  The full-span log-linear(FSLL) model introduced in this paper is considered
an $n$-th order Boltzmann machine, where $n$ is the number of all variables in
the target system. Let $X=(X_0,...,X_{n-1})$ be finite discrete random
variables that can take $|X|=|X_0|...|X_{n-1}|$ different values. The FSLL
model has $|X|-1$ parameters and can represent arbitrary positive distributions
of $X$. The FSLL model is a "highest-order" Boltzmann machine; nevertheless, we
can compute the dual parameters of the model distribution, which plays
important roles in exponential families, in $O(|X|\log|X|)$ time. Furthermore,
using properties of the dual parameters of the FSLL model, we can construct an
efficient learning algorithm. The FSLL model is limited to small probabilistic
models up to $|X|\approx2^{25}$; however, in this problem domain, the FSLL
model flexibly fits various true distributions underlying the training data
without any hyperparameter tuning. The experiments presented that the FSLL
successfully learned six training datasets such that $|X|=2^{20}$ within one
minute with a laptop PC.
