ChainModels

This package provides utilities to deal with multivariate functions factorized over a one-dimensional chain, i.e. where each variable $x_i$ interacts only with $x_{i-1}$ and $x_{i+1}$

$$f(x_1,\ldots,x_L) = e^{f_1(x_1,x_2)} e^{f_2(x_2,x_3)} \cdots e^{f_{L-1}(x_{L-1},x_L)} = \prod_{i=1}^{L-1} e^{f_i(x_i,x_{i+1})}$$

where $x_i \in \mathcal{X}_i=\{1,2,\ldots,q_i\}$ and $f_i : \mathcal{X}_i \times \mathcal{X}_{i+1} \to \mathbb{R}$. Once properly normalized, these give probability distributions

$$p(x_1,\ldots,x_L) = \frac1Z\prod_{i=1}^{L-1} e^{f_i(x_i,x_{i+1})}$$

with $Z = \sum\limits_{x_1,\ldots,x_L}\prod\limits_{i=1}^{L-1} e^{f_i(x_i,x_{i+1})}$ the normalization constant.

In other words, a probability distribution is a chain model if its factor graph is a simple path:

Provided functionalities

For a chain of length $L$ with variables taking one of $q$ values, the following can be performed efficiently:

Operation	Cost
Compute normalization $Z$	$\mathcal O (Lq^2)$
Compute marginals $p(x_i)$	$\mathcal O (Lq^2)$
Compute neighbor marginals $p(x_i,x_{i+1})$	$\mathcal O (Lq^2)$
Compute pair marginals $p(x_i,x_j)$	$\mathcal O (L^2q^2)$
Draw a sample from $p$	$\mathcal O (Lq^2)$
Compute the entropy $S[p]=-\sum_xp(x)\log p(x) $	$\mathcal O (Lq^2)$
Compute the log-likelihood $\log L=\sum\limits_{\mu=1}^N\log p(x^{(\mu)})$ of $N$ samples	$\mathcal O (Lq^2 + NL)$
Compute the gradient of the log-likelihood $\frac{d\log L}{d f_i(x_i,x_{i+1})}$	$\mathcal O (Lq^2 + NLq^2)$

Examples of chain models

• Markov Chains
• One-dimensional Ising models

Efficient operations

The efficiency of the operations mentioned above relies on some strategic pre-computations. For example, partial normalizations from the left and from the right ($l$ and $r$, respectively)

$$\begin{eqnarray} l_{i}(x_{i+1}) =& \log\sum\limits_{x_1,\ldots,x_i}\prod\limits_{j=1}^i e^{f_j(x_j,x_{j+1})}\\\ r_{i}(x_{i-1}) =& \log\sum\limits_{x_i,\ldots,x_L}\prod\limits_{j=i-1}^L e^{f_j(x_j,x_{j+1})} \end{eqnarray}$$

can be used to compute normalization, single-variable and nearest-neighbor marginals

$$\begin{eqnarray} Z =& \sum\limits_{x_i} e^{l_{i-1}(x_i)+r_{i+1}(x_i)}\quad\forall i\\\ p(x_i) =& \frac1Z e^{l_{i-1}(x_i)+r_{i+1}(x_i)}\\\ p(x_i,x_{i+1}) =& \frac1Z e^{l_{i-1}(x_i) + f_i(x_i,x_{i+1}) + r_{i+2}(x_{i+1})}\\\ \end{eqnarray}$$

Notes

• The exponential parametrization is favorable because it puts no constraint on the values taken by the $f_i$'s, which can be positive or negative. One might as well parametrize directly as $f(x)=\prod\limits_{i=1}^{L-1} g_i(x_i,x_{i+1})$ with $g_i(x_i,x_{i+1})=e^{f_i(x_i,x_{i+1})}$, but must always ensure $g_i \ge 0$.

Example

Install with

import Pkg; Pkg.add("https://github.com/stecrotti/ChainModels.jl.git")

Create a ChainModel and compute some stuff

using ChainModels

L = 100
q = fill(20, L)
f = [randn(q[i], q[i+1]) for i in 1:L-1]
p = ChainModel(f)

Z = normalization(p)
marg = marginals(p)
neigmarg = neighbor_marginals(p)
pairmarg = pair_marginals(p)
S = entropy(p)
x = rand(p, 500)
logL = loglikelihood(p, x)