# Dependent sampling from the uniform distribution on a polytope.

### Description

Let p=(p1,...,pn) be a probability distribution which belongs to a lower dimensional polytope of the n-dimensional simplex. The polytope is defined by a collection of linear equality and inequality constraints. A dependent sequence of the p's are generated by a Markov chain using the Metropolis-Hastings algorithm whose stationary distribution is the uniform distribution over the polytope. This is done by generating k blocks of size step where the last member of each is returned.

### Usage

1 | ```
constrppprob(A1,A2,A3,b1,b2,b3,initsol,step,k)
``` |

### Arguments

`A1` |
The matrix for the equality constraints.This must always contain the constraint that the sum of the pi's is one. |

`A2` |
The matrix for the <= inequality constraints. This must always contain the constraints -pi <= 0, i.e. that the pi's must be nonnegative. |

`A3` |
The matrix for the >= inequality constraints. If there are no such constraints A3 must be set equal to NULL. |

`b1` |
The rhs vector for A1, each component must be nonnegative. |

`b2` |
The rhs vector for A2, each component must be nonnegative. |

`b3` |
The rhs vector for A3, each component must be nonnegative. If A3 is NULL then b3 must be NULL. |

`initsol` |
A vector which lies in the interior of the polytope. |

`step` |
The number of p's generated in a block before the last member of a block is returned. |

`k` |
The total number of blocks generated and hence the number of p's returned. |

### Value

The returned value is a k by n matrix of probability vectors.

### Examples

1 2 3 4 5 6 7 8 |