logsumb

Logsumb is a term used in statistical computing to denote the log-sum-exp operation, a mathematical function that maps a list or vector of real numbers to the logarithm of the sum of their exponentials. In formal terms, for a finite set of values x1, x2, ..., xn, logsumb(X) equals log(sum_i exp(x_i)). This operation is widely used in the domain of log-domain probability calculations, where working with logarithms helps prevent numerical underflow when dealing with very small probabilities.

Definition and stable computation: For X = {x1, ..., xn}, logsumb(X) = log(sum_i exp(x_i)). A common approach for numerical stability is to use the maximum value m = max_i x_i, computing logsumb(X) as m + log(sum_i exp(x_i - m)). This “log-sum-exp trick” avoids overflow and reduces rounding errors during computation.

Relation to softmax and applications: Logsumb is closely related to the softmax function, since softmax_i(x) = exp(x_i) / sum_j exp(x_j), and taking the logarithm of the denominator yields the log-sum-exp. It appears in log-likelihood calculations, hidden Markov models, conditional random fields, and neural networks, particularly in loss functions and attention mechanisms where stable normalization over log-probabilities is required.

Implementations and variants: Many numerical libraries provide implementations of log-sum-exp under the name logsumexp or via related functions such as logaddexp. Examples include SciPy’s scipy.special.logsumexp and PyTorch’s logsumexp, as well as framework-specific utilities for stable log-domain arithmetic.

Etymology and usage: The term logsumb is a shorthand used in some texts and software to refer to this operation, though the standard terminology remains log-sum-exp. It is considered a foundational tool for stable computations in log-domain probability and statistical modeling.