ml-finance-python

README.md

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 # Chapter 9: Bayesian Machine Learning
 
 Bayesian statistics allows us to quantify uncertainty about future events and refine our estimates in a principled way as new information arrives. This dynamic approach adapts well to the evolving nature of financial markets. It is particularly useful when there are fewer relevant data and we require methods that systematically integrate prior knowledge or assumptions.
 
 Bayesian approaches to machine learning allow for richer insights into the uncertainty around statistical metrics, parameters estimates and predictions. The applications range from more granular risk management to dynamic updates of predictive models that incorporate changes in the market environment. The Black-Litterman approach to asset allocation (see [Chapter 5](../05_strategy_evaluation)) can be interpreted as a Bayesian model. It computes the expected return as an average of the market equilibrium and the investor’s views, weighted by each asset’s volatility, cross-asset correlations and the confidence in each forecast. More specifically, this chapter covers:
 - How Bayesian statistics apply to machine learning
 - How to use probabilistic programming with PyMC3
 - How to define and train machine learning models
 - How to run state-of-the-art sampling methods to conduct approximate inference
 - How to apply Bayesian machine learning to compute dynamic Sharpe ratios, build Bayesian classifiers, and estimate stochastic volatility
 
 
 ## How Bayesian Machine Learning Works
 
 Classical statistics is also called frequentist because it interprets probability as the relative frequency of an event over the long run, i.e. after observing a large number of trials. In the context of probabilities, an event is a combination of one or more elementary outcomes of an experiment, such as any of six equal results in rolls of two dice or an asset price dropping by 10% or more on a given day). 
 
 Bayesian statistics, in contrast, views probability as a measure of the confidence or belief in the occurrence of an event. The Bayesian perspective on probability leaves more room for subjective views and, consequently, differences in opinions than the frequentist interpretation. This difference is most striking for events that do not happen often enough to arrive at an objective measure of long-term frequency.
 
 This chapter first introduces the Bayes Theorem that crystallizes the concept of updating beliefs by combining prior assumptions with new empirical evidence and compare the resulting parameter estimates with their frequentist counterparts. We then demonstrate two approaches to Bayesian statistical inference that produce insights into the posterior distribution of the latent, i.e. unobserved parameters, such as their expected values, under different circumstances:  
 1. Conjugate priors facilitate the updating process by providing a closed-form solution, but exact, analytical methods are not always available.
 2. Approximate inference simulates the distribution that results from combining assumptions and data and uses samples from this distribution to compute statistical insights.
 
 #### References
 
 - [Bayesian Methods for Hackers](http://camdavidsonpilon.github.io/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/)
 - [Andrew Gelman's Blog](https://andrewgelman.com/)
 - [Thomas Wiecki's Blog](https://twiecki.github.io/)
 
 
 ### How to update assumptions from empirical evidence
 
 The theorem that Reverend Thomas Bayes came up with over 250 years ago uses fundamental probability theory to prescribe how probabilities or beliefs should change as relevant new information arrives as captured by John Maynard Keynes’ quote “When the facts change, I change my mind. What do you do, sir?”.
 
 - [Bayes' rule: Guide](https://arbital.com/p/bayes_rule/?l=1zq)
 - [Bayesian Updating with Continuous Priors](https://ocw.mit.edu/courses/mathematics/18-05-introduction-to-probability-and-statistics-spring-2014/readings/MIT18_05S14_Reading13a.pdf), MIT Open Courseware, 18.05 Introduction to Probability and Statistics
 
 ### Exact Inference: Maximum a Posterior Estimation
 
 Practical applications of Bayes’ rule to exactly compute posterior probabilities are quite limited because the computation of the evidence term in the denominator is quite challenging.
 
 #### How to keep inference simple: Conjugate Priors
 
 A prior distribution is conjugate with respect to the likelihood when the resulting posterior is of the same type of distribution as the prior except for different parameters. The conjugacy of prior and likelihood implies a closed-form solution for the posterior that facilitates the update process and avoids the need to use numerical methods to approximate the posterior.
 
 #### How to dynamically estimate the probabilities of asset price moves
 
 The notebook [updating_conjugate_priors](01_updating_conjugate_priors.ipynb) demonstrates how to use a conjugate prior to update price movement estimates from S&P 500 samples.
 
 ### Approximate Inference: Stochastic vs Deterministic Approaches
 
 For most models of practical relevance, it will not be possible to derive the exact posterior distribution analytically and compute expected values for the latent parameters.
 
 Although for some applications the posterior distribution over unobserved parameters
 will be of interest, most often it is primarily required to evaluate expectations, e.g. to make predictions. In such situations, we can rely on approximate inference:
 - Stochastic techniques based on Markov Chain Monte Carlo (MCMC) sampling have popularized the use of Bayesian methods across many domains. They generally have the property to converge to the exact result. In practice, sampling methods can be computationally demanding and are often limited to small-scale problems. 
     - [A Conceptual Introduction to Hamiltonian Monte Carlo](https://arxiv.org/pdf/1701.02434.pdf), Michael Betancourt, 2018
     - [The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo](https://arxiv.org/abs/1111.4246), Matthew D. Hoffman, Andrew Gelman, 2011
     - [ML, MAP, and Bayesian — The Holy Trinity of Parameter Estimation and Data Prediction](https://engineering.purdue.edu/kak/Trinity.pdf)
 
 - Deterministic methods called variational inference or variational Bayes are based on analytical approximations to the posterior distribution and can scale well to large applications. They make simplifying assumptions, e.g., that the posterior factorizes in a particular way or it has a specific parametric form such as a Gaussian. Hence, they do not generate exact results and can be used as complements to sampling methods.
     - [Variational Inference: A Review for Statisticians](https://arxiv.org/pdf/1601.00670.pdf), David Blei et al, 2018
 
 
 ## Probabilistic Programming with PyMC3
 
 Probabilistic programming provides a language to describe and fit probability distributions so that we can design, encode and automatically estimate and evaluate complex models. It aims to abstract away some of the computational and analytical complexity to allow us to focus on the conceptually more straightforward and intuitive aspects of Bayesian reasoning and inference.
 The field has become quite dynamic since new languages emerged since Uber open-sourced Pyro (based on PyTorch) and Google more recently added a probability module to TensorFlow. 
 
 - [Probabilistic Programming](http://www.probabilistic-programming.org/wiki/Home) Community Repository with links to papers and software
 - [Stan](https://mc-stan.org/)
 - [Edward](http://edwardlib.org/)
 - [TensorFlow Probability](https://github.com/tensorflow/probability)
 - [Pyro](http://pyro.ai/)
 
 ### Bayesian ML with Theano
 
 PyMC3 was released in January 2017 to add Hamiltonian MC methods to the Metropolis-Hastings sampler used in PyMC2 (released 2012). PyMC3 uses [Theano](http://www.deeplearning.net/software/theano/) as its computational backend for dynamic C compilation and automatic differentiation. Theano is a matrix-focused and GPU-enabled optimization library developed at Yoshua Bengio’s Montreal Institute for Learning Algorithms (MILA) that inspired TensorFlow. MILA recently ceased to further develop Theano due to the success of newer deep learning libraries (see chapter 16 for details). PyMC4, planned for 2019, will use TensorFlow instead, with presumably limited impact on the API.
 
 ### The PyMC3 Workflow
 
 PyMC3 aims for intuitive and readable, yet powerful syntax that reflects how statisticians describe models. The modeling process generally follows these three steps:
 1) Encode a probability model by defining: 
     1) The prior distributions that quantify knowledge and uncertainty about latent variables
     2) The likelihood function that conditions the parameters on observed data
 2) Analyze the posterior using one of the options described in the previous section:
     1) Obtain a point estimate using MAP inference
     2) Sample from the posterior using MCMC methods
     3)Approximate the posterior using variational Bayes
 3) Check your model using various diagnostic tools
 4) Generate predictions
 
 - [Documentation](https://docs.pymc.io/)
 - [Probabilistic Programming in Python using PyMC](https://arxiv.org/abs/1507.08050), Salvatier et al 2015
 - [Theano: A Python framework for fast computation of mathematical expressions](https://pdfs.semanticscholar.org/6b57/0069f14c7588e066f7138e1f21af59d62e61.pdf), Al-Rfou et al, 2016
 - [Bayesian Methods for Hackers](https://github.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers)
 - [Bad Traces, or, Don't Use Metropolis](https://colindcarroll.com/2018/01/01/bad-traces-or-dont-use-metropolis/)
 
 ### Model Definition: Bayesian Logistic Regression
 
 As discussed in [Linear Models](../07_linear_models), logistic regression estimates a linear relationship between a set of features and a binary outcome, mediated by a sigmoid function to ensure the model produces probabilities. The frequentist approach resulted in point estimates for the parameters that measure the influence of each feature on the probability that a data point belongs to the positive class, with confidence intervals based on assumptions about the parameter distribution. 
 
 Bayesian logistic regression, in contrast, estimates the posterior distribution over the parameters itself. The posterior allows for more robust estimates of what is called a Bayesian credible interval for each parameter with the benefit of more transparency about the model’s uncertainty.
 
 The notebook [bayesian_logistic_regression](02_bayesian_logistic_regression.ipynb) demonstrates the PYMC3 workflow, including
 - MAP Inference
 - Markov Chain Monte Carlo Estimate 
     - Metropolis-Hastings
     - NUTS Sampler
 - Variational Inference
 - Model Diagnostics
     - Energy Plot
     - Forest Plot
     - Posterior Predictive Credible Intervals (CI)
 - Prediction
 - MCMC Sampler Animation
     
 
 ### Practical Applications
 
 - [Tackling the Poor Assumptions of Naive Bayes Text Classifiers](http://people.csail.mit.edu/jrennie/papers/icml03-nb.pdf), Rennie, et al, MIT SAIL, 2003
 - [On Discriminative vs Generative Classifiers: A comparison of logistic regression and naive Bayes](https://ai.stanford.edu/~ang/papers/nips01-discriminativegenerative.pdf), Jordan, Ng, 2002
 - [Bayesian estimation supersedes the t test](http://www.indiana.edu/~kruschke/BEST/BEST.pdf), John K. Kruschke, Journal of Experimental Psychology, 2012
 - [Automatic Differentiation Variational Inference](https://arxiv.org/pdf/1603.00788.pdf)
 
 #### Bayesian Linear Regression for Pairs Trading
 
 The chapter [Time Series Models](../08_time_series_models) introduced pairs trading as a popular algorithmic trading strategy that relies on cointegration of two or more assets. Given such assets, we need to estimate the hedging ratio to decide on the relative magnitude of long and short positions. A basic approach uses linear regression. 
 The notebook [linear_regression](04_linear_regression.ipynb) illustrates how Bayesian linear regression tracks changes in the relationship between two assets over time.
 
 #### Bayesian Time Series Models
 
 PyMC3 includes AR(p) models that allow us to gain similar insights into the parameter uncertainty as for the previous models. The notebook [bayesian_time_series](05_bayesian_time_series.ipynb)notebook illustrates time series model for one and more lags.
 
 #### Stochastic Volatility Models
 As discussed in the chapter [Time Series Models](../08_time_series_models), asset prices have time-varying volatility. In some periods, returns are highly variable, while in others very stable. Stochastic volatility models model this with a latent volatility variable, modeled as a stochastic process. The  No-U-Turn Sampler was introduced using such a model, and the notebook [stochastic_volatility](06_stochastic_volatility.ipynb) illustrates this use case.