BIC sign conventions

Section of Model selection for stochastic models, part of Statistical learning

BIC sign conventions

Type Section
Taught in Model selection for stochastic models
Part Statistical learning

One to six full-covariance mixtures are fitted to the same two petal measurements. The package reports 2\ell-d\log n, larger meaning better. The table also computes -2\ell+d\log n, smaller meaning better. The check is that the two columns sum to zero. As on the mixture page, iris_x holds the four iris measurements and columns 3 and 4 are the petals. There are n=150 flowers.

set.seed(6006)
fits <- lapply(1:6, function(g)
  mclust::Mclust(iris_x[, 3:4], G = g,
                 modelNames = "VVV", verbose = FALSE))
bic_table <- data.frame(
  Components = 1:6,
  Parameters = vapply(fits, function(f) f$df, numeric(1)),
  BIC = vapply(fits, function(f)
    -2 * f$loglik + f$df * log(nrow(iris)), numeric(1)))
print(bic_table, row.names = FALSE)
#> Components Parameters   BIC
#>          1          5 570.6
#>          2         11 364.6
#>          3         17 355.8
#>          4         23 361.1
#>          5         29 372.3
#>          6         35 399.7
stopifnot(max(abs(bic_table$BIC +
                  vapply(fits, function(f) f$bic, numeric(1)))) < 1e-7)

The parameter counts 5,11,17,23,29,35 are the full-covariance counts d=Hp+Hp(p+1)/2+(H-1) at p=2. On this list the BIC of -2\ell+d\log n selects 3 components, with value 355.8. That number describes these two measurements under this candidate family. Reading it as a count of biological species requires a separate scientific argument. A Gaussian component can approximate one piece of a non-Gaussian species distribution, and a different constraint on the covariances defines a different list of models.

Key idea

BIC is minus twice the maximised log-likelihood plus d\log n. It approximates minus twice the log marginal likelihood of a regular model, which is why each additional parameter must buy at least \tfrac12\log n units of log-likelihood. Fit each candidate and choose the smallest BIC. The mclust column 2\ell-d\log n is the negative of that quantity, so it chooses the largest value. AIC, with penalty 2d, and likelihood cross-validation estimate a predictive target and can prefer a different candidate. For mixtures the Laplace derivation is a regular-model argument and needs the qualification above.

Used in

See also

Bayesian information criterion · Akaike information criterion · Laplace approximation · Dropping terms that do not scale with n · Reading the BIC · Cross-validation of the likelihood · Kullback–Leibler divergence

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