Bregman-divergence-guided Legendre exponential dispersion model with finite cumulants (K-LED)

Exponential dispersion model is a useful framework in machine learning and statistics. Primarily, thanks to the additive structure of the model, it can be achieved without difficulty to estimate parameters including mean. However, tight conditions on cumulant function, such as analyticity, strict convexity, and steepness, reduce the class of exponential dispersion model. In this work, we present relaxed exponential dispersion model K-LED (Legendre exponential dispersion model with K cumulants). The cumulant function of the proposed model is a convex function of Legendre type having continuous partial derivatives of K-th order on the interior of a convex domain. Most of the K-LED models are developed via Bregman-divergence-guided log-concave density function with coercivity shape constraints. The main advantage of the proposed model is that the first cumulant (or the mean parameter space) of the 1-LED model is easily computed through the extended global optimum property of Bregman divergence. An extended normal distribution is introduced as an example of 1-LED based on Tweedie distribution. There is an equivalence between a subclass of quasi-likelihood function and a regular 2 -LED model, of which the canonical parameter space is open. A typical example is a regular 2 -LED model with power variance function, i.e., a variance is in proportion to the power of the mean of observations. This model is equivalent to a subclass of beta-divergence (or a subclass of quasi-likelihood function with power variance function). Furthermore, a new parameterized K-LED model is proposed. The cumulant function of this model is the convex extended logistic loss function which is generated by extended log and exp functions. The proposed model includes Bernoulli distribution and Poisson distribution depending on the selection of parameters of the convex extended logistic loss function. V arious probability distributions, such as normal distribution, Poisson distribution, gamma distribution, and Bernoulli distribution, are formulated into the exponential families [4], [11], [29] with sufficient statistics by virtue of the Fisher-Neyman factorization theorem [22]. As a consequence of the additive structure of the exponential families, it is easy to estimate parameters, such as mean and variance, of probability distributions. Numerous applications of the exponential families are introduced in [3], [26], [31], [34], [46].