Stable log-mean-exp computation

Computes the log of the mean of exponentials in a numerically stable way. This is equivalent to log(mean(exp(x))) but avoids numerical overflow/underflow by subtracting the maximum value before exponentiation.

Usage

calc_log_mean_exp(x)

Arguments

x

Numeric vector of log-values. -Inf is a valid log-value (a zero likelihood) and enters the mean as exp(-Inf) = 0; NA/NaN entries (failed evaluations) are dropped.

Value

Numeric scalar; the log-mean-exp of the non-missing entries. -Inf if every non-missing entry is -Inf, Inf if any entry is Inf, and NA_real_ if no entry is non-missing.

Details

The log-mean-exp for a vector \(x\) is computed as: $$ \mathrm{LME}(x) = \max(x) + \log\left(\frac{1}{|x|}\sum_i e^{x_i - \max(x)}\right) $$

This is numerically stable because:

• The maximum value is subtracted before exponentiation, preventing overflow

• At least one exponentiated term equals 1, preventing underflow

• Equivalent to log(mean(exp(x))) but without numerical issues

Dropping -Inf instead would shrink the denominator \(|x|\) and bias the result upward: averaging one zero-likelihood replicate out of \(m\) would gain \(\log(m/(m-1))\).

Examples

# Basic usage
x <- c(-100, -101, -99)
calc_log_mean_exp(x)  # -99.69
#> [1] -99.69101

# Compare with the naive form (underflows for very negative values)
log(mean(exp(x)))
#> [1] -99.69101

# -Inf counts as a zero likelihood; NA is dropped
calc_log_mean_exp(c(0, -Inf))        # log(0.5)
#> [1] -0.6931472
calc_log_mean_exp(c(-50, -60, NA))   # log-mean-exp of -50 and -60
#> [1] -50.6931

# Empty, all-missing, or all -Inf input
calc_log_mean_exp(c())           # NA
#> [1] NA
calc_log_mean_exp(c(NA, NaN))    # NA
#> [1] NA
calc_log_mean_exp(c(-Inf, -Inf)) # -Inf
#> [1] -Inf