Multiset - Cumulant Generating Function

Cumulant Generating Function

A non-negative integer, n, can be represented by the monomial xn . A finite multiset of non-negative integers, say {2, 2, 2, 3, 5}, can likewise be represented by a polynomial f(x), say f(x) = 3·x2 + x3 + x5 .

It is convenient to consider the cumulant generating function g(t) = log(f(et)), say g(t) = log(3·e2·t + e3·t + e5·t) .

The cardinality of the multiset is eg(0) = f(1), say 3 + 1 + 1 = 5.

The derivative g is g '(t) = f(et)−1·f '(et)·et, say g '(t) = (3·e2·t + e3·t + e5·t)−1·(6·e2·t + 3·e3·t + 5·e5·t) .
- The mean value of the multiset is μ = g '(0) = f(1)−1·f '(1), say μ = (3+1+1)−1·(6+3+5) = 2.8 .
- The variance of the multiset is σ2 = g ' '(0) .

The numbers ( μ, σ2, ··· ) = ( g '(0), g ' '(0), ··· ) are called cumulants.

The infinite set of non-negative integers {0, 1, 2, ···} is represented by the formal power series 1 + x + x2 + ··· = (1 − x)−1. The mean value and standard deviation are undefined. Nevertheless it has a cumulant-generating function, g(t) = −log(1−et). The derivative of this cumulant-generating function is g '(t) = (e−t−1)−1.

A finite multiset of real numbers, A = { A_i }, is represented by the cumulant generating function

This representation is unique: different multisets have different cumulant generating functions. See partition function (statistical mechanics) for the case where the numbers in question are the energy levels of a physical system.

The cumulant-generating function of a multiset of n real numbers having mean μ and standard deviation σ is: g(t) = log(n) + μ·t + 2−1·(σ·t)2 + ···, and the derivative is simply: g '(t) = μ + σ2·t + ···

The cumulant-generating function of set, {k}, consisting of a single real number, k, is g(t) = k·t, and the derivative is the number itself: g '(t) = k . So the concept of the derivative of the cumulant generating function of a multiset of real numbers is a generalization of the concept of a real number.

The cumulant-generating function of a constant multiset, {k, k, k, k, ···, k} of n elements all equal to the same real number k, is g(t) = log(n)+k·t, and the derivative is the number itself: g '(t) = k, irrespective of n.

The cumulant-generating function of the multiset of sums of elements of two multisets of numbers is the sum of the two cumulant-generating functions:

$\begin{align} g_{A+B}(t) & = \log \left(\sum_i \sum_j e^{t\cdot(A_i+B_j)}\right) = \log \left(\sum_i\sum_j e^{t\cdot A_i}\cdot e^{t\cdot B_j}\right) \\ & = \log \left(\sum_i e^{t\cdot A_i}\cdot \sum_j e^{t\cdot B_j}\right) = \log \left(\sum_i e^{t\cdot A_i}\right)+ \log\left(\sum_j e^{t\cdot B_j}\right) \\ & = g_A(t) + g_B(t). \end{align}$

There is unfortunately no general formula for computing the cumulant generating function of a product

but the special case of a constant times a multiset of numbers is:

The multiset 2·A = {2·A_i} is not the same multiset as 2×A =A+A = {A_i+A_j}. For example, 2·{+1,−1} = {+2,−2} while 2×{+1,−1} = {+1,−1} + {+1,−1} = {+1+1,+1−1,−1+1,−1−1} = {+2,0,0,−2}.

The standard normal distribution is like a limit of big multisets of numbers.

This limit does not make sense as a multiset of numbers, but the derivative of the cumulant generating functions of the multisets in question makes sense, and the limit is well defined.

$\begin{align} \lim_{k \rarr \infty} g'_{k^{-1}\cdot (k^2\times \{+1,-1\})}(t) & = \lim_{k \rarr \infty} \frac{d(k^2\cdot \log(e^{+t\cdot k^{-1}}+e^{-t\cdot k^{-1}}))}{dt} \\ & = \lim_{k \rarr \infty} \frac{d(k^2\cdot \log(2)+2^{-1}\cdot t^2+\cdots)}{dt}=t. \end{align}$

The constant term k2·log(2) vanishes by differentiation. The terms ··· vanish in the limit. So for the standard normal distribution, having mean 0 and standard deviation 1, the derivative of the cumulant generating function is simply g '(t) = t . For the normal distribution having mean μ and standard deviation σ, the derivative of the cumulant generating function is g '(t) = μ + σ2·t .