Bayesian Beta Distributed Coin Inference¶
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build a fully bayesian beta distributed coin inference¶
This notebook is based on examples from Benjamin Datko (https://gist.github.com/bdatko).
The basic idea of this notebook is to show you could assess the probability for a coin, knowing a sequence of heads/tails.
In [1]:
import itertools
import time
from pylab import *
import matplotlib.pyplot as plt
import numpy as np
import scipy.stats
import pyAgrum as gum
import pyAgrum.lib.notebook as gnb
In [2]:
gum.config["notebook","default_graph_size"]="12!"
gum.config["notebook","default_graph_inference_size"]="12!"
Fill Beta parameters with a re-parameterization¶
We propose a model where : mu and nu are the parameters of a beta which gives the distribution for the coins.
- below are some useful definitions $$ \alpha = \mu \nu$$ $$ \beta = (1 - \mu) \nu $$
$$ \mu = \frac{\alpha}{\alpha + \beta} $$
- like in Wikipedia article, we will have a uniform prior on μ and an expoential prior on ν
In [3]:
# the sequence of COINS
serie=[1,0,0,0,1,0,1,1,0,1,0,0,1,0,0,1]
In [4]:
NB_ = 200
vmin, vmax = 0.001, 0.999
pmin_mu, pmax_mu = 0.001, 0.999
pmin_nu, pmax_nu = 1,50
size_ = 16
In [5]:
bn=gum.BayesNet("SEQUENCE OF COINS MODEL")
mu = bn.add(gum.NumericalDiscreteVariable("mu","mean of the Beta distribution",pmin_mu,pmax_mu,NB_))
nu = bn.add(gum.NumericalDiscreteVariable("nu","'sample size' of the Beta where nu = a + b > 0",pmin_nu,pmax_nu,NB_))
bias=bn.add(gum.NumericalDiscreteVariable("bias","The bias of the coin",vmin,vmax,NB_))
hs=[bn.add(gum.RangeVariable(f"H{i}","The hallucinations of coin flips",0,1)) for i in range(size_)]
bn.addArc(mu,bias)
bn.addArc(nu,bias)
for h in hs:
bn.addArc(bias,h)
print(bn)
bn
BN{nodes: 19, arcs: 18, domainSize: 10^11.7196, dim: 7963598, mem: 61Mo 89Ko 128o}
Out[5]:
In [6]:
bn.cpt(nu).fillFromDistribution(scipy.stats.expon,loc=2,scale=5)
bn.cpt(mu).fillFromDistribution(scipy.stats.uniform,loc=pmin_mu,scale=pmax_mu-pmin_mu)
gnb.flow.clear()
gnb.flow.add(gnb.getProba(bn.cpt(nu)),caption="Distribution for nu")
gnb.flow.add(gnb.getProba(bn.cpt(mu)),caption="Distribution for mu")
gnb.flow.display()
In [7]:
# https://scicomp.stackexchange.com/a/10800
t_start = time.time()
bn.cpt("bias").fillFromDistribution(scipy.stats.beta,a="mu*nu",b="(1-mu)*nu")
end_time = time.time() - t_start
print(f"Filling {NB_}^3 parameters in {end_time:5.3f}s")
Filling 200^3 parameters in 8.676s
In [8]:
for h in hs:
bn.cpt(h).fillFromDistribution(scipy.stats.bernoulli,p="bias")
Evidence without evidence¶
In [9]:
gnb.showInference(bn)
In [10]:
print(bn)
BN{nodes: 19, arcs: 18, domainSize: 10^11.7196, dim: 7963598, mem: 61Mo 89Ko 128o}
Evidence with the sequence¶
In [11]:
coin_evidence={f"H{i}":serie[i] for i in range(len(serie))}
gnb.showInference(bn,evs=coin_evidence)