that require labels to be sampled and G is the number of active clusters s.t. sampled labels can take values in `1:G`. Typically `N > G`. - `slice`: A logical indicating whether or not the indicator correction for slice sampling has been applied to `probs`. Defaults to `FALSE` but is `TRUE` for the `"IMIFA"` and `"IMFA"` methods under `mcmc_IMIFA`. Details of this correction are given in Murphy et. al. (2020). When set to `TRUE`, this results in a speed-improvement when `probs` contains non-finite values (e.g. `-Inf`, corresponding to zero on the probability scale). ## Returns A vector of N sampled cluster labels, with the largest label no greater than G. ## Details Computation takes place on the log scale for stability/underflow reasons (to ensure negligible probabilities won't cause computational difficulties); in any case, many functions for calculating multivariate normal densities already output on the log scale. ## Note Though the function is available for standalone use, note that no checks take place, in order to speed up repeated calls to the function inside `mcmc_IMIFA`. If the normalising constant is required for another reason, e.g. to compute the log-likelihood, it can be calculated by summing the output obtained by calling `rowLogSumExps` on `probs`. ## Examples ```r # Create weights for 3 components G <- 3 weights <- seq_len(G) # Call gumbel_max() repeatedly to obtain samples of the labels, zs iters <- 10000 zs <- replicate(iters, gumbel_max(probs=log(weights))) # Compare answer to the normalised weights tabulate(zs, nbins=G)/iters (normalised <- as.numeric(weights/sum(weights))) # Simulate a matrix of Dirichlet weights & the associated vector of N labels N <- 400 G <- 8 sizes <- seq(from=85, to=15, by=-10) weights <- matrix(rDirichlet(N * G, alpha=1, nn=sizes), byrow=TRUE, nrow=N, ncol=G) (zs <- gumbel_max(probs=log(weights))) ``` ## References Murphy, K., Viroli, C., and Gormley, I. C. (2020) Infinite mixtures of infinite factor analysers, **Bayesian Analysis**, 15(3): 937-963. . Yellott, J. I. Jr. (1977) The relationship between Luce's choice axiom, Thurstone's theory of comparative judgment, and the double exponential distribution, **Journal of Mathematical Psychology**, 15(2): 109-144. ## See Also `mcmc_IMIFA`, `rowLogSumExps` ## Author(s) Keefe Murphy -

gumbel_max function