Log-likelihood is the natural logarithm of the likelihood: the probability of observed data given candidate parameter values. Parameter estimation chooses values that make the observed data most likely. Because the logarithm preserves ordering, maximizing likelihood and maximizing log-likelihood produce the same estimate, while the logarithmic form is usually easier to manipulate and compute. section 2.3.4
A practical estimation workflow treats log-likelihood as an optimization objective:
- Specify a probabilistic model and its parameter vector, such as regression weights or neural-network parameters. section 2.3.4
- Maximize log-likelihood, or equivalently minimize negative log-likelihood loss. section 2.3.4
- Use a closed-form solution when available, as in some linear-regression cases; otherwise use numerical optimization. section 2.3.4
- Add regularization when optimization should discourage excessively large parameter estimates. section 2.3.4
A high log-likelihood supports fit to the observed distribution, but statistical modeling and causal modeling target different objects: observational or joint distributions versus the data-generating process. That distinction matters when interpreting estimates causally. section 2.3.4
