# Bayesian Beta Distributed Coin Inference

## 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_)]

for h in hs:
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.display()


Distribution for nu

Distribution for mu
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 10.688s

In [8]:

for h in hs:
bn.cpt(h).fillFromDistribution(scipy.stats.bernoulli,p="bias")


## Evidence without evidence

In [9]:

gnb.showInference(bn)

In [11]:

print(bn)

BN{nodes: 19, arcs: 18, domainSize: 10^11.7196, dim: 7963598, mem: 61Mo 89Ko 128o}


## Evidence with the sequence

In [12]:

coin_evidence={f"H{i}":serie[i] for i in range(len(serie))}

gnb.showInference(bn,evs=coin_evidence)

In [13]:

ie=gum.LazyPropagation(bn)
ie.setEvidence(coin_evidence)
ie.makeInference()

In [14]:

from scipy.ndimage import center_of_mass

idx= ie.posterior('bias').argmax()[0][0]['bias']
map_bias = bn['bias'].label(idx)

com = center_of_mass(ie.posterior('nu').toarray())[0]

idx = ie.posterior('mu').argmax()[0][0]['mu']
map_mu = bn['mu'].label(idx)

print(f"MAP for mu : {map_mu}")
print(f"center of mass for nu : {com}")
print(f"MAP for bias : {map_bias}")

MAP for mu : 0.4473
center of mass for nu : 26.67889867211198
MAP for bias : 0.4373


## Smaller serie

In [15]:

# With a smaller serie
serie=[1,0,0,0,0,0,1,]

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(len(serie))]

for h in hs:

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)

bn.cpt("bias").fillFromDistribution(scipy.stats.beta,a="mu*nu",b="(1-mu)*nu")

for h in hs:
bn.cpt(h).fillFromDistribution(scipy.stats.bernoulli,p="bias")

coin_evidence={f"H{i}":serie[i] for i in range(len(serie))}

gnb.showInference(bn,evs=coin_evidence)

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