In rje42/imset: Methods for imsets with mixed graphs

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Imsets

This package allows for the construction, manipulation and testing of independence models represented by imsets.

An imset is an integer-valued multiset, or a function from the powerset of a collection of variables $V$ to the integers. They can be used to represent arbitrary conditional independence models, by assessing whether or not certain imsets are structural.

Basic Definitions

The identifier imset $\delta_A$ is a vector indexed by subsets of $V$ with a 1 in the entry that corresponds to $A$, and 0 in all other entries.

A semi-elementary imset is isomorphic to a triple $A,B,C$ of disjoint sets where the order of the first two elements is arbitrary. It is defined as \begin{align} u_{\langle A,B | C\rangle} := \delta_{C} - \delta_{AC} - \delta_{BC} + \delta_{ABC} \end{align}

An imset $u$ is said to be structural if there is some natural number $k \in \mathbb{N}$ such that $k\cdot u$ can be written as a sum of semi-elementary imsets with natural numbers as coefficients. In other words, if \begin{align} \exists k \in \mathbb{N} : \qquad k \cdot u = \sum_{i} k_i \cdot u_{\langle A_i,B_i | C_i\rangle} \qquad \text{for } k_i \in \mathbb{N}. \label{eqn:struc} \end{align} An imset is said to represent an independence $A \perp B \mid C$ if there exists a $k \in \mathbb{N}$ such that \begin{align} k \cdot u - u_{\langle A,B | C\rangle} \end{align} is structural. Note that therefore the imset itself must be structural, and will represent any of the independences in the expansion on the right-hand side of (\ref{eqn:struc}).

The Package

You will need the libraries rje, contingency and lpSolve which are available both from CRAN and from the GitHub page rje42. You will also need the packages MixedGraphs, ADMGs2 and imset, which are currently only available from rje42.

# devtools::install_github("rje42/rje")
# devtools::install_github("rje42/contingency")
# devtools::install_github("rje42/MixedGraphs")
# devtools::install_github("rje42/ADMGs2")
# devtools::install_github("rje42/imset")
# devtools::install_github("rje42/lpSolve")

We can then load the package in the usual manner.

library(imset)

Then to define an imset, we can use the elem_imset function.

elem_imset(1,2)
elem_imset(1,2,4)
elem_imset(1,2,4,n=6)

Note that the argument n modifies the total number of variables, which is otherwise assumed to be the largest number specified.

We can test whether a given imset is structural using is_structural.

up <- elem_imset(1,2) + elem_imset(1,3,2)
um <- elem_imset(1,2) - elem_imset(1,3,2)
is_structural(up)
is_structural(um)

Note that if the imset is structural, then a valid decomposition is returned. It is clear that this is not unique in general.

Standard Imsets

The standard imset for a DAG is defined in the usual way. We can create a DAG using the graphCr function from the MixedGraphs package.

dag1 <- graphCr("1 -> 3 <- 2, 3 -> 4")
u1 <- standard_imset(dag1)
u1

The characteristic imset can be obtained by applying char_imset to either the graph or the standard imset.

We can also check whether the standard imset defines the model for the DAG.

defines_mod(dag1, u1)

And then test another imset which we know does not:

defines_mod(dag1, elem_imset(1,4))

rje42/imset documentation built on March 20, 2023, 9:55 a.m.