TemperingConductance.py

From Werner KRAUTH

This is the algorithm TemperingConductance.py, which computes the conductance of the Hildebrand problem.It was presented in the 8th lecture of the 2022 course "Advanced topics in MCMC"

# Computing the conductance of the  Metropolis algorithm for the V-shaped
# stationary distribution. This, for the moment, is through complete enumeration
# of the power set of the set of all samples. A rigorous argument can be made
# also.
#
import random
import numpy as np
import scipy.linalg as la
from itertools import chain, combinations

def powerset(iterable): #powerset, but without the empty set.
    s = list(iterable)
    return chain.from_iterable(combinations(s, r) for r in range(1, len(s) + 1))


def OutsideFlow(S):
    Flow = 0.0
    Weight = 0.0
    for i in S:
        Weight += PiStat[Table[i]]
        for j in range(2 * n):
            if j not in S:
                Flow += PiStat[Table[i]] * P[i, j]
    return Weight, Flow / Weight

n =  8
for ReplicaChange in [k/100 for k in range(50 + 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
    Position = (1, 0)
    PTrans   = np.eye(2 * n)
    for x in range(1, n + 1):
        for Rep in [0, 1]:
            i = Table.index((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
#
# compute the conductance.
#
    x = powerset([i for i in range(2 * n)])
    MinFlow = 100.00
    for i in x:
        S = set(list(i))
        Weight, Flow =  OutsideFlow(S)
        if Weight <= 0.5:
            if Flow < MinFlow:
                MinFlow = Flow
                MinS = S
    print(MinS, MinFlow, '2 / 2n + 1 / n^2 = ', 1.0 / (2.0 * n) + 1.0 / (n ** 2))