expLL

expLL is a term encountered in statistics and machine learning to denote the exponential of a log-likelihood function. If ℓ(θ; x) denotes the log-likelihood of parameter vector θ given data x, then expLL(θ; x) = exp(ℓ(θ; x)) = L(θ; x), the likelihood function. Because the logarithm is strictly increasing, maximizing ℓ is equivalent to maximizing L; the two forms contain the same information regarding the best-fitting θ, but log-likelihood is preferred for numerical stability and additive properties.

In practice, expLL equals the likelihood; it is widely used in likelihood ratio tests and model comparison, and it is related to Bayesian posteriors through proportional relationships. However, direct computations of L can suffer from underflow, hence optimization is often performed in the log domain. Exponentiating back to obtain expLL is typically avoided during fitting but can be useful for interpretation or certain diagnostic calculations.

Examples help illustrate the relationship. For independent observations from a normal distribution with known variance, L(μ) = ∏ f(x_i; μ) = ∏ (1/√(2πσ^2)) exp(-(x_i-μ)^2/(2σ^2)); the corresponding ℓ(μ) is a constant minus the sum of squared deviations, and exp(ℓ(μ)) equals the product of the Gaussian densities, i.e., the likelihood. In exponential-family models, expℓ is closely tied to natural parameters and sufficient statistics, making the interpretation of θ in terms of data summaries natural.

Limitations include numerical underflow when expℓ is very small. Practitioners typically work in the log-likelihood form for optimization and only exponentiate when necessary, using techniques like the log-sum-exp trick to compare multiple models without instability.