advanced-algorithmic-trading

40
4.5.2 Inferring a Binomial Proportion with PyMC3
We are now going to carry out the same analysis using the numerical Markov Chain Monte Carlo
method instead.
Firstly, we need to install PyMC3:
pip install pymc3
Once installed, the next task is to import the necessary libraries, which include Matplotlib,
Numpy, Scipy and PyMC3 itself. We also set the graphical style of the Matplotlib output to be
similar to the ggplot2 graphing library from the R statistical language:
import matplotlib.pyplot as plt
import numpy as np
import pymc3
import scipy.stats as stats
plt.style.use("ggplot")
The next step is to set the prior parameters, as well as the number of coin flip trials carried
out and heads returned. We also specify, for completeness, the parameters of the analyticallycalculated
posterior beta distribution, which we will use for comparison with our MCMC approach.
In addition we specify that we want to carry out 100,000 iterations of the Metropolis
algorithm:
# Parameter values for prior and analytic posterior
n = 50
z = 10
alpha = 12
beta = 12
alpha_post = 22
beta_post = 52
# How many iterations of the Metropolis
# algorithm to carry out for MCMC
iterations = 100000
Now we actually define our beta distribution prior and Bernoulli likelihood model. PyMC3
has a very clean API for carrying this out. It uses a Python with context to assign all of
the parameters, step sizes and starting values to a pymc3.Model instance (which I have called
basic_model, as per the PyMC3 tutorial).
Firstly, we specify the theta parameter as a beta distribution, taking the prior alpha and
beta values as parameters. Remember that our particular values of α = 12 and β = 12 imply a
prior mean µ = 0.5 and a prior s.d. σ = 0.1.
We then define the Bernoulli likelihood function, specifying the fairness parameter p=theta,
the number of trials n=n and the observed heads observed=z, all taken from the parameters
specified above.
At this stage we can find an optimal starting value for the Metropolis algorithm using the
PyMC3 Maximum A Posteriori (MAP) optimisation, the details of which we will omit here.