Bayesian Methods for Hackers: Probabilistic Programming and Bayesian Inference

Probabilistic Programming & Bayesian Methods for Hackers

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Those two parts combined constitute the ARMA 4,2 process. The LSTM will identify the structure in the time series while the Bayesian model will provided the probabilist estimates. In a first step, we will train a LSTM with a linear last layer which will mimic the Bayesian linear regression. We will describe the full model in the third section. The next three sections will be about.

Since the process is ARMA 4,2 , we have only a short term dependency, no seasonality nor cycles and no trend. If we had those two components, further preprocessing would have been needed, but we want to stay in the happy path for this example. The LSTM was trained with sequences of seven time steps with the mean squared error loss.

We used early stopping to prevent overfitting. We can see in the next figures that our predictions on the training set are close to the true values and that the errors can be well fitted using the Normal distribution.

Bayesian Methods for Hackers: Probabilistic Programming and Bayesian Inference

We now have an accurate predictor for our time series that gives only point-wise predictions. In the next section, we will include it in a Bayesian model to obtain probabilistic predictions. First of all, we will look at a graphical representation of our model. White circles are stochastic nodes, shaded circle are observations and square node are the results of deterministic transformations.

The full arrows points to the parameters of the stochastic node and the dashed line are deterministic transformations. We will start at y which is the value that we want to predict. The main difference between the last layer of the LSTM model and the Bayesian model is that the weights and bias will be represented by a Normal distribution instead of being point estimates i.

At this point, z can be considered like a deterministic transformation done by the LSTM of the observed data. Since the uncertainty of the weights will not depend on the specific data values themselves, we only characterise the model uncertainty. The ADVI method has the advantage of being scalable, but only gives an approximate posterior.

The MCMC method is slower but converges to the exact posterior. In a further post, we will examine those in more details. We have obtained distributions for the parameters which were point estimates in the previous step. We are now ready to make probabilistic predictions.

The forecasts are made using the Posterior Predictive Checks i. We can notice that most predictions are close to the true value and that most predictions fall in the confidence interval. We trained at LSTM on a time series to obtain accurate prediction. We could have stopped there, but we wanted a probabilistic prediction. Since the Bayesian model parameters are represented by distributions, we could characterise the model uncertainty. The Bayesian model is then used to make probabilistic prediction using the posterior predictive checks. A lot of ideas are put together in this post.

As stated earlier, this is part of a series of posts on probabilistic modeling so we will tackle some of the pieces individually in further posts. References and suggested readings:. Gelman, J.

Quick Reference

Carlin, H. Stern, D. Dunson, A.

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It can be downloaded here. New to Python or Jupyter, and help with the namespaces? Check out this answer. These are not only designed for the book, but they offer many improvements over the default settings of matplotlib and the Jupyter notebook. The in notebook style has not been finalized yet. This book has an unusual development design. The content is open-sourced, meaning anyone can be an author. Authors submit content or revisions using the GitHub interface. I like it!

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We would like to thank the Python community for building an amazing architecture. We would like to thank the statistics community for building an amazing architecture. Similarly, the book is only possible because of the PyMC library. One final thanks.

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Download ZIP. Sign in Sign up. Launching GitHub Desktop Go back. Launching Xcode Launching Visual Studio Latest commit c2a8 Jul 12, Nov 1, Prologue Fix a typo in the prologue. Bayesian Methods for Hackers Using Python and PyMC The Bayesian method is the natural approach to inference, yet it is hidden from readers behind chapters of slow, mathematical analysis.

Printed Version by Addison-Wesley. You signed in with another tab or window. Reload to refresh your session. You signed out in another tab or window. Fix typo. May 5, Apr 23, Upgrades to exponential landscape, Jul 10, Final modifications to Format. Nov 29, Fixed log-loss latex formula in Chapter 5. Jan 27, Update Chapter 7 notebook formats to version 4. Nov 4, Fix typos in code examples. Nov 1, Fix a typo in the prologue. Nov 7, Not adding or removing semicolons. Mar 8, May 15, Do not track IPython notebook checkpoints. Jun 30, Jul 8,

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