is the parameter space. Using the inverse function theorem, it can be shown that is well-defined in an open neighborhood about with probability going to one, and is a consistent estimate of . As a consequence there exists a sequence such that asymptotically almost surely, and . A similar result can be established using Rolle's theorem. The second derivative evaluated at , known as Fisher information, determines the curvature of the likelihood surface, and thus indicates the precision of the estimate. Exponential families The log-likelihood is also particularly useful for exponential families of distributions, which include many of the common parametric probability distributions. The probability distribution function (and thus likelihood function) for exponential families contain products of factors involving exponentiation. The logarithm of such a function is a sum of products, again easier to differentiate than the original function. An exponential family is one whose probability density function is of the form (for some functions, writing for the inner product): Each of these terms has an interpretation, but simply switching from probability to likelihood and taking logarithms yields the sum: The and each correspond to a change of coordinates, so in these coordinates, the log-likelihood of an exponential family is given by the simple formula: In words, the log-likelihood of an exponential family is inner product of the natural parameter and the sufficient statistic , minus the normalization factor (log-partition function) . Thus for example the maximum likelihood estimate can be computed by taking derivatives of the sufficient statistic and the log-partition function . Example: the gamma distribution The gamma distribution is an exponential family with two parameters, and . The likelihood function is Finding the maximum likelihood estimate of for a single observed value looks rather daunting. Its logarithm is much simpler to work with: To maximize the log-likelihood, we first take the partial derivative with respect to : If there are a number of independent observations , then the joint log-likelihood will be the sum of individual log-likelihoods, and the derivative of this sum will be a sum of derivatives of each individual log-likelihood: To complete the maximization procedure for the joint log-likelihood, the equation is set to zero and solved for : Here denotes the maximum-likelihood estimate, and is the sample mean of the observations. Background and interpretation Historical remarks The term "likelihood" has been in use in English since at least late Middle English. Its formal