gumbel_max {IMIFA}

Simulate Cluster Labels from Unnormalised Log-Probabilities using the Gumbel-Max Trick

Description

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.

Usage

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).

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.

Value

A vector of N sampled cluster labels, with the largest label no greater than G.

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.

Author(s)

Keefe Murphy - <[e-mail]>

References

Murphy, K., Viroli, C., and Gormley, I. C. (2020) Infinite mixtures of infinite factor analysers, Bayesian Analysis, 15(3): 937-963. <doi:10.1214/19-BA1179>.

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

Examples

# 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)))