I am with you. There are countless reasons why we should learn Bayesian statisticsin particular, Bayesian statistics is emerging as a powerful framework to express and understand next-generation deep neural networks. I believe that for the things we have to learn before we can do them, we learn by doing them.

Prior to memorizing the endless terminologies, we will code the solutions and visualize the results, and using the terminologies and theories to explain the models along the way.

PyMC3 is a Python library for probabilistic programming with a very simple and intuitive syntax. ArviZa Python library that works hand-in-hand with PyMC3 and can help us interpret and visualize posterior distributions.

And we will apply Bayesian methods to a practical problem, to show an end-to-end Bayesian analysis that move from framing the question to building models to eliciting prior probabilities to implementing in Python the final posterior distribution. Bayesian models are also known as probabilistic models because they are built using probabilities. Therefore, the answers we get are distributions not point estimates.

Step 1: Establish a belief about the data, including Prior and Likelihood functions.

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Step 2, Use the data and probability, in accordance with our belief of the data, to update our model, check that our model agrees with the original data. Step 3, Update our view of the data based on our model.

### Markov Chain Monte Carlo in Python

Since I am interested in using machine learning for price optimization, I decide to apply Bayesian methods to a Spanish High Speed Rail tickets pricing data set that can be found here.

Appreciate The Gurus team for scraping the data set. Also fill the other two categorical columns with the most common values. The KDE plot of the rail ticket price shows a Gaussian-like distributionexcept for about several dozens of data points that are far away from the mean.

Since we do not know the mean or the standard deviation, we must set priors for both of them. Therefore, a reasonable model could be as follows. We will perform Gaussian inferences on the ticket price data. Choices for ticket price likelihood function:. Using PyMC3, we can write the model as follows:.

The y specifies the likelihood. This is the way in which we tell PyMC3 that we want to condition for the unknown on the knows data. We plot the gaussian model trace. This runs on a Theano graph under the hood. There are a couple of things to notice here:. We can plot a joint distributions of parameters.

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This means we probably do not have collinearity in the model. This is good. We can also have a detailed summary of the posterior distribution for each parameter. We can also see the above summary visually by generating a plot with the mean and Highest Posterior Density HPD of a distribution, and to interpret and report the results of a Bayesian inference. We can verify the convergence of the chains formally using the Gelman Rubin test.

Values close to 1.The holy trinity when it comes to being Bayesian. Of course then there is the mad men old professors who are becoming irrelevant who actually do their own Gibbs sampling. You specify the generative model for the data. You feed in the data as observations and then it samples from the posterior of the data for you. Stan was the first probabilistic programming language that I used. The documentation is absolutely amazing.

The examples are quite extensive. PyMC3 on the other hand was made with Python user specifically in mind.

You can see below a code example. The pm. As far as documentation goes, not quite extensive as Stan in my opinion but the examples are really good.

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The reason PyMC3 is my go to Bayesian tool is for one reason and one reason alone, the pm. Variational inference is one way of doing approximate Bayesian inference. Both Stan and PyMC3 has this. Getting a just a bit into the maths what Variational inference does is maximise a lower bound to the log probability of data log p y.

Last I checked with PyMC3 it can only handle cases when all hidden variables are global I might be wrong here. The second term can be approximated with. This is the essence of what has been written in this paper by Matthew Hoffman. This is where GPU acceleration would really come into play. However, I must say that Edward is showing the most promise when it comes to the future of Bayesian learning due to alot of work done in Bayesian Deep Learning. So in conclusion, PyMC3 for me is the clear winner these days.

I would love to see Edward or PyMC3 moving to a Keras or Torch backend just because it means we can model and debug better. Happy modelling! Oh and clap! Sign in.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The dark mode beta is finally here.

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When I check the trace in this case samples from the posterior probabilityI notice that 2 chains are created:. I read the tutorial on PyMC3 and looked through the API but it is unclear to me what a chain represents in this case I asked for samples from the posterior but I got 2 chains, each one with samples from the posterior.

A chain is a single run of MCMC. So if you have six 2-d parameters in your model and ask for samples, you will get six 2x arrays for each chain.

When running MCMC, it is a best practice to use multiple chains, as they can help diagnose problems. For example, the Gelman-Rubin diagnostic requires multiple chains, and runs automatically using joblibwhich tries to use multiple cores if possible if you use more than 1 chain in PyMC3. As a concrete example of when you might want multiple chains, consider sampling from a multimodal distribution.

Even the NUTS sampler may not visit both modes in a single chain, but you could diagnose this using multiple chains. Note that PyMC3 usually combines the chains when you work with them e. This does lead to some confusing behavior in that asking for samples actually gets you on most systems, where you get 4 chains by default.

In my research lab, in podcasts, in articles, every time I heard the phrase I would nod and think that sounds pretty cool with only a vague idea of what anyone was talking about. Exasperated, I turned to the best method to learn any new skill: apply it to a problem.

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Using some of my sleep data I had been meaning to explore and a hands-on application-based book Bayesian Methods for Hackersavailable free onlineI finally learned Markov Chain Monte Carlo through a real-world project. As usual, it was much easier and more enjoyable to understand the technical concepts when I applied them to a problem rather than reading them as abstract ideas on a page. This article walks through the introductory implementation of Markov Chain Monte Carlo in Python that finally taught me this powerful modeling and analysis tool.

The full code and data for this project is on GitHub. I encourage anyone to take a look and use it on their own data. This article focuses on applications and results, so there are a lot of topics covered at a high level, but I have tried to provide links for those wanting to learn more!

My Garmin Vivosmart watch tracks when I fall asleep and wake up based on heart rate and motion. The objective of this project was to use the sleep data to create a model that specifies the posterior probability of sleep as a function of time.

As time is a continuous variable, specifying the entire posterior distribution is intractable, and we turn to methods to approximate a distribution, such as Markov Chain Monte Carlo MCMC. Before we can start with MCMC, we need to determine an appropriate function for modeling the posterior probability distribution of sleep.

One simple way to do this is to visually inspect the data. The observations for when I fall asleep as a function of time are shown below. Every data point is represented as a dot, with the intensity of the dot showing the number of observations at the specific time.

My watch records only the minute at which I fall asleep, so to expand the data, I added points to every minute on both sides of the precise time. If my watch says I fell asleep at PM, then every minute before is represented as a 0 awake and every minute after gets a 1 asleep.

This expanded the roughly 60 nights of observations into data points. We can see that I tend to fall asleep a little after PM but we want to create a model that captures the transition from awake to asleep in terms of a probability.

We could use a simple step function for our model that changes from awake 0 to asleep 1 at one precise time, but this would not represent the uncertainty in the data. I do not go to sleep at the same time every night, and we need a function to that models the transition as a gradual process to show the variability. The best choice given the data is a logistic function which is smoothly transitions between the bounds of 0 and 1.The computational issue is the difficulty of evaluating the integral in the denominator.

There are many ways to address this difficulty, inlcuding:. If we use a beta distribution as the prior, then the posterior distribution has a closed form solution. This is shown in the example below. Some general points:. One advantage of this is that the prior does not have to be conjugate although the example below uses the same beta prior for ease of comaprsionand so we are not restricted in our choice of an approproirate prior distribution.

For example, the prior can be a mixture distribution or estimated empirically from data. All ocde will be built from the ground up to ilustrate what is involved in fitting an MCMC model, but only toy examples will be shown since the goal is conceptual understanding. More realiztic computational examples will be shown in the next lecture using the pymc and pystan packages. In Bayesian statistics, we want to estiamte the posterior distribution, but this is often intractable due to the high-dimensional integral in the denominator marginal likelihood.

A few other ideas we have encountered that are also relevant here are Monte Carlo integration with inddependent samples and the use of proposal distributions e. With vanilla Monte Carlo integration, we need the samples to be independent draws from the posterior distribution, which is a problem if we do not actually know what the posterior distribution is because we cannot integrte the marginal likelihood.

With MCMC, we draw samples from a simple proposal distribution so that each draw depends only on the state of the previous draw i. Under certain condiitons, the Markov chain will have a unique stationary distribution. In addition, not all samples are used - instead we set up acceptance criteria for each draw based on comparing successive states with respect to a target distribution that enusre that the stationary distribution is the posterior distribution of interest.

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After some time, the Markov chain of accepted draws will converge to the staionary distribution, and we can use those samples as correlated draws from the posterior distribution, and find functions of the posterior distribution in the same way as for vanilla Monte Carlo integration. There are several flavors of MCMC, but the simplest to understand is the Metropolis-Hastings random walk algorithm, and we will start there.

To carry out the Metropolis-Hastings algorithm, we need to draw random samples from the folllowing distributions. If the proposal distribution is not symmetrical, we need to weight the accceptanc probablity to maintain detailed balance reversibilty of the stationary distribution, and insetad calculate. Here are initial concepts to help your intuition about why this is so:. Trace plots are often used to informally assess for stochastic convergence. Rigorous demonstration of convergence is an unsolved problem, but simple ideas such as running mutliple chains and checking that they are converging to similar distribtions are often employed in practice.

There are two main ideas - first that the samples generated by MCMC constitute a Markov chain, and that this Markov chain has a unique stationary distribution that is always reached if we geenrate a very large number of samples. The seocnd idea is to show that this stationary distribution is exactly the posterior distribution that we are looking for. We will only give the intuition here as a refreseher.

If it is posssible to go from any state to any other state, then the matrix is irreducible.GitHub is home to over 40 million developers working together. Join them to grow your own development teams, manage permissions, and collaborate on projects.

Python 5k 1. PyMC3 educational resources. A high-level probabilistic programming interface for TensorFlow Probability. Experimental code for porting PyMC to alternative backends. Uncertainty quantification book chapter. Skip to content. Sign up. Pinned repositories. Type: All Select type. All Sources Forks Archived Mirrors.

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You signed in with another tab or window. Reload to refresh your session. You signed out in another tab or window.BayesPy provides tools for Bayesian inference with Python. The user constructs a model as a Bayesian network, observes data and runs posterior inference. The goal is to provide a tool which is efficient, flexible and extendable enough for expert use but also accessible for more casual users.

Currently, only variational Bayesian inference for conjugate-exponential family variational message passing has been implemented. Future work includes variational approximations for other types of distributions and possibly other approximate inference methods such as expectation propagation, Laplace approximations, Markov chain Monte Carlo MCMC and other methods.

Contributions are welcome.

## Tutorial Notebooks

It is implemented in Java and released under revised BSD license. The framework allows easy learning of a wide variety of models using variational Bayesian learning. NET framework for machine learning. It provides message-passing algorithms and statistical routines for performing Bayesian inference. It is partly closed source and licensed for non-commercial use only. It is released under the Academic Free License.

It is released under the Apache License. There was no information about the license. BayesPy latest. Version 0. This can be used for simple regularization.

## Hands On Bayesian Statistics with Python, PyMC3 & ArviZ

Skip some failing image comparison unit tests. Image comparison tests will be deprecated at some point.

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Allow VB iteration without maximum number of iteration steps Add ellipse patch creation from covariance or precision The bug caused basically all models with Take node to be incorrect. Fix ndim handling in GaussianGamma and Wishart Support lists and other array-convertible formats in several nodes.

Added Gaussian arrays not just scalars or vectors. Added Gaussian Markov chains with time-varying or swithing dynamics. Added discrete Markov chains enabling hidden Markov models. Added joint Gaussian-Wishart and Gaussian-gamma nodes. Added deterministic gating node. Added deterministic general sum-product node. Added new plotting functions: pdf, Hinton diagram.