averaging priors vs averaging posteriors

conciliatepriors.py

from random import *
from math import *

def randomOpen():
    while True:
        r = random()
        if 0 < r:
            return r

class Grid(object):
    def __init__(self, EH=None, NEH=None, ENH=None, NENH=None):
        if EH is None:
            EH = randomOpen()
            NEH = randomOpen()
            ENH = randomOpen()
            NENH = randomOpen()
            s = EH+NEH+ENH+NENH
            EH /= s
            NEH /= s
            ENH /= s
            NENH /= s
        self.EH = EH
        self.NEH = NEH
        self.ENH = ENH
        self.NENH = NENH
        
    def average(self, a):
        EH=self.EH+a.EH
        NEH=self.NEH+a.NEH
        ENH=self.ENH+a.ENH
        NENH=self.NENH+a.NENH
        s = EH+NEH+ENH+NENH
        
        return Grid(EH=EH/s,NEH=NEH/s,ENH=ENH/s,NENH=NENH/s)
        
    def posteriorH(self):
        return self.EH/(self.EH+self.ENH)
        
maxDelta = 0
avgDelta = 0
N = 100000
        
for i in range(N):
    g1 = Grid()
    g2 = Grid()
    p1 = g1.posteriorH()
    p2 = g2.posteriorH()
    p3 = g1.average(g2).posteriorH()
    p4 = (p1+p2)/2
    delta = abs(p4-p3)
    maxDelta = max(maxDelta,delta)
    avgDelta += delta
 
avgDelta /= N 
print(maxDelta,avgDelta)