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Gibbs-Posterior Model Weights

Source: R/calc_model_weights_gibbs.R
calc_model_weights_gibbs.Rd

Converts a numeric vector of model scores (treated as loss-like, i.e., lower-is-better) into normalized posterior weights using the Gibbs (generalized Bayes) formulation: $$w_m(\eta) = \frac{\exp(-\eta \, x_m)}{\sum_{j=1}^K \exp(-\eta \, x_j)} \,,$$ where \(x_m\) is the loss (or information criterion) for model \(m\) and \(\eta \ge 0\) is the inverse temperature (sharpness) parameter.

Usage

calc_model_weights_gibbs(x, eta, verbose = FALSE, ...)

Arguments

x

Numeric vector of model scores (treated as loss-like; lower is better).

eta

Non-negative scalar inverse temperature \(\eta\). Higher values concentrate weight on the best models; lower values flatten weights. Typical choices: 0.5 for \(\Delta\)AIC (Akaike weights), 1 for \(-\log L\).

verbose

Logical; if TRUE, prints brief diagnostics (range of x, applied shift, entropy, and effective number of models).

...

Reserved for backward compatibility. The deprecated argument temperature is accepted here and mapped to eta with a warning.

Value

A numeric vector of length length(x) containing weights that sum to 1. Names are preserved when present.

Details

This function is agnostic to the specific form of \(x\); smaller values are assumed to indicate better models. Common choices include:

  • \(x = \Delta \mathrm{AIC}\) \; (use eta = 1/2 to recover Akaike weights),

  • \(x = -\log L\) \; (use eta = 1 for normalized likelihood weights),

  • Cross-validated losses (e.g., RMSE, MSLE), or any proper scoring rule.

If you start from a higher-is-better score (e.g., log-likelihood), pass x = -score so that lower-is-better holds.

Numerical stability: the implementation subtracts \(\min(x)\) prior to exponentiation; this shift leaves the normalized weights unchanged but helps avoid underflow/overflow.

Special cases:

  • eta = 0: uniform weights.

  • eta = Inf: hard selection (ties share weight equally).

  • All x equal: uniform weights.

References

Bissiri, P. G., Holmes, C. C., & Walker, S. G. (2016). A general framework for updating belief distributions. Journal of the Royal Statistical Society: Series B, 78(5), 1103–1130. doi:10.1111/rssb.12158

Examples

# Three models with Delta AIC = (0, 2, 6): Akaike weights via eta = 0.5
calc_model_weights_gibbs(x = c(0, 2, 6), eta = 0.5)
#> [1] 0.70538451 0.25949646 0.03511903

# From negative log-likelihoods: normalized likelihood weights with eta = 1
calc_model_weights_gibbs(x = c(120.3, 121.1, 124.8), eta = 1)
#> [1] 0.68472611 0.30766727 0.00760662

# Using a CV loss (lower is better) with moderate sharpness
set.seed(1)
losses <- c(A = 0.83, B = 0.81, C = 0.92)
calc_model_weights_gibbs(losses, eta = 3, verbose = TRUE)
#> Gibbs weights with eta = 3
#> Input range: min = 0.81, max = 0.92, spread = 0.11
#> Stabilization: subtract min(x) = 0.81 before exponentiation.
#> Entropy = 1.0890; effective number of models = 2.971
#>         A         B         C 
#> 0.3539552 0.3758426 0.2702022 

# Eta extremes
calc_model_weights_gibbs(c(1, 2, 5), eta = 0)      # uniform
#> [1] 0.3333333 0.3333333 0.3333333
calc_model_weights_gibbs(c(1, 2, 5), eta = Inf)    # hard selection
#> [1] 1 0 0