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For two probability mass functions $P,Q$, a decision region $B_n$ accepts $P^{\otimes n}$. Its two error probabilities are $P^{\otimes n}(B_n^c)$ and $Q^{\otimes n}(B_n)$.

For probability mass functions $P,Q$ with $P$ absolutely continuous with respect to $Q$,

The zero-field ferromagnetic Curie–Weiss model is an Ising model on $n$ spins with an equal interaction between every pair. For $s_i\in \{-1,1\}$ and $\beta >0$, its normalized probability mass function is $Z_{n,\beta }^{-1}\exp [\beta ({\sum }_i{s}_i{)}^2/(2n)]$. The partition function $Z_{n,\beta }$ sums these weights over all configurations. The square includes constant diagonal terms; these affect only normalization. Spin inversion symmetry makes every spin's expected value zero, while grouping configurations by their spin magnetization givesHere $m=-1+2k/n$, $0\le k\le n$.

A random variable supported on a finite or countable subset of the real line. Its law is specified by a probability mass function. Countable support need not consist of integers: a variable constantly equal to $3/2$ is discrete. Integer-indexed tail-sum formulas therefore require a separate integer-valued hypothesis.

For a probability mass function $Q$ of full support on a finite alphabet and a real function $g$,where $D_e$ uses natural logarithms. The unique maximizing probability mass function is the exponential tilt $P^{\ast }(x)=Q(x)e^{g(x)}/{\mathbb{E}}_Q{e}^g$.