advanced-algorithmic-trading

Chapter 4
Markov Chain Monte Carlo
In previous chapters we introduced Bayesian Statistics and considered how to infer a binomial
proportion using the concept of conjugate priors. We briefly mentioned that not all models can
make use of conjugate priors and thus calculation of the posterior distribution would need to be
approximated numerically.
In this chapter we introduce the main family of algorithms, known collectively as Markov
Chain Monte Carlo (MCMC), that allow us to approximate the posterior distribution as calculated
by Bayes’ Theorem. In particular, we consider the Metropolis Algorithm, which is easily
stated and relatively straightforward to understand. It serves as a useful starting point when
learning about MCMC before delving into more sophisticated algorithms such as Metropolis-
Hastings, Gibbs Samplers, Hamiltonian Monte Carlo and the No-U-Turn Sampler (NUTS).
Once we have described how MCMC works, we will carry it out using the open-source Pythonbased
PyMC3 library. The library takes care of the underlying implementation details allowing
us to concentrate specifically on modelling.
4.1 Bayesian Inference Goals
As quants our goal in studying Bayesian Statistics is to ultimately produce quantitative trading
strategies, using models derived from Bayesian methods. In order to reach that goal we need to
consider a reasonable amount of Bayesian Statistics theory. So far we have:
• Introduced the philosophy of Bayesian Statistics, making use of Bayes’ Theorem to update
our prior beliefs on probabilities of outcomes based on new data
• Used conjugate priors as a means of simplifying the computation of the posterior distribution
in the case of inference on a binomial proportion
In this chapter we are going to discuss MCMC as a means of computing the posterior distribution
when conjugate priors are not applicable.
Subsequent to a discussion on the Metropolis algorithm using PyMC3, we will consider more
sophisticated samplers and then apply them to more complex models. Ultimately we will arrive
at the point where our models are useful enough to provide insight into asset returns prediction.
At that stage we will be able to begin building a trading model from our Bayesian analysis.
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