This is Markov.info, produced by Macaulay2, version 1.20

Markov -- Markov ideals, arising from Bayesian networks in statistics
*********************************************************************

Description
===========

This package is used to construct ideals corresponding to discrete graphical
models, as described in several places, including the paper: Garcia, Stillman
and Sturmfels, "The algebraic geometry of Bayesian networks", J. Symbolic
Comput., 39(3-4):331–355, 2005.

The paper also constructs Gaussian ideals, as described in the paper by Seth
Sullivant: "Algebraic geometry of Gaussian Bayesian networks", Adv. in Appl.
Math. 40 (2008), no. 4, 482–513.

Here is a typical use of this package.  We create the ideal in 16 variables
whose zero set represents the probability distributions on four binary random
variables which satisfy the conditional independence statements coming from the
"diamond" graph 4 --> 2,3 --> 1.

+-----------------------------------------------------------------------------+
|i1 : R = markovRing(2,2,2,2)                                                 |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : G = makeGraph{{},{1},{1},{2,3}}                                         |
|                                                                             |
|o2 = Graph{1 => set {}    }                                                  |
|           2 => set {1}                                                      |
|           3 => set {1}                                                      |
|           4 => set {2, 3}                                                   |
|                                                                             |
|o2 : Graph                                                                   |
+-----------------------------------------------------------------------------+
|i3 : S = globalMarkovStmts G                                                 |
|                                                                             |
|o3 = {{{1}, {4}, {2, 3}}, {{2}, {3}, {4}}}                                   |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+
|i4 : I = markovIdeal(R,S)                                                    |
|                                                                             |
|o4 = ideal (- p       p        + p       p       , - p       p        +      |
|               1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2     1,1,2,2 2,1,2,1        |
|     ------------------------------------------------------------------------|
|     p       p       , - p       p        + p       p       , -              |
|      1,1,2,1 2,1,2,2     1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2                 |
|     ------------------------------------------------------------------------|
|     p       p        + p       p       , - p       p        +               |
|      1,2,2,2 2,2,2,1    1,2,2,1 2,2,2,2     1,1,2,1 1,2,1,1                 |
|     ------------------------------------------------------------------------|
|     p       p        + p       p        - p       p        -                |
|      1,1,1,1 1,2,2,1    1,2,2,1 2,1,1,1    1,2,1,1 2,1,2,1                  |
|     ------------------------------------------------------------------------|
|     p       p        - p       p        + p       p        +                |
|      1,1,2,1 2,2,1,1    2,1,2,1 2,2,1,1    1,1,1,1 2,2,2,1                  |
|     ------------------------------------------------------------------------|
|     p       p       , - p       p        + p       p        +               |
|      2,1,1,1 2,2,2,1     1,1,2,2 1,2,1,2    1,1,1,2 1,2,2,2                 |
|     ------------------------------------------------------------------------|
|     p       p        - p       p        - p       p        -                |
|      1,2,2,2 2,1,1,2    1,2,1,2 2,1,2,2    1,1,2,2 2,2,1,2                  |
|     ------------------------------------------------------------------------|
|     p       p        + p       p        + p       p       )                 |
|      2,1,2,2 2,2,1,2    1,1,1,2 2,2,2,2    2,1,1,2 2,2,2,2                  |
|                                                                             |
|o4 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+

Sometime an ideal can be simplified by changing variables.  Very often, by
using marginMap (missing documentation), such ideals can be transformed to
binomial ideals.  This is the case here.

+-----------------------------------------------------------------------------+
|i5 : F = marginMap(1,R)                                                      |
|                                                                             |
|o5 = map (R, R, {p        - p       , p        - p       , p        -        |
|                  1,1,1,1    2,1,1,1   1,1,1,2    2,1,1,2   1,1,2,1          |
|                                                                             |
|o5 : RingMap R <--- R                                                        |
|-----------------------------------------------------------------------------|
|p       , p        - p       , p        - p       , p        - p       ,     |
| 2,1,2,1   1,1,2,2    2,1,2,2   1,2,1,1    2,2,1,1   1,2,1,2    2,2,1,2      |
|-----------------------------------------------------------------------------|
|p        - p       , p        - p       , p       , p       , p       ,      |
| 1,2,2,1    2,2,2,1   1,2,2,2    2,2,2,2   2,1,1,1   2,1,1,2   2,1,2,1       |
|-----------------------------------------------------------------------------|
|p       , p       , p       , p       , p       })                           |
| 2,1,2,2   2,2,1,1   2,2,1,2   2,2,2,1   2,2,2,2                             |
+-----------------------------------------------------------------------------+
|i6 : J = F I;                                                                |
|                                                                             |
|o6 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i7 : netList pack(2,J_*)                                                     |
|                                                                             |
|     +-------------------------------------+---------------------------------|
|o7 = |- p       p        + p       p       |- p       p        + p       p   |
|     |   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2|   1,1,2,2 2,1,2,1    1,1,2,1 2,1|
|     +-------------------------------------+---------------------------------|
|     |- p       p        + p       p       |- p       p        + p       p   |
|     |   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2|   1,2,2,2 2,2,2,1    1,2,2,1 2,2|
|     +-------------------------------------+---------------------------------|
|     |- p       p        + p       p       |- p       p        + p       p   |
|     |   1,1,2,1 1,2,1,1    1,1,1,1 1,2,2,1|   1,1,2,2 1,2,1,2    1,1,1,2 1,2|
|     +-------------------------------------+---------------------------------|
|-----------------------------------------------------------------------------|
|----+                                                                        |
|    |                                                                        |
|,2,2|                                                                        |
|----+                                                                        |
|    |                                                                        |
|,2,2|                                                                        |
|----+                                                                        |
|    |                                                                        |
|,2,2|                                                                        |
|----+                                                                        |
+-----------------------------------------------------------------------------+

This ideal has 5 primary components.  The first is the one that has statistical
significance. The significance of the other components is still poorly
understood.

+-----------------------------------------------------------------------------+
|i8 : time netList primaryDecomposition J                                     |
|     -- used 2.30701 seconds                                                 |
|                                                                             |
|     +-----------------------------------------------------------------------|
|o8 = |ideal (p       , p       , p       , p       , p       p        - p    |
|     |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,|
|     +-----------------------------------------------------------------------|
|     |ideal (p       , p       , p       , p       , p       p        - p    |
|     |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,|
|     +-----------------------------------------------------------------------|
|     |ideal (p       , p       , p       , p       , p       p        - p    |
|     |        1,2,1,2   1,2,1,1   1,1,1,2   1,1,1,1   1,2,2,2 2,2,2,1    1,2,|
|     +-----------------------------------------------------------------------|
|     |ideal (p       , p       , p       , p       , p       p        - p    |
|     |        1,1,2,2   1,1,2,1   1,1,1,2   1,1,1,1   1,2,2,2 2,2,2,1    1,2,|
|     +-----------------------------------------------------------------------|
|     |ideal (p       p        - p       p       , p       p        - p       |
|     |        1,2,2,2 2,2,2,1    1,2,2,1 2,2,2,2   1,2,1,2 2,2,1,1    1,2,1,1|
|     +-----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|   p       , p       p        - p       p       )                            |
|2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                             |
|-----------------------------------------------------------------------------|
|   p       , p       p        - p       p       )                            |
|1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                             |
|-----------------------------------------------------------------------------|
|   p       , p       p        - p       p       )                            |
|2,1 2,2,2,2   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2                             |
|-----------------------------------------------------------------------------|
|   p       , p       p        - p       p       )                            |
|2,1 2,2,2,2   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2                             |
|-----------------------------------------------------------------------------|
|p       , p       p        - p       p       , p       p        - p       p  |
| 2,2,1,2   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|     , p       p        - p       p       , p       p        - p       p     |
|1,1,2   1,1,2,2 1,2,1,2    1,1,1,2 1,2,2,2   1,1,2,1 1,2,1,1    1,1,1,1 1,2,2|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|  , p       p       p       p        - p       p       p       p       , p   |
|,1   2,1,1,1 2,1,2,2 2,2,1,2 2,2,2,1    2,1,1,2 2,1,2,1 2,2,1,1 2,2,2,2   1,1|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|    p       p       p        - p       p       p       p       , p       p   |
|,1,1 2,1,2,2 2,2,1,2 2,2,2,1    1,1,1,2 2,1,2,1 2,2,1,1 2,2,2,2   1,1,2,2 2,1|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|    p       p        - p       p       p       p       , p       p       p   |
|,1,1 2,2,1,2 2,2,2,1    1,1,2,1 2,1,1,2 2,2,1,1 2,2,2,2   1,1,1,1 1,1,2,2 2,2|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|    p        - p       p       p       p       , p       p       p       p   |
|,1,2 2,2,2,1    1,1,1,2 1,1,2,1 2,2,1,1 2,2,2,2   1,2,1,2 2,1,1,1 2,1,2,2 2,2|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|     - p       p       p       p       , p       p       p       p        - p|
|,2,1    1,2,1,1 2,1,1,2 2,1,2,1 2,2,2,2   1,1,1,1 1,2,1,2 2,1,2,2 2,2,2,1    |
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|       p       p       p       , p       p       p       p        - p       p|
|1,1,1,2 1,2,1,1 2,1,2,1 2,2,2,2   1,2,2,2 2,1,1,2 2,1,2,1 2,2,1,1    1,2,2,1 |
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|       p       p       , p       p       p       p        - p       p       p|
|2,1,1,1 2,1,2,2 2,2,1,2   1,1,2,1 1,2,2,2 2,1,1,2 2,2,1,1    1,1,2,2 1,2,2,1 |
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|-----------------------------------------------------------------------------|
|       p       , p       p       p       p        - p       p       p       p|
|2,1,1,1 2,2,1,2   1,2,1,1 1,2,2,2 2,1,1,2 2,1,2,1    1,2,1,2 1,2,2,1 2,1,1,1 |
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|--------+                                                                    |
|        |                                                                    |
|        |                                                                    |
|--------+                                                                    |
|        |                                                                    |
|        |                                                                    |
|--------+                                                                    |
|        |                                                                    |
|        |                                                                    |
|--------+                                                                    |
|        |                                                                    |
|        |                                                                    |
|--------+                                                                    |
|       )|                                                                    |
|2,1,2,2 |                                                                    |
|--------+                                                                    |
+-----------------------------------------------------------------------------+

Caveat
======

The parts of the package involving graphs might eventually be changed to use a
package dealing specifically with graphs.  This might change the interface to
this package.

Authors
=======

  * Luis Garcia-Puente
  * Mike Stillman

Version
=======

This documentation describes version 1.3 of Markov.

Source code
===========

The source code from which this documentation is derived is in the file
Markov.m2 (see common/share/Macaulay2/Markov.m2 ).

Exports
=======

  * Types
      * Graph (missing documentation)
  * Functions and commands
      * displayGraph (missing documentation)
      * gaussIdeal -- correlation ideal of a Bayesian
        network of joint Gaussian variables
      * gaussMinors (missing documentation)
      * gaussRing -- ring of gaussian correlations on n
        random variables
      * gaussTrekIdeal (missing documentation)
      * globalMarkovStmts (missing documentation)
      * hideMap (missing documentation)
      * localMarkovStmts (missing documentation)
      * makeGraph (missing documentation)
      * marginMap (missing documentation)
      * markovIdeal (missing documentation)
      * markovMatrices (missing documentation)
      * markovRing -- ring of probability distributions on
        several discrete random variables
      * pairMarkovStmts (missing documentation)
      * removeRedundants (missing documentation)
      * writeDotFile (missing documentation)
  * Methods
      * "gaussIdeal(Ring,Graph)" -- see gaussIdeal, --
        correlation ideal of a Bayesian network of joint Gaussian variables
      * "gaussIdeal(Ring,List)" -- see gaussIdeal, --
        correlation ideal of a Bayesian network of joint Gaussian variables
      * "gaussRing(ZZ)" -- see gaussRing, -- ring of gaussian
        correlations on n random variables
  * Other things
      * dotBinary (missing documentation)

For the programmer
==================

The object Markov, is a package (Macaulay2Doc).

displayGraph
************

Ways to use displayGraph :
==========================

  * displayGraph(Graph) (missing documentation)
  * displayGraph(String,Graph) (missing documentation)
  * displayGraph(String,String,Graph) (missing documentation)

For the programmer
==================

The object displayGraph (missing documentation) is a method function (Macaulay2Doc).

dotBinary
*********

For the programmer
==================

The object dotBinary (missing documentation) is a string (Macaulay2Doc).

gaussIdeal -- correlation ideal of a Bayesian network of joint Gaussian variables
*********************************************************************************

Synopsis
========

  * Usage:
        gaussIdeal(R,G)
  * Inputs:
      * R, a ring (Macaulay2Doc), created with gaussRing,
      * G, an instance of the type Graph (missing documentation) or a
        list (Macaulay2Doc),
  * Outputs:
      * the ideal in R of the relations in the correlations of the random
        variables implied by the independence statements of the graph G or the
        list of independence statements G

Description
===========

These ideals were first written down by Seth Sullivant, in "Algebraic geometry
of Gaussian Bayesian networks". The routines in this package involving Gaussian
variables are all based on that paper.
+-----------------------------------------------------------------------------+
|i1 : R = gaussRing 5;                                                        |
+-----------------------------------------------------------------------------+
|i2 : G = makeGraph {{2},{3},{4,5},{5},{}}                                    |
|                                                                             |
|o2 = Graph{1 => set {2}   }                                                  |
|           2 => set {3}                                                      |
|           3 => set {4, 5}                                                   |
|           4 => set {5}                                                      |
|           5 => set {}                                                       |
|                                                                             |
|o2 : Graph                                                                   |
+-----------------------------------------------------------------------------+
|i3 : (globalMarkovStmts G)/print;                                            |
|{{1, 2}, {4, 5}, {3}}                                                        |
|{{1}, {3, 4, 5}, {2}}                                                        |
+-----------------------------------------------------------------------------+
|i4 : J = gaussIdeal(R,G)                                                     |
|                                                                             |
|o4 = ideal (- s   s    + s   s   , - s   s    + s   s   , - s   s    +       |
|               1,5 2,4    1,4 2,5     1,5 3,4    1,4 3,5     2,5 3,4         |
|     ------------------------------------------------------------------------|
|     s   s   , s   s    - s   s   , s   s    - s   s   , s   s    - s   s   ,|
|      2,4 3,5   1,4 2,3    1,3 2,4   1,4 3,3    1,3 3,4   2,4 3,3    2,3 3,4 |
|     ------------------------------------------------------------------------|
|     s   s    - s   s   , s   s    - s   s   , s   s    - s   s   , -        |
|      1,5 2,3    1,3 2,5   1,5 3,3    1,3 3,5   2,5 3,3    2,3 3,5           |
|     ------------------------------------------------------------------------|
|     s   s    + s   s   , - s   s    + s   s   , - s   s    + s   s   ,      |
|      1,4 2,3    1,3 2,4     1,5 2,3    1,3 2,5     1,5 2,4    1,4 2,5       |
|     ------------------------------------------------------------------------|
|     s   s    - s   s   , s   s    - s   s   , s   s    - s   s   )          |
|      1,3 2,2    1,2 2,3   1,4 2,2    1,2 2,4   1,5 2,2    1,2 2,5           |
|                                                                             |
|o4 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+

A list of independence statements (as for example returned by
globalMarkovStmts) can be provided instead of a graph.

The ideal corresponding to a conditional independence statement {A,B,C} (where
A,B,C, are disjoint lists of integers in the range 1..n (n is the number of
random variables) is the #C+1 x #C+1 minors of the submatrix of the generic
symmetric matrix M = (s_(i,j)), whose rows are in A union C, and whose columns
are in B union C.  In general, this does not need to be a prime ideal.
+-----------------------------------------------------------------------------+
|i5 : I = gaussIdeal(R,{{{1,2},{4,5},{3}}, {{1},{2},{3,4,5}}})                |
|                                                                             |
|                                                                             |
|o5 = ideal (- s   s    + s   s   , - s   s    + s   s   , - s   s    +       |
|               1,5 2,4    1,4 2,5     1,5 3,4    1,4 3,5     2,5 3,4         |
|     ------------------------------------------------------------------------|
|                                                                             |
|     s   s   , s   s    - s   s   , s   s    - s   s   , s   s    - s   s   ,|
|      2,4 3,5   1,4 2,3    1,3 2,4   1,4 3,3    1,3 3,4   2,4 3,3    2,3 3,4 |
|     ------------------------------------------------------------------------|
|                                                                             |
|     s   s    - s   s   , s   s    - s   s   , s   s    - s   s   ,          |
|      1,5 2,3    1,3 2,5   1,5 3,3    1,3 3,5   2,5 3,3    2,3 3,5           |
|     ------------------------------------------------------------------------|
|              2                                                    2         |
|     s   s   s    - s   s   s   s    - s   s   s   s    + s   s   s    -     |
|      1,5 2,5 3,4    1,5 2,4 3,4 3,5    1,4 2,5 3,4 3,5    1,4 2,4 3,5       |
|     ------------------------------------------------------------------------|
|                                                                   2         |
|     s   s   s   s    + s   s   s   s    + s   s   s   s    - s   s   s    + |
|      1,5 2,5 3,3 4,4    1,5 2,3 3,5 4,4    1,3 2,5 3,5 4,4    1,2 3,5 4,4   |
|     ------------------------------------------------------------------------|
|                                                                             |
|     s   s   s   s    + s   s   s   s    - s   s   s   s    -                |
|      1,5 2,4 3,3 4,5    1,4 2,5 3,3 4,5    1,5 2,3 3,4 4,5                  |
|     ------------------------------------------------------------------------|
|                                                                             |
|     s   s   s   s    - s   s   s   s    - s   s   s   s    +                |
|      1,3 2,5 3,4 4,5    1,4 2,3 3,5 4,5    1,3 2,4 3,5 4,5                  |
|     ------------------------------------------------------------------------|
|                                  2              2                           |
|     2s   s   s   s    + s   s   s    - s   s   s    - s   s   s   s    +    |
|       1,2 3,4 3,5 4,5    1,3 2,3 4,5    1,2 3,3 4,5    1,4 2,4 3,3 5,5      |
|     ------------------------------------------------------------------------|
|                                                2                            |
|     s   s   s   s    + s   s   s   s    - s   s   s    - s   s   s   s    + |
|      1,4 2,3 3,4 5,5    1,3 2,4 3,4 5,5    1,2 3,4 5,5    1,3 2,3 4,4 5,5   |
|     ------------------------------------------------------------------------|
|     s   s   s   s   )                                                       |
|      1,2 3,3 4,4 5,5                                                        |
|                                                                             |
|o5 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i6 : codim I                                                                 |
|                                                                             |
|o6 = 5                                                                       |
+-----------------------------------------------------------------------------+

See also
========

  * makeGraph (missing documentation)
  * globalMarkovStmts (missing documentation)
  * localMarkovStmts (missing documentation)
  * gaussRing -- ring of gaussian correlations on n random
    variables
  * gaussMinors (missing documentation)
  * gaussTrekIdeal (missing documentation)

Ways to use gaussIdeal :
========================

  * "gaussIdeal(Ring,Graph)"
  * "gaussIdeal(Ring,List)"

For the programmer
==================

The object gaussIdeal, is a method function (Macaulay2Doc).

gaussMinors
***********

Ways to use gaussMinors :
=========================

  * gaussMinors(Matrix,List) (missing documentation)

For the programmer
==================

The object gaussMinors (missing documentation) is a method function (Macaulay2Doc).

gaussRing -- ring of gaussian correlations on n random variables
****************************************************************

Synopsis
========

  * Usage:
        gaussRing n
  * Inputs:
      * n, an integer (Macaulay2Doc), the number of random variables
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing (missing documentation) => ..., default value QQ, a
        coefficient field or ring
      * Variable (missing documentation) => ..., default value "s", a symbol,
        the variables in the ring will be s_(1,1),...
  * Outputs:
      * a ring (Macaulay2Doc), a ring with indeterminates s_(i,j),
        1 <= i <= j <= n

Description
===========

The routines gaussMinors (missing documentation), see gaussIdeal,
gaussTrekIdeal (missing documentation) all require that the ring
be created by this function.

+-----------------------------------------------------------------------------+
|i1 : R = gaussRing 5;                                                        |
+-----------------------------------------------------------------------------+
|i2 : gens R                                                                  |
|                                                                             |
|o2 = {s   , s   , s   , s   , s   , s   , s   , s   , s   , s   , s   , s   ,|
|       1,1   1,2   1,3   1,4   1,5   2,2   2,3   2,4   2,5   3,3   3,4   3,5 |
|     ------------------------------------------------------------------------|
|     s   , s   , s   }                                                       |
|      4,4   4,5   5,5                                                        |
|                                                                             |
|o2 : List                                                                    |
+-----------------------------------------------------------------------------+
|i3 : genericSymmetricMatrix(R,5)                                             |
|                                                                             |
|o3 = | s_(1,1) s_(1,2) s_(1,3) s_(1,4) s_(1,5) |                             |
|     | s_(1,2) s_(2,2) s_(2,3) s_(2,4) s_(2,5) |                             |
|     | s_(1,3) s_(2,3) s_(3,3) s_(3,4) s_(3,5) |                             |
|     | s_(1,4) s_(2,4) s_(3,4) s_(4,4) s_(4,5) |                             |
|     | s_(1,5) s_(2,5) s_(3,5) s_(4,5) s_(5,5) |                             |
|                                                                             |
|             5       5                                                       |
|o3 : Matrix R  <--- R                                                        |
+-----------------------------------------------------------------------------+

See also
========

  * gaussMinors (missing documentation)
  * gaussIdeal -- correlation ideal of a Bayesian network of
    joint Gaussian variables
  * gaussTrekIdeal (missing documentation)

Ways to use gaussRing :
=======================

  * "gaussRing(ZZ)"

For the programmer
==================

The object gaussRing, is a method function with options (Macaulay2Doc).

gaussTrekIdeal
**************

Ways to use gaussTrekIdeal :
============================

  * gaussTrekIdeal(Ring,Graph) (missing documentation)

For the programmer
==================

The object gaussTrekIdeal (missing documentation) is a method function (Macaulay2Doc).

globalMarkovStmts
*****************

Ways to use globalMarkovStmts :
===============================

  * globalMarkovStmts(Graph) (missing documentation)

For the programmer
==================

The object globalMarkovStmts (missing documentation) is a method function (Macaulay2Doc).

Graph
*****

Methods that use an object of class Graph :
===========================================

  * displayGraph(Graph) (missing documentation)
  * displayGraph(String,Graph) (missing documentation)
  * displayGraph(String,String,Graph) (missing documentation)
  * "gaussIdeal(Ring,Graph)" -- see gaussIdeal, --
    correlation ideal of a Bayesian network of joint Gaussian variables
  * gaussTrekIdeal(Ring,Graph) (missing documentation)
  * globalMarkovStmts(Graph) (missing documentation)
  * localMarkovStmts(Graph) (missing documentation)
  * pairMarkovStmts(Graph) (missing documentation)
  * writeDotFile(String,Graph) (missing documentation)

For the programmer
==================

The object Graph (missing documentation) is a type (Macaulay2Doc),
with ancestor classes HashTable (Macaulay2Doc), < Thing (Macaulay2Doc).

hideMap
*******

Ways to use hideMap :
=====================

  * hideMap(ZZ,Ring) (missing documentation)

For the programmer
==================

The object hideMap (missing documentation) is a method function (Macaulay2Doc).

localMarkovStmts
****************

Ways to use localMarkovStmts :
==============================

  * localMarkovStmts(Graph) (missing documentation)

For the programmer
==================

The object localMarkovStmts (missing documentation) is a method function (Macaulay2Doc).

makeGraph
*********

Ways to use makeGraph :
=======================

  * makeGraph(List) (missing documentation)

For the programmer
==================

The object makeGraph (missing documentation) is a method function (Macaulay2Doc).

marginMap
*********

Ways to use marginMap :
=======================

  * marginMap(ZZ,Ring) (missing documentation)

For the programmer
==================

The object marginMap (missing documentation) is a method function (Macaulay2Doc).

markovIdeal
***********

Ways to use markovIdeal :
=========================

  * markovIdeal(Ring,List) (missing documentation)

For the programmer
==================

The object markovIdeal (missing documentation) is a method function (Macaulay2Doc).

markovMatrices
**************

Ways to use markovMatrices :
============================

  * markovMatrices(Ring,List) (missing documentation)

For the programmer
==================

The object markovMatrices (missing documentation) is a method function (Macaulay2Doc).

markovRing -- ring of probability distributions on several discrete random variables
************************************************************************************

Synopsis
========

  * Usage:
        markovRing(d1,d2,...,dr)
  * Inputs:
      * di, an integer (Macaulay2Doc), Each d_i should be a positive
        integer
  * Outputs:
      * R, a ring (Macaulay2Doc), A polynomial ring with
        d1*d2*...*dr variables $p_{(i1,...,ir)}$, with each i_j satisfying 1 <=
        i_j <= d_j.
  * Consequences:
      * Information about this sequence of integers is placed into the ring,
        and is used by other functions in this package.  Also, at most one ring
        for each such sequence is created: the results are cached.

Description
===========

+-----------------------------------+
|i1 : R = markovRing(2,3,4,5);      |
+-----------------------------------+
|i2 : numgens R                     |
|                                   |
|o2 = 120                           |
+-----------------------------------+
|i3 : R_0, R_1, R_119               |
|                                   |
|o3 = (p       , p       , p       )|
|       1,1,1,1   1,1,1,2   2,3,4,5 |
|                                   |
|o3 : Sequence                      |
+-----------------------------------+
|i4 : coefficientRing R             |
|                                   |
|o4 = QQ                            |
|                                   |
|o4 : Ring                          |
+-----------------------------------+

Caveat
======

Currently, the user has no choice about the names of the variables. Also, the
base field is set to be QQ, without option of changing it. These will hopefully
change in a later version.

For the programmer
==================

The object markovRing, is a function closure (Macaulay2Doc).

pairMarkovStmts
***************

Ways to use pairMarkovStmts :
=============================

  * pairMarkovStmts(Graph) (missing documentation)

For the programmer
==================

The object pairMarkovStmts (missing documentation) is a method function (Macaulay2Doc).

removeRedundants
****************

For the programmer
==================

The object removeRedundants (missing documentation) is a function closure (Macaulay2Doc).

writeDotFile
************

Ways to use writeDotFile :
==========================

  * writeDotFile(String,Graph) (missing documentation)

For the programmer
==================

The object writeDotFile (missing documentation) is a method function (Macaulay2Doc).