Homework 18, Due Wednesday Nov 25, 2009.
(but you may hand this assignment IN CLASS 
on Monday November 30.)
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Devise a Markov Chain Monte Carlo simulation
to generate a sample of 1000 (approximately) 
independent random points uniformly distributed 
in the region A of R^7 (7-dimensional Euclidean 
space defined as the intersection of the 15 
linear inequalities (for x =c(x1,x2,x3,x4,x5,x6,x7):

carr %*% x > 
    c(-3,-2,0,-1,-10,3,-8,-5,-2,-4,-1,-4,0,2,-5,-2)

where   carr   is given by

> carr
       [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]
 [1,] -0.04 -1.18 -1.28  1.98  0.04 -1.48 -0.34
 [2,]  1.54 -0.86  0.32  1.68  0.06 -1.24 -1.84
 [3,]  1.96 -0.64 -0.02 -0.34  1.14 -1.02  0.40
 [4,] -0.34 -0.42 -1.24  1.30  1.94 -1.50 -0.04
 [5,] -1.48 -1.26 -0.22 -1.92 -1.18 -1.66  0.54
 [6,] -0.68  1.24  0.82 -1.12  0.08  1.74  1.16
 [7,] -1.96 -1.64 -1.96 -0.86 -1.98 -0.06  2.00
 [8,] -1.98 -0.04  0.66  1.20 -0.96 -0.96 -1.84
 [9,]  1.76  0.16  1.04 -1.32 -1.48 -1.04  0.32
[10,]  0.12 -0.46 -0.06 -0.68 -1.48 -0.56  0.14
[11,] -1.52 -1.60  0.88  1.58 -0.04 -0.62  1.00
[12,] -0.66 -1.78  0.68  0.22 -1.70  1.66  0.28
[13,] -1.00  0.30  1.04  1.26 -0.52 -0.08  0.98
[14,]  0.78  1.14 -0.34 -0.78  0.84 -0.56  1.36
[15,] -1.16 -1.72  0.48  1.04 -0.44  0.82 -1.94
[16,]  0.13  0.44 -0.64 -0.15 -0.09  0.27 -0.52


NOTE that the point rep(1,7) satisfies 
all of these inequalities, so that the 
intersection is the (open, convex) interior of 
a polyhedron in R^7.

HINT: one way to define a "proposal" transition 
density q(y|x) is to find the density of the 
following two successive steps:
  (1) choose a direction v in R^7 at random (i.e., a 
unit vector uniformly distributed on the surface of 
the unit sphere) , and 
  (2) make the successor state to x uniformly 
distributed on the segment  (x+t*v, tmin < t < tmax)
where tmax is the smallest positive number such that
x+tmax*v  falls of=n the boundary of the region A, 
and tmin is the largest negative number such that
x+tmin*v  falls of=n the boundary of the region A.

(As we will show in class: this transition density 
defines a reversible chain on A with respect to the 
uniform density: so Metropolis Hastings with this 
proposal density always accepts the newly simulated 
point !) 

After coding a Metropolis-Hastings algorithm:
perform several checks to make sure that your 
coded proposal steps ALWAYS take you from one 
point in the interior of A to another, and that 
the resulting Markov chain (after a long burn-in 
period which you specify) has converged to a 
stationary and "mixing" Markov sequence.