From Werner KRAUTH
#
# Simulated tempering for the V-shaped stationary distribution --- Metropolis
# algorithm
#
import random
import pylab
import numpy as np
for n in [10, 20, 40, 80, 160, 320]:
ReplicaChange = 0.1
const = 4.0 / n ** 2
PiStat = {}
Table = []
for x in range(1, n + 1):
Table.append((x, 0))
Table.append((x, 1))
#
# factor of 1/2 because the total must be normalized
#
PiStat[(x, 0)] = 1.0 / float(n) / 2.0
PiStat[(x, 1)] = const * abs( (n + 1) / 2 - x) / 2.0
PiStat[(0, 0)] = 0.0
PiStat[(0, 1)] = 0.0
PiStat[(n + 1, 0)] = 0.0
PiStat[(n + 1, 1)] = 0.0
PTrans = np.eye(2 * n)
Pi = np.zeros([2 * n])
for x in range(1, n + 1):
for Rep in [0, 1]:
i = Table.index((x, Rep))
Pi[i] = PiStat[(x, Rep)]
for Dir in [-1, 1]:
if PiStat[(x + Dir, Rep)] > 0.0:
j = Table.index((x + Dir, Rep))
PTrans[i, j] = min(1.0, PiStat[(x + Dir, Rep)] / PiStat[(x, Rep)]) / 2.0
PTrans[i, i] -= PTrans[i, j]
PReplica = np.zeros((2 * n,2 * n))
for x in range(1, n + 1):
i = Table.index((x,0))
j = Table.index((x,1))
PReplica[i, j] = ReplicaChange * min(1.0, PiStat[(x, 1)] / PiStat[(x, 0)])
PReplica[i, i] = 1.0 - PReplica[i, j]
PReplica[j, i] = ReplicaChange * min(1.0, PiStat[(x, 0)] / PiStat[(x, 1)])
PReplica[j, j] = 1.0 - PReplica[j, i]
P = PTrans @ PReplica
Pit = np.zeros([2 * n])
Pit[0] = 1.0
xvalues = []
yvalues = []
iter = 0
while True:
iter += 1
Pit = Pit @ P
TVD = sum(np.absolute(Pi - Pit) / 2.0)
xvalues.append(iter / float(n ** 2))
yvalues.append(TVD)
if TVD < 0.1: break
pylab.plot(xvalues,yvalues, label='$n =$ '+str(n))
pylab.legend(loc='upper right')
pylab.xlabel("$t/ n^2$ (rescaled time) ")
pylab.ylabel("TVD")
pylab.title("TVD rev tempering on the path graph of $n$ sites")
pylab.show()