├── .gitignore ├── LICENSE ├── README.md ├── data ├── cty.dat ├── melanoma_data.py ├── nashville_precip.txt ├── radon.csv ├── srrs2.dat └── vlbw.csv ├── environment.yml └── notebooks ├── 0. Basic Bayesian Inference.ipynb ├── 1. Introduction to PyMC3.ipynb ├── 2. Markov Chain Monte Carlo.ipynb ├── 3. Model Building with PyMC3.ipynb ├── 4. Case Studies.ipynb ├── 5. Model Checking.ipynb ├── 6. Hierarchical Models.ipynb ├── 7. Approximation Methods.ipynb └── images ├── 123.png ├── bayes.png ├── bayes_formula.png ├── binary_doubling.png ├── f.png ├── fisher.png ├── hmc.png ├── ivh.gif ├── nuts.png ├── prob_model.png ├── radon_entry.jpg └── test_stats.png /.gitignore: -------------------------------------------------------------------------------- 1 | # Byte-compiled / optimized / DLL files 2 | __pycache__/ 3 | *.py[cod] 4 | *$py.class 5 | 6 | # C extensions 7 | *.so 8 | 9 | # Distribution / packaging 10 | .Python 11 | env/ 12 | build/ 13 | develop-eggs/ 14 | dist/ 15 | downloads/ 16 | eggs/ 17 | .eggs/ 18 | lib/ 19 | lib64/ 20 | parts/ 21 | sdist/ 22 | var/ 23 | *.egg-info/ 24 | .installed.cfg 25 | *.egg 26 | 27 | # PyInstaller 28 | # Usually these files are written by a python script from a template 29 | # before PyInstaller builds the exe, so as to inject date/other infos into it. 30 | *.manifest 31 | *.spec 32 | 33 | # Installer logs 34 | pip-log.txt 35 | pip-delete-this-directory.txt 36 | 37 | # Unit test / coverage reports 38 | htmlcov/ 39 | .tox/ 40 | .coverage 41 | .coverage.* 42 | .cache 43 | nosetests.xml 44 | coverage.xml 45 | *,cover 46 | .hypothesis/ 47 | 48 | # Translations 49 | *.mo 50 | *.pot 51 | 52 | # Django stuff: 53 | *.log 54 | 55 | # Sphinx documentation 56 | docs/_build/ 57 | 58 | # PyBuilder 59 | target/ 60 | 61 | #Ipython Notebook 62 | .ipynb_checkpoints 63 | *sqlite -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | CC0 1.0 Universal 2 | 3 | Statement of Purpose 4 | 5 | The laws of most jurisdictions throughout the world automatically confer 6 | exclusive Copyright and Related Rights (defined below) upon the creator and 7 | subsequent owner(s) (each and all, an "owner") of an original work of 8 | authorship and/or a database (each, a "Work"). 9 | 10 | Certain owners wish to permanently relinquish those rights to a Work for the 11 | purpose of contributing to a commons of creative, cultural and scientific 12 | works ("Commons") that the public can reliably and without fear of later 13 | claims of infringement build upon, modify, incorporate in other works, reuse 14 | and redistribute as freely as possible in any form whatsoever and for any 15 | purposes, including without limitation commercial purposes. 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Affirmer understands and acknowledges that Creative Commons is not a 112 | party to this document and has no duty or obligation with respect to this 113 | CC0 or use of the Work. 114 | 115 | For more information, please see 116 | 117 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Tutorial: Probabilistic Programming with Python 2 | 3 | **EU Summer School, June 26, 2018** 4 | 5 | [![Binder](http://mybinder.org/badge.svg)](http://mybinder.org/repo/fonnesbeck/PyMC3_EUSS) 6 | 7 | 8 | This intermediate-level course will provide students with hands-on experience applying practical Bayesian statistical modeling methods on real data. PyMC3 is a high-level Python library for building statistical models using probabilistic programming, and fitting them using modern Bayesian computational methods. I will provide an introduction to Bayesian inference and prediction, followed by a tutorial on probabilistic programming with PyMC3, including the use of Markov chain Monte Carlo (MCMC) and Variational Inference (VI), using real-world datasets. The last part of the course will focus on modeling strategies and how to avoid various pitfalls when applying Bayesian statistics to your own work. The course will assume familiarity with Python and with basic statistics (e.g. probability), but does not require previous experience with Bayesian methods or probabilistic programming. 9 | 10 | ## Syllabus 11 | 12 | ### Basic Bayesian Inference 13 | 14 | * Bayesian vs. frequentist world views 15 | * Bayesian inference in three steps 16 | * Probability distributions 17 | * Parameter estimation model 18 | 19 | ### Introduction to PyMC3 20 | 21 | * Variable types 22 | * Probability models 23 | * Simple case studies 24 | 25 | ### Markov Chain Monte Carlo 26 | 27 | * Metropolis sampling 28 | * Gradient-based sampling methods 29 | 30 | ### Model Building with PyMC3 31 | 32 | * Specifying priors and likelihoods 33 | * Deterministic variables 34 | * Factor potentials 35 | * Custom variables 36 | * Step methods 37 | * Generalized linear models 38 | * Missing Data 39 | 40 | ### Case Studies 41 | 42 | * Comparing two groups with continuous outcomes 43 | * Comparing two groups with binary outcomes 44 | 45 | ### Model Checking and Output Processing 46 | 47 | * Storage backends 48 | * Convergence diagnostics 49 | * Goodness of fit 50 | * Plotting and summarization 51 | 52 | ### Hierarchical Models 53 | 54 | * Partial pooling 55 | * Non-centered parameterization 56 | * Contextual effects 57 | 58 | ### Approximation Methods 59 | 60 | * MAP 61 | * Variational inference 62 | * ADVI 63 | 64 | 65 | ## Software Installation 66 | 67 | Running PyMC3 requires a working Python3 interpreter, preferably Python 3.6. A complete Python installation for Mac OSX, Linux and Windows can most easily be obtained by downloading and installing the free [`Anaconda Python Distribution`](https://www.continuum.io/downloads) by ContinuumIO. **If possible, please have your Python environment ste up prior to the course.** 68 | 69 | `PyMC3` can be installed using `conda`, a package management tool that is bundled with Anaconda. PyMC3 also depends on several third-party Python packages which will be automatically installed when installing via `conda`. The four required dependencies are: `Theano`, `NumPy`, `SciPy`, `Matplotlib`, and `joblib`. To take full advantage of PyMC3, the optional dependencies `seaborn`, `pandas` and `Patsy` should also be installed. You can install PyMC3 and its dependencies by cloning this repository: 70 | 71 | ``` 72 | git clone https://github.com/fonnesbeck/PyMC3_EUSS.git 73 | ``` 74 | 75 | Then move into the directory created by the clone, and install the required packages using `conda`: 76 | 77 | ```bash 78 | cd PyMC3_EUSS 79 | conda env create -f environment.yml 80 | ``` 81 | 82 | This will create a *virtual environment* called `pymc_tutorial` that includes the dependencies for PyMC3 that is completely separate from any other Python installations you may have on your machine. To activate this environment to run the course materials, you can run the following command from the terminal: 83 | 84 | ```bash 85 | source activate pymc_tutorial 86 | ``` 87 | 88 | **If you would rather not install the software yourself, you can use the MyBinder.org link at the top of the page to run the course materials on a remote server** 89 | 90 | You can update the course materials at any time by pulling from the course repository. From your course directory, type: 91 | 92 | ```bash 93 | git pull 94 | ``` 95 | 96 | Note that this will overwrite any changes you have made to notebooks that need to be updated. 97 | -------------------------------------------------------------------------------- /data/melanoma_data.py: -------------------------------------------------------------------------------- 1 | from numpy import reshape, sum 2 | 3 | melanoma_data = reshape([1.57808, 0.00000, 2, 1.27, 35.9945, 1, 1.48219, 4 | 0.00000, 2, 0.76, 41.9014, 1, 0.0, 7.33425, 1, 35.00, 70.2164, 2, 2.23288, 5 | 0.00000, 1, 1.70, 33.7096, 1, 0.0, 9.38356, 2, 1.00, 47.9726, 1, 3.27671, 6 | 0.00000, 1, 11.00, 31.8219, 2, 0.0, 9.64384, 1, 6.50, 32.9479, 1, 1.66575, 7 | 0.00000, 2, 3.62, 35.9205, 1, 0.94247, 0.00000, 1, 8.50, 40.5068, 2, 8 | 1.68767, 0.00000, 2, 4.20, 57.0384, 1, 2.34247, 0.00000, 2, 5.00, 62.0630, 1, 9 | 0.89863, 0.00000, 1, 2.25, 56.5342, 1, 0.0, 9.03288, 2, 2.30, 22.9945, 2, 10 | 0.0, 9.63014, 2, 10.00, 18.4712, 1, 0.52603, 0.00000, 1, 3.35, 41.2521, 1, 11 | 1.82192, 0.00000, 2, 3.80, 29.5178, 1, 0.93425, 0.00000, 1, 0.75, 59.0493, 2, 12 | 0.0, 8.98630, 2, 0.50, 32.2877, 1, 3.35068, 0.00000, 1, 0.46, 26.4822, 1, 13 | 8.67397, 0.00000, 1, 2.55, 55.0411, 1, 0.41096, 0.00000, 2, 1.95, 55.1233, 2, 14 | 2.78630, 0.00000, 1, 2.50, 22.4055, 2, 2.56438, 0.00000, 1, 2.10, 50.8466, 1, 15 | 0.0, 8.75342, 2, 9.00, 56.0274, 1, 0.56986, 0.00000, 2, 5.00, 55.4767, 1, 16 | 0.0, 8.40000, 1, 0.55, 41.2411, 1, 0.0, 7.25205, 1, 12.50, 32.7425, 1, 17 | 4.38630, 0.00000, 2, 1.16, 45.3479, 1, 0.0, 8.36712, 2, 4.25, 42.8438, 2, 18 | 0.0, 8.99178, 2, 15.00, 51.1068, 1, 0.86575, 0.00000, 2, 0.72, 30.1808, 1, 19 | 0.0, 4.76986, 1, 1.50, 58.7014, 2, 1.15616, 0.00000, 2, 6.50, 51.5397, 1, 20 | 0.0, 7.28767, 1, 2.75, 27.1973, 1, 3.13151, 0.00000, 1, 3.83, 67.6740, 1, 21 | 0.0, 8.55068, 2, 1.80, 64.4274, 2, 0.0, 8.45753, 2, 4.75, 35.4411, 1, 22 | 4.59452, 0.00000, 1, 5.80, 35.9452, 1, 2.88219, 0.00000, 2, 0.51, 48.1370, 1, 23 | 0.89589, 0.00000, 1, 3.25, 58.6082, 1, 1.76164, 0.00000, 2, 0.90, 40.0137, 2, 24 | 0.0, 7.81370, 1, 3.45, 26.0055, 1, 0.0, 8.33425, 2, 1.38, 36.9616, 1, 25 | 2.62192, 0.00000, 1, 5.28, 25.9068, 2, 0.16164, 0.00000, 2, 3.00, 63.8055, 1, 26 | 0.0, 8.24658, 1, 2.20, 29.6986, 2, 1.52603, 0.00000, 1, 7.00, 61.6384, 1, 27 | 5.30959, 0.00000, 1, 4.00, 49.9918, 1, 0.87123, 0.00000, 2, 2.36, 37.1068, 1, 28 | 0.41644, 0.00000, 1, 1.06, 53.4658, 2, 4.24110, 0.00000, 1, 6.50, 57.7425, 2, 29 | 0.13699, 0.00000, 1, 10.00, 29.1479, 1, 7.07671, 0.00000, 2, 1.20, 59.2466, 30 | 1, 0.13151, 0.00000, 2, 15.00, 61.3507, 2, 0.0, 8.02740, 1, 0.49, 33.9205, 31 | 2, 0.0, 6.16164, 2, 1.60, 43.1918, 1, 1.29863, 0.00000, 2, 11.50, 34.1890, 32 | 2, 1.29041, 0.00000, 2, 1.90, 58.3808, 2, 0.0, 7.99726, 1, 4.80, 21.9479, 33 | 2, 0.0, 8.34795, 1, 0.55, 35.1151, 1, 0.0, 7.30137, 2, 6.50, 31.6493, 1, 34 | 2.32877, 0.00000, 2, 12.00, 56.1890, 1, 0.56438, 0.00000, 1, 7.00, 60.7123, 35 | 1, 5.62740, 0.00000, 2, 6.50, 58.8329, 2, 1.23014, 0.00000, 1, 1.60, 36 | 44.4849, 2, 0.0, 7.94521, 1, 1.15, 51.1315, 2, 5.06301, 0.00000, 1, 2.65, 37 | 34.2164, 1, 3.27671, 0.00000, 2, 2.00, 35.2301, 1, 0.0, 0.60822, 2, 2.50, 38 | 32.7425, 2, 0.65753, 0.00000, 1, 4.38, 38.0986, 2, 0.84110, 0.00000, 2, 39 | 2.93, 45.7699, 1, 0.0, 8.40000, 2, 3.00, 44.2000, 1, 0.18356, 0.00000, 1, 40 | 2.50, 71.3260, 1, 2.62466, 0.00000, 2, 2.30, 59.0795, 1, 0.0, 7.96438, 2, 41 | 2.00, 35.3836, 2, 0.0, 7.77808, 1, 0.75, 58.0438, 2, 0.22192, 0.00000, 1, 42 | 5.00, 43.2164, 1, 2.33973, 0.00000, 1, 10.00, 60.4932, 1, 0.52329, 0.00000, 43 | 1, 0.87, 32.4795, 2, 0.0, 8.04110, 2, 1.33, 60.2986, 1, 0.0, 7.83288, 1, 44 | 5.60, 47.1342, 1, 0.64110, 0.00000, 1, 2.55, 42.3233, 1, 0.38356, 0.00000, 45 | 1, 6.50, 54.4164, 1, 0.0, 7.82192, 2, 1.20, 51.4219, 1, 0.51781, 0.00000, 46 | 2, 3.00, 46.5973, 1, 0.0, 8.09863, 2, 2.55, 58.3562, 1, 0.0, 8.16712, 2, 47 | 1.61, 25.6712, 2, 4.42740, 0.00000, 1, 1.40, 29.1726, 1, 0.88493, 0.00000, 48 | 1, 2.25, 18.6795, 1, 2.78356, 0.00000, 1, 4.50, 60.9671, 2, 2.64658, 49 | 0.00000, 2, 0.81, 63.8849, 2, 0.0, 8.21370, 2, 1.30, 37.9808, 2, 0.0, 50 | 7.41918, 2, 3.20, 32.3507, 2, 0.99726, 0.00000, 1, 1.29, 42.9589, 1, 51 | 5.88493, 0.00000, 2, 4.40, 40.9562, 1, 0.41644, 0.00000, 1, 6.00, 61.9753, 1, 52 | 3.53699, 0.00000, 1, 3.93, 55.3315, 2, 0.0, 7.56164, 1, 0.60, 36.0767, 1, 53 | 0.0, 7.53151, 1, 0.75, 50.6795, 1, 0.27671, 0.00000, 1, 0.73, 66.6986, 1, 54 | 0.76986, 0.00000, 2, 0.20, 29.3479, 2, 0.0, 7.62192, 2, 3.88, 33.1863, 1, 55 | 0.0, 7.79726, 1, 2.48, 48.6356, 2, 0.64110, 0.00000, 1, 2.50, 29.4877, 1, 56 | 1.14521, 0.00000, 2, 10.00, 42.6685, 1, 2.01644, 0.00000, 1, 16.00, 24.4055, 57 | 2, 2.84384, 0.00000, 1, 4.00, 40.3890, 1, 0.0, 7.00000, 2, 1.35, 45.4192, 58 | 1, 1.27397, 0.00000, 2, 3.00, 65.3945, 1, 0.0, 7.09589, 1, 10.72, 47.5753, 59 | 2, 2.04110, 0.00000, 1, 1.50, 58.4438, 2, 0.83562, 0.00000, 1, 3.50, 60 | 59.2767, 2, 0.92329, 0.00000, 1, 1.10, 30.2630, 2, 0.07397, 0.00000, 1, 61 | 1.00, 40.7370, 1, 0.0, 7.30685, 2, 5.10, 44.7452, 1, 2.07671, 0.00000, 2, 62 | 0.50, 67.8329, 1, 0.0, 7.70959, 2, 4.03, 27.7452, 1, 0.0, 6.15890, 1, 63 | 1.80, 25.9260, 2, 0.0, 6.89315, 2, 3.50, 31.2740, 1, 3.30685, 0.00000, 1, 64 | 1.15, 58.8822, 2, 0.36164, 0.00000, 1, 1.75, 57.0575, 1, 1.97808, 0.00000, 65 | 2, 2.50, 59.8137, 1, 1.23836, 0.00000, 2, 2.10, 77.5151, 1, 0.10685, 66 | 0.00000, 1, 1.35, 43.4219, 1, 0.0, 7.63836, 1, 4.50, 52.2082, 1, 2.06301, 67 | 0.00000, 1, 0.50, 36.3205, 2, 0.0, 7.42466, 2, 2.30, 25.9781, 1, 0.50959, 68 | 0.00000, 1, 4.00, 49.4411, 1, 0.65753, 0.00000, 1, 5.40, 57.9589, 1, 0.0, 69 | 6.93151, 1, 6.00, 65.5644, 1, 0.0, 7.23288, 2, 5.10, 72.3425, 1, 6.01096, 70 | 0.00000, 1, 4.50, 68.8548, 1, 0.33699, 0.00000, 1, 1.45, 50.4438, 2, 0.0, 71 | 6.47123, 2, 3.38, 48.2877, 1, 0.94795, 0.00000, 1, 3.00, 46.9479, 2, 72 | 2.91781, 0.00000, 2, 1.20, 33.6000, 2, 1.59726, 0.00000, 2, 7.30, 51.1644, 2, 73 | 0.84932, 0.00000, 2, 1.67, 47.7836, 1, 1.38356, 0.00000, 1, 4.00, 53.8795, 2, 74 | 3.81644, 0.00000, 2, 2.10, 38.7068, 2, 0.0, 7.06849, 1, 10.00, 69.3205, 2, 75 | 0.0, 7.04110, 2, 3.50, 66.0219, 1, 1.00274, 0.00000, 2, 1.10, 36.0329, 2, 76 | 0.0, 6.34795, 2, 0.40, 63.4603, 1, 1.18082, 0.00000, 1, 0.70, 48.8986, 2, 77 | 0.97534, 0.00000, 1, 5.00, 45.0575, 1, 2.16712, 0.00000, 1, 0.85, 57.6712, 2, 78 | 0.0, 6.85479, 1, 4.80, 45.2000, 1, 1.38356, 0.00000, 1, 1.20, 49.0438, 1, 79 | 1.71507, 0.00000, 2, 1.30, 51.4630, 1, 0.79452, 0.00000, 2, 5.80, 34.5479, 1, 80 | 0.0, 6.86301, 2, 6.00, 47.6438, 2, 0.0, 6.50411, 1, 3.00, 38.7233, 2, 81 | 0.42466, 0.00000, 2, 1.88, 54.0658, 1, 0.98630, 0.00000, 1, 2.60, 45.7397, 1, 82 | 0.0, 6.13699, 2, 2.70, 47.2822, 2, 3.80000, 0.00000, 2, 6.00, 62.6411, 1, 83 | 0.0, 6.48493, 1, 4.00, 62.0192, 2, 0.0, 6.96438, 2, 1.71, 41.0904, 2, 0.0, 84 | 6.78082, 2, 1.60, 50.2712, 2, 0.56164, 0.00000, 2, 1.50, 49.5288, 2, 85 | 2.67123, 0.00000, 1, 3.00, 70.8192, 1, 1.56712, 0.00000, 2, 0.90, 59.0712, 1, 86 | 2.07397, 0.00000, 2, 4.00, 53.9041, 1, 0.33973, 0.00000, 1, 2.80, 44.7342, 1, 87 | 3.37808, 0.00000, 2, 0.80, 22.1397, 1, 3.15068, 0.00000, 1, 0.70, 72.8575, 1, 88 | 0.0, 6.81096, 2, 0.90, 61.4521, 1, 3.20822, 0.00000, 2, 12.00, 61.2904, 1, 89 | 0.62740, 0.00000, 1, 5.78, 34.7507, 1, 1.64384, 0.00000, 1, 0.60, 67.4164, 2, 90 | 1.40822, 0.00000, 1, 12.00, 53.2493, 1, 0.0, 6.06575, 1, 4.00, 49.0082, 1, 91 | 1.66301, 0.00000, 2, 0.45, 56.7699, 1, 1.36986, 0.00000, 2, 1.30, 34.0247, 2, 92 | 5.46849, 0.00000, 1, 0.81, 34.3014, 2, 0.42740, 0.00000, 1, 3.20, 45.0712, 2, 93 | 1.13973, 0.00000, 2, 4.00, 54.7671, 2, 1.73699, 0.00000, 2, 4.77, 42.8548, 2, 94 | 0.0, 5.54521, 2, 2.20, 36.6301, 2, 0.85205, 0.00000, 1, 3.00, 43.2466, 1, 95 | 0.43014, 0.00000, 1, 3.00, 53.3562, 1, 1.20822, 0.00000, 2, 0.80, 35.3534, 1, 96 | 4.36164, 0.00000, 1, 4.00, 36.5233, 1, 0.52877, 0.00000, 2, 5.00, 52.7863, 1, 97 | 0.0, 6.51507, 1, 2.00, 24.4329, 2, 2.89863, 0.00000, 2, 3.85, 58.7178, 1, 98 | 0.0, 6.20274, 2, 0.76, 45.5479, 1, 1.21644, 0.00000, 2, 0.75, 43.3014, 2, 99 | 0.0, 6.00000, 2, 6.50, 51.4055, 2, 0.0, 6.25479, 1, 0.85, 38.9671, 2, 0.0, 100 | 6.49863, 1, 4.30, 68.2658, 1, 1.13699, 0.00000, 2, 2.10, 59.4493, 2, 101 | 1.69589, 0.00000, 1, 1.50, 30.0192, 1, 0.0, 6.41096, 2, 2.00, 22.1562, 2, 102 | 0.0, 6.02192, 1, 11.00, 54.7671, 1, 3.04932, 0.00000, 2, 4.88, 45.0384, 1, 103 | 0.0, 5.62740, 2, 5.20, 39.7589, 1, 0.72603, 0.00000, 1, 3.04, 41.3808, 1, 104 | 0.73425, 0.00000, 2, 8.00, 34.9671, 1, 1.47945, 0.00000, 2, 1.60, 46.3479, 1, 105 | 0.37808, 0.00000, 2, 1.10, 29.9233, 2, 0.0, 5.75890, 2, 3.00, 32.8740, 1, 106 | 1.48219, 0.00000, 2, 10.00, 39.5397, 2, 0.0, 5.88493, 1, 1.95, 55.4822, 1, 107 | 0.0, 1.80274, 1, 2.00, 32.3562, 1, 1.40548, 0.00000, 2, 3.70, 41.8027, 2, 108 | 0.0, 4.74795, 1, 2.90, 35.3452, 2, 0.0, 5.24658, 1, 1.80, 50.4795, 1, 109 | 0.29041, 0.00000, 1, 6.00, 61.3507, 2, 0.0, 5.83836, 1, 1.50, 67.3562, 1, 110 | 0.0, 5.32055, 2, 1.75, 53.8548, 2, 5.16712, 0.00000, 2, 5.00, 78.7315, 2, 111 | 0.0, 5.59178, 2, 0.63, 62.7233, 1, 0.0, 5.77808, 1, 1.15, 65.1507, 1, 112 | 0.53425, 0.00000, 2, 1.50, 34.8274, 1, 0.0, 2.22466, 1, 0.98, 33.8466, 2, 113 | 3.59726, 0.00000, 1, 5.00, 67.8822, 1, 0.0, 5.32329, 1, 5.50, 66.0712, 2, 114 | 1.78630, 0.00000, 2, 1.00, 55.0658, 2, 0.70411, 0.00000, 2, 10.00, 50.5123, 115 | 1, 0.0, 4.94795, 2, 5.00, 42.4055, 2, 0.0, 5.45479, 2, 3.75, 58.1068, 2, 116 | 4.32877, 0.00000, 1, 10.00, 26.0137, 1, 1.16164, 0.00000, 2, 3.00, 54.4685, 117 | 1, 0.0, 5.20274, 2, 8.00, 54.0630, 2, 0.0, 4.40822, 1, 1.64, 34.5589, 1, 118 | 1.41096, 0.00000, 1, 4.95, 58.5068, 1, 0.0, 4.92877, 2, 1.45, 63.9370, 1, 119 | 0.0, 5.42192, 2, 12.00, 49.8274, 2, 0.98904, 0.00000, 1, 2.05, 50.5562, 1, 120 | 0.36438, 0.00000, 1, 3.60, 40.4795, 2, 0.0, 4.38082, 1, 8.30, 61.7479, 2, 121 | 0.77260, 0.00000, 2, 0.45, 41.6712, 1, 4.90959, 0.00000, 2, 3.00, 25.5096, 1, 122 | 1.26849, 0.00000, 1, 4.40, 61.2000, 1, 0.58082, 0.00000, 2, 1.10, 53.1260, 1, 123 | 0.0, 4.95616, 1, 1.05, 40.4658, 1, 0.0, 5.12329, 1, 1.71, 60.3068, 1, 0.0, 124 | 4.74795, 1, 6.30, 48.7425, 2, 0.0, 4.90685, 2, 0.50, 46.7562, 2, 1.41918, 125 | 0.00000, 1, 5.10, 34.8932, 2, 0.44110, 0.00000, 1, 6.00, 33.3096, 1, 0.0, 126 | 4.29863, 2, 1.50, 35.7589, 1, 0.0, 4.63836, 2, 0.36, 49.8575, 1, 0.0, 127 | 4.81370, 1, 3.00, 57.3726, 2, 4.50137, 0.00000, 2, 1.24, 29.7726, 2, 128 | 3.92329, 0.00000, 2, 0.70, 51.8822, 2, 0.0, 4.86027, 2, 0.80, 65.3123, 2, 129 | 0.52603, 0.00000, 1, 1.00, 52.0658, 2, 2.10685, 0.00000, 2, 3.38, 60.9534, 2, 130 | 0.0, 4.24384, 1, 1.52, 32.6055, 2, 3.39178, 0.00000, 1, 2.20, 51.5123, 2, 131 | 0.0, 4.36164, 2, 2.10, 48.6548, 1, 0.0, 4.81918, 2, 1.40, 43.8438, 2], 132 | (255, 6)) 133 | 134 | # Censoring indicator 135 | censored = (melanoma_data[:, 0] == 0).astype(int) 136 | # Time 137 | t = sum(melanoma_data[:, 0:2], 1) 138 | # Treatment 139 | treat = melanoma_data[:, 2].astype(int) - 1 140 | # Breslow scale 141 | breslow = melanoma_data[:, 3] 142 | # Age and sex 143 | age = melanoma_data[:, 4] 144 | sex = melanoma_data[:, 5].astype(int) - 1 145 | -------------------------------------------------------------------------------- /data/nashville_precip.txt: -------------------------------------------------------------------------------- 1 | Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 2 | 1871 2.76 4.58 5.01 4.13 3.30 2.98 1.58 2.36 0.95 1.31 2.13 1.65 3 | 1872 2.32 2.11 3.14 5.91 3.09 5.17 6.10 1.65 4.50 1.58 2.25 2.38 4 | 1873 2.96 7.14 4.11 3.59 6.31 4.20 4.63 2.36 1.81 4.28 4.36 5.94 5 | 1874 5.22 9.23 5.36 11.84 1.49 2.87 2.65 3.52 3.12 2.63 6.12 4.19 6 | 1875 6.15 3.06 8.14 4.22 1.73 5.63 8.12 1.60 3.79 1.25 5.46 4.30 7 | 1876 6.41 2.22 5.28 3.62 3.40 5.65 7.15 5.77 2.52 2.68 1.26 0.95 8 | 1877 4.05 1.06 4.98 9.47 1.25 6.02 3.25 4.16 5.40 2.61 4.93 2.49 9 | 1878 3.34 2.10 3.48 6.88 2.33 3.28 9.43 5.02 1.28 2.17 3.20 6.04 10 | 1879 6.32 3.13 3.81 2.88 2.88 2.50 8.47 4.62 5.18 2.90 5.85 9.15 11 | 1880 3.74 12.37 8.16 5.26 4.13 3.97 5.69 2.22 5.39 7.24 5.77 3.32 12 | 1881 3.54 5.48 2.79 5.12 3.67 3.70 0.86 1.81 6.57 4.80 4.89 4.85 13 | 1882 14.51 8.61 9.38 3.59 7.38 2.54 4.06 5.54 1.61 1.11 3.60 1.52 14 | 1883 3.76 7.90 3.98 9.12 4.82 3.82 4.94 4.47 2.23 5.27 3.11 4.97 15 | 1884 7.20 8.18 8.89 3.51 3.58 6.53 3.18 2.81 2.36 2.43 1.57 3.78 16 | 1885 6.29 2.00 2.33 3.75 4.36 3.72 5.26 1.02 5.60 2.99 2.73 2.90 17 | 1886 5.18 3.82 4.76 2.36 2.10 7.69 1.90 5.50 3.68 0.51 5.76 1.48 18 | 1887 5.13 8.47 3.36 2.67 3.43 2.31 3.77 2.89 6.85 1.92 2.29 5.31 19 | 1888 6.29 3.78 6.46 4.18 2.97 4.68 2.36 7.03 3.82 2.82 4.33 1.77 20 | 1889 3.83 1.84 2.47 2.83 5.30 5.33 2.74 1.57 6.81 1.54 6.88 1.17 21 | 1890 8.10 10.95 8.64 3.84 4.16 2.23 0.46 6.59 5.86 3.01 2.01 4.12 22 | 1891 6.15 6.96 10.31 2.24 2.39 6.50 1.49 3.72 1.25 0.84 6.71 4.26 23 | 1892 2.81 2.73 4.10 7.45 4.03 5.01 5.13 3.39 4.78 0.25 3.91 6.43 24 | 1893 1.27 4.88 3.37 4.11 7.31 4.74 2.12 1.92 6.43 3.68 2.97 3.50 25 | 1894 4.28 8.65 2.69 4.05 2.53 3.55 5.45 2.43 3.07 0.53 1.92 2.81 26 | 1895 5.71 0.98 5.09 3.07 2.05 2.90 7.14 1.40 6.69 1.57 2.14 4.09 27 | 1896 1.37 3.65 6.45 2.92 4.05 1.82 7.33 1.40 2.74 0.98 5.71 1.79 28 | 1897 3.13 3.84 8.49 5.79 1.22 1.82 8.53 2.34 0.19 0.92 2.83 4.93 29 | 1898 9.46 0.63 5.36 3.16 1.80 4.97 4.50 6.56 4.87 3.21 3.09 2.41 30 | 1899 5.59 5.19 7.81 3.25 3.36 0.75 6.44 2.53 1.50 1.83 1.55 4.64 31 | 1900 2.61 3.80 2.20 4.04 1.86 10.35 2.87 1.24 4.55 3.93 8.87 2.22 32 | 1901 3.45 1.60 2.39 4.99 4.11 0.48 2.59 8.24 4.27 0.63 0.85 4.51 33 | 1902 4.68 2.20 6.96 3.08 4.36 2.77 1.89 2.69 2.82 1.83 4.04 6.58 34 | 1903 1.80 4.81 5.17 4.00 5.64 7.58 3.04 1.42 0.13 2.06 3.47 3.91 35 | 1904 3.93 1.24 5.20 2.35 2.97 6.38 4.62 2.80 0.81 0.29 1.66 5.01 36 | 1905 3.17 2.66 5.10 1.50 6.00 7.31 3.23 2.97 6.12 3.56 1.35 4.15 37 | 1906 3.26 0.79 6.49 1.75 3.80 2.62 3.61 2.94 10.95 2.28 5.88 5.11 38 | 1907 2.60 3.53 3.58 4.66 6.01 2.67 0.80 2.53 1.95 2.01 4.39 2.69 39 | 1908 3.02 4.15 4.16 4.43 2.80 2.73 3.32 1.69 1.89 0.44 3.31 2.10 40 | 1909 2.34 7.08 4.94 4.12 4.57 5.56 4.93 2.73 4.23 1.01 2.58 2.90 41 | 1910 3.45 4.87 0.85 6.10 5.81 6.61 4.45 1.73 0.74 3.20 1.17 3.92 42 | 1911 2.00 4.68 2.22 5.11 1.67 4.58 3.34 5.17 1.89 2.21 4.38 11.01 43 | 1912 2.92 4.21 6.00 11.73 4.02 5.66 5.37 3.06 2.46 2.45 0.65 5.14 44 | 1913 12.30 4.71 4.54 1.65 2.66 0.90 4.09 0.85 1.79 2.93 1.84 2.45 45 | 1914 1.56 2.03 4.33 3.83 3.01 2.95 2.58 8.64 1.46 2.80 2.13 5.06 46 | 1915 5.89 1.01 2.14 0.72 4.94 1.42 2.03 6.03 4.63 0.42 6.75 6.44 47 | 1916 7.62 1.19 3.60 2.49 5.37 4.62 4.17 4.27 1.92 2.67 1.00 4.29 48 | 1917 7.27 2.76 8.06 4.05 4.75 8.03 3.25 3.02 1.51 2.25 0.85 1.46 49 | 1918 7.43 1.54 1.86 3.39 3.61 2.70 3.63 3.05 3.75 3.44 1.36 4.56 50 | 1919 4.71 2.19 8.67 2.66 8.67 3.96 1.83 6.80 1.33 8.35 7.89 3.28 51 | 1920 7.35 1.92 3.25 8.58 3.18 3.81 3.00 6.85 4.15 2.75 2.60 2.99 52 | 1921 3.11 4.70 5.95 3.50 1.15 2.29 4.56 2.85 3.72 2.93 5.68 1.76 53 | 1922 2.90 3.89 9.32 4.53 4.39 5.37 6.15 3.83 3.28 0.75 1.80 6.29 54 | 1923 5.89 4.36 7.69 4.26 4.31 4.42 2.13 9.60 1.44 1.23 2.87 4.32 55 | 1924 5.40 3.44 1.74 3.55 6.39 0.91 4.36 2.59 2.64 0.03 1.25 5.36 56 | 1925 2.70 4.88 3.34 3.74 1.95 2.18 1.74 1.02 3.52 6.99 4.52 1.10 57 | 1926 4.48 2.06 3.88 2.45 2.15 2.17 3.68 8.30 2.52 4.64 4.64 13.53 58 | 1927 3.27 4.26 9.66 7.38 3.63 4.49 1.59 1.86 1.65 3.75 4.48 2.69 59 | 1928 2.55 2.64 3.26 3.22 2.89 11.64 3.62 4.62 0.68 6.82 2.63 1.38 60 | 1929 3.85 3.81 6.48 3.93 6.46 2.02 6.59 0.51 4.85 4.05 4.79 3.33 61 | 1930 3.82 3.68 5.14 1.16 5.23 1.59 1.22 4.69 3.88 2.11 3.19 2.09 62 | 1931 1.31 5.05 3.49 2.44 1.92 2.31 2.47 3.94 1.54 1.86 3.75 6.33 63 | 1932 7.74 5.80 4.89 7.20 1.32 2.91 3.40 1.70 4.32 4.94 1.88 4.43 64 | 1933 3.51 6.21 6.14 4.17 9.94 1.76 5.37 2.79 3.66 1.23 1.47 6.40 65 | 1934 2.99 2.61 7.99 2.24 1.61 5.79 3.85 1.87 4.52 2.02 1.45 2.57 66 | 1935 6.96 3.63 6.96 4.51 3.51 2.81 4.50 2.29 2.33 1.51 2.77 0.91 67 | 1936 3.52 1.52 8.40 3.70 1.41 0.21 8.33 0.59 1.86 3.38 3.52 4.72 68 | 1937 14.75 1.90 1.57 3.73 5.33 2.65 2.49 2.17 2.05 4.31 2.06 3.12 69 | 1938 5.81 1.82 4.78 2.23 4.35 4.98 5.93 2.38 3.40 0.47 3.06 2.00 70 | 1939 6.80 8.87 4.83 3.96 1.98 4.85 3.11 1.87 0.95 1.13 1.48 2.68 71 | 1940 1.13 5.06 7.63 5.24 3.32 2.84 2.17 1.33 0.87 1.30 3.71 2.44 72 | 1941 1.81 0.64 1.71 2.40 0.87 2.61 7.12 3.64 0.62 2.12 2.81 3.89 73 | 1942 2.81 3.50 3.82 4.22 2.39 1.87 2.77 8.31 1.82 3.05 2.27 6.06 74 | 1943 1.44 1.88 8.43 2.97 5.29 1.61 2.51 1.55 5.98 0.81 1.61 3.44 75 | 1944 2.74 7.14 5.81 3.85 4.33 1.33 0.87 6.26 4.40 0.68 2.40 7.10 76 | 1945 3.47 7.50 1.72 5.51 5.70 2.73 5.15 1.83 3.40 2.09 9.04 3.91 77 | 1946 8.97 4.28 5.48 3.45 3.73 1.30 4.86 2.22 4.13 1.80 5.11 4.04 78 | 1947 7.62 1.07 2.69 3.56 6.47 2.44 4.56 3.05 0.47 2.10 3.21 2.48 79 | 1948 5.59 7.73 6.49 1.15 1.65 1.64 1.65 2.07 3.31 1.59 7.85 5.58 80 | 1949 7.12 2.94 5.46 5.27 5.65 7.16 4.25 2.87 0.76 4.38 0.54 5.68 81 | 1950 13.92 7.78 3.77 1.58 4.10 4.69 7.75 6.66 3.59 1.65 6.60 2.17 82 | 1951 10.54 2.94 5.16 4.75 0.83 7.40 3.79 0.95 2.92 4.35 4.16 10.60 83 | 1952 5.10 4.10 9.87 2.01 2.81 0.78 1.30 4.12 3.65 1.36 2.33 2.38 84 | 1953 7.09 3.31 5.52 4.53 5.57 1.90 6.62 1.20 1.00 0.42 0.70 3.45 85 | 1954 7.65 4.00 4.22 4.42 3.80 2.42 0.71 1.61 4.28 2.93 1.12 5.57 86 | 1955 1.25 6.76 9.87 4.62 2.92 2.94 1.76 4.02 4.26 2.04 3.78 1.21 87 | 1956 5.67 10.31 4.08 4.23 2.87 2.42 1.94 1.89 0.28 1.99 1.73 6.53 88 | 1957 9.39 5.24 2.85 2.79 8.23 7.06 1.83 2.55 4.09 3.89 6.31 5.84 89 | 1958 2.60 1.61 3.92 6.35 2.72 3.30 6.15 5.10 3.04 1.24 3.38 1.49 90 | 1959 3.26 4.60 3.71 2.49 5.20 2.41 3.90 2.54 3.73 6.13 4.96 5.04 91 | 1960 3.22 5.40 4.37 2.04 3.16 9.37 1.46 1.72 3.96 1.38 2.72 3.62 92 | 1961 1.44 5.33 6.52 4.50 4.36 2.96 5.34 2.62 0.35 1.12 3.87 6.50 93 | 1962 6.51 9.07 5.89 6.91 1.87 7.29 1.97 2.45 8.03 2.29 3.37 1.92 94 | 1963 1.60 2.83 10.03 3.37 2.47 3.09 5.33 7.63 3.43 NA 2.43 2.15 95 | 1964 3.70 3.26 4.01 5.86 5.04 1.21 2.16 4.56 2.65 1.83 3.67 5.15 96 | 1965 2.98 4.71 6.13 5.72 3.12 2.74 3.32 2.53 5.02 0.57 1.82 1.01 97 | 1966 3.93 3.63 1.39 5.08 3.99 1.09 2.70 5.29 3.87 2.50 2.76 5.69 98 | 1967 1.62 1.78 4.44 3.40 6.98 4.23 7.46 2.06 1.93 1.57 3.87 5.88 99 | 1968 3.50 0.64 4.47 3.57 6.28 2.26 6.87 0.69 2.76 3.92 5.39 3.58 100 | 1969 4.96 4.48 2.12 6.03 4.81 3.34 5.33 2.27 2.06 2.01 1.83 8.03 101 | 1970 1.16 4.36 3.87 6.81 5.90 6.73 3.61 2.99 2.76 2.94 2.20 3.60 102 | 1971 2.66 4.70 2.95 3.34 2.93 3.47 5.00 5.87 2.11 1.27 1.18 5.17 103 | 1972 5.15 3.45 4.34 3.58 3.52 2.54 6.40 4.30 3.71 4.06 5.22 8.14 104 | 1973 3.40 3.63 9.88 7.00 5.72 4.80 7.67 1.79 1.56 3.32 7.78 3.23 105 | 1974 9.45 3.01 5.25 3.97 5.04 6.80 2.10 4.13 10.44 1.47 6.23 2.81 106 | 1975 4.67 5.22 12.35 3.55 6.52 2.22 2.96 4.69 5.42 5.86 3.00 4.12 107 | 1976 4.11 2.28 5.32 1.53 6.19 4.72 4.01 8.05 5.08 5.17 1.30 1.81 108 | 1977 2.53 3.27 5.83 7.87 1.65 4.29 1.15 4.65 5.04 4.22 5.96 4.25 109 | 1978 5.95 1.57 4.88 2.42 8.03 1.46 4.03 3.81 1.37 2.28 4.01 13.63 110 | 1979 7.13 4.01 4.92 7.80 8.18 2.79 4.27 4.59 11.44 3.97 5.98 5.04 111 | 1980 2.59 1.38 7.27 3.67 6.14 2.89 3.53 1.24 1.09 1.17 2.55 1.40 112 | 1981 1.60 3.83 3.38 4.78 3.05 8.05 3.49 3.10 1.37 2.82 3.83 2.38 113 | 1982 6.50 4.80 3.00 4.36 4.19 2.28 5.47 3.46 3.23 1.91 3.87 6.36 114 | 1983 2.56 2.93 3.44 6.80 11.04 3.93 1.71 1.36 0.45 2.77 6.98 7.75 115 | 1984 1.79 2.38 5.14 8.41 9.68 4.49 6.63 2.42 0.97 6.00 6.20 2.38 116 | 1985 3.02 3.30 2.70 2.91 2.65 1.53 2.00 3.91 2.52 1.59 3.81 0.98 117 | 1986 0.19 3.59 2.29 0.52 3.36 2.38 0.77 3.38 2.19 2.19 7.43 3.31 118 | 1987 1.61 4.87 1.18 1.03 4.41 2.82 2.56 0.73 1.95 0.21 3.40 5.46 119 | 1988 3.73 2.02 2.18 2.09 1.86 0.45 3.26 2.39 2.45 1.54 5.49 3.95 120 | 1989 4.52 9.36 5.31 2.68 4.61 7.87 3.18 3.67 6.30 3.62 3.94 1.97 121 | 1990 2.76 4.73 3.26 1.60 2.80 2.37 4.86 3.12 2.13 4.41 4.29 10.76 122 | 1991 2.92 5.44 4.25 3.35 5.63 1.25 2.82 1.79 5.47 3.88 2.87 7.27 123 | 1992 2.97 2.60 4.50 0.77 3.12 4.31 5.89 3.25 3.45 1.62 4.48 2.88 124 | 1993 2.76 3.33 5.50 3.33 4.50 5.31 3.64 1.76 2.90 2.20 2.53 6.62 125 | 1994 4.36 6.18 7.56 5.72 3.76 8.08 4.82 5.05 4.20 3.31 4.04 2.69 126 | 1995 5.61 1.81 3.87 3.95 7.66 3.69 1.95 3.40 5.00 5.60 3.98 2.32 127 | 1996 3.82 2.46 5.15 3.68 4.48 3.68 5.45 1.09 4.88 3.16 6.00 4.77 128 | 1997 4.19 3.10 9.64 2.42 4.92 6.66 3.26 3.52 5.75 2.71 6.59 2.19 129 | 1998 3.68 4.11 3.13 6.31 4.46 11.95 4.63 2.93 1.39 1.59 1.30 6.53 130 | 1999 9.28 2.33 4.27 2.29 4.35 3.56 3.19 3.05 1.97 2.04 2.99 2.50 131 | 2000 3.52 3.75 3.34 6.23 7.66 1.74 2.25 1.95 1.90 0.26 6.39 3.44 132 | 2001 3.21 8.54 2.73 2.42 5.54 4.47 2.77 4.07 1.79 4.61 5.09 3.32 133 | 2002 4.93 1.99 9.40 4.31 3.98 3.76 5.64 3.13 6.29 4.48 2.91 5.81 134 | 2003 1.59 8.47 2.30 4.69 10.73 7.08 2.87 3.88 8.70 1.80 4.17 3.19 135 | 2004 3.60 5.77 4.81 6.69 6.90 3.39 3.19 4.24 4.55 4.90 5.21 5.93 136 | 2005 4.42 3.84 3.90 6.93 1.03 2.70 2.39 6.89 1.44 0.02 3.29 2.46 137 | 2006 6.57 2.69 2.90 4.14 4.95 2.19 2.64 5.20 4.00 2.98 4.05 3.41 138 | 2007 3.32 1.84 2.26 2.75 3.30 2.37 1.47 1.38 1.99 4.95 6.20 3.83 139 | 2008 4.76 2.53 5.56 7.20 5.54 2.21 4.32 1.67 0.88 5.03 1.75 6.72 140 | 2009 4.59 2.85 2.92 4.13 8.45 4.53 6.03 2.14 11.08 6.49 0.67 3.99 141 | 2010 4.13 2.77 3.52 3.48 16.43 4.96 5.86 6.99 1.17 2.49 5.41 1.87 142 | 2011 2.31 5.54 4.59 7.51 4.38 5.04 3.46 1.78 6.20 0.93 6.15 4.25 -------------------------------------------------------------------------------- /data/vlbw.csv: -------------------------------------------------------------------------------- 1 | "","birth","exit","hospstay","lowph","pltct","race","bwt","gest","inout","twn","lol","magsulf","meth","toc","delivery","apg1","vent","pneumo","pda","cld","pvh","ivh","ipe","year","sex","dead" 2 | "1",81.5110016,81.6039963,34,NA,100,"white",1250,35,"born at Duke",0,NA,NA,0,0,"abdominal",8,0,0,0,0,NA,NA,NA,81.5119629,"female",0 3 | "2",81.5139999,81.5390015,9,7.25,244,"white",1370,32,"born at Duke",0,NA,NA,1,0,"abdominal",7,0,0,0,0,NA,NA,NA,81.5147095,"female",0 4 | "3",81.552002,81.552002,-2,7.05999756,114,"black",620,23,"born at 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"664",87.4329987,87.5289993,35,7.17999649,289,"black",910,28,"transported",0,NA,0,0,0,"vaginal",4,0,0,0,0,"definite","definite","definite",87.4353027,"female",0 666 | "665",87.435997,NA,NA,NA,NA,NA,1490,NA,NA,0,0,NA,NA,NA,"abdominal",NA,NA,NA,NA,NA,"absent","absent","absent",87.4380493,NA,0 667 | "666",87.4550018,87.5370026,30,7.2899971,144,"black",1320,30,"born at Duke",0,5,NA,1,1,"vaginal",8,0,0,0,0,"absent","absent","absent",87.4572144,"male",0 668 | "667",87.4660034,87.6819992,79,7.23999786,278,"white",805,26,"transported",0,22,0,0,0,"vaginal",6,1,1,0,1,"definite","possible","absent",87.4681396,"male",0 669 | "668",87.4690018,NA,NA,7.19999695,290,"white",1200,30,"born at Duke",0,6,NA,0,0,"vaginal",8,1,0,1,1,"absent","absent","absent",87.4708862,"male",0 670 | "669",87.4720001,87.8000031,120,NA,NA,"white",1150,29,"born at Duke",0,0,0,0,0,"abdominal",5,1,0,1,1,"absent","absent","absent",87.4736328,"female",0 671 | "670",87.4720001,87.5479965,28,7.14999771,173,"white",860,29,"born at Duke",1,0,0,0,0,"abdominal",5,1,0,1,NA,"absent","absent","absent",87.4736328,"female",1 672 | "671",87.4830017,87.7180023,86,7.18999863,136,"white",1050,28,"born at Duke",0,6,0,0,0,"vaginal",5,1,0,1,1,"absent","absent","absent",87.4845581,"male",0 673 | -------------------------------------------------------------------------------- /environment.yml: -------------------------------------------------------------------------------- 1 | name: pymc_tutorial 2 | 3 | channels: 4 | - conda-forge 5 | 6 | dependencies: 7 | - python=3.6 8 | - ipython 9 | - jupyterlab 10 | - seaborn 11 | - patsy 12 | - pandas 13 | - theano 14 | - pymc3 -------------------------------------------------------------------------------- /notebooks/0. Basic Bayesian Inference.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Basic Bayesian Inference\n", 8 | "\n", 9 | "Statistical inference is a process of learning from incomplete or imperfect (error-contaminated) data. Can account for this \"imperfection\" using either a sampling model or a measurement error model.\n", 10 | "\n", 11 | "### Statistical hypothesis testing\n", 12 | "\n", 13 | "The *de facto* standard for statistical inference is statistical hypothesis testing. The goal of hypothesis testing is to evaluate a **null hypothesis**. There are two possible outcomes:\n", 14 | "\n", 15 | "- reject the null hypothesis\n", 16 | "- fail to reject the null hypothesis\n", 17 | "\n", 18 | "Rejection occurs when a chosen test statistic is higher than some pre-specified threshold valuel; non-rejection occurs otherwise.\n", 19 | "\n", 20 | "![](images/test_stats.png)\n", 21 | "\n", 22 | "Notice that neither outcome says anything about the quantity of interest, the **research hypothesis**. \n", 23 | "\n", 24 | "Setting up a statistical test involves several subjective choices by the user that are rarely justified based on the problem or decision at hand:\n", 25 | "\n", 26 | "- statistical test to use\n", 27 | "- null hypothesis to test\n", 28 | "- significance level\n", 29 | "\n", 30 | "Choices are often based on arbitrary criteria, including \"statistical tradition\" (Johnson 1999). The resulting evidence is indirect, incomplete, and typically overstates the evidence against the null hypothesis (Goodman 1999).\n", 31 | "\n", 32 | "Most importantly to applied users, the results of statistical hypothesis tests are very easy to misinterpret. \n", 33 | "\n", 34 | "### Estimation \n", 35 | "\n", 36 | "Instead of testing, a more informative and effective approach for inference is based on **estimation** (be it frequentist or Bayesian). That is, rather than testing whether two groups are different, we instead pursue an estimate of *how different* they are, which is fundamentally more informative. \n", 37 | "\n", 38 | "Additionally, we include an estimate of **uncertainty** associated with that difference which includes uncertainty due to our lack of knowledge of the model parameters (*epistemic uncertainty*) and uncertainty due to the inherent stochasticity of the system (*aleatory uncertainty*)." 39 | ] 40 | }, 41 | { 42 | "cell_type": "markdown", 43 | "metadata": {}, 44 | "source": [ 45 | "# An Introduction to Bayesian Statistical Analysis\n", 46 | "\n", 47 | "Though many of you will have taken a statistics course or two during your undergraduate (or graduate education, most of those who have will likely not have had a course in *Bayesian* statistics. Most introductory courses, particularly for non-statisticians, still do not cover Bayesian methods at all. Even today, Bayesian courses (similarly to statistical computing courses!) are typically tacked onto the curriculum, rather than being integrated into the program.\n", 48 | "\n", 49 | "In fact, Bayesian statistics is not just a particular method, or even a class of methods; it is an entirely **different paradigm** for doing statistical analysis.\n", 50 | "\n", 51 | "> Practical methods for making inferences from data using probability models for quantities we observe and about which we wish to learn.\n", 52 | "*-- Gelman et al. 2013*\n", 53 | "\n", 54 | "A Bayesian model is described by parameters, uncertainty in those parameters is described using probability distributions.\n", 55 | "\n", 56 | "All conclusions from Bayesian statistical procedures are stated in terms of **probability statements**\n", 57 | "\n", 58 | "![prob model](images/prob_model.png)\n", 59 | "\n", 60 | "This confers several benefits to the analyst, including:\n", 61 | "\n", 62 | "- ease of interpretation, summarization of uncertainty\n", 63 | "- can incorporate uncertainty in parent parameters\n", 64 | "- easy to calculate summary statistics" 65 | ] 66 | }, 67 | { 68 | "cell_type": "markdown", 69 | "metadata": {}, 70 | "source": [ 71 | "### Bayesian vs Frequentist Statistics: *What's the difference?*\n", 72 | "\n", 73 | "Any statistical inferece paradigm, Bayesian or otherwise, involves at least the following: \n", 74 | "\n", 75 | "1. Some **unknown quantities** about which we are interested in learning or testing. We call these *parameters*.\n", 76 | "2. Some **data** which have been observed, and hopefully contain information about.\n", 77 | "3. One or more **models** that relate the data to the parameters, and is the instrument that is used to learn.\n", 78 | "\n" 79 | ] 80 | }, 81 | { 82 | "cell_type": "markdown", 83 | "metadata": {}, 84 | "source": [ 85 | "### The Frequentist World View\n", 86 | "\n", 87 | "![Fisher](images/fisher.png)\n", 88 | "\n", 89 | "- The **data** that have been observed are considered **random**, because they are realizations of random processes, and hence will vary each time one goes to observe the system.\n", 90 | "- Model **parameters** are considered **fixed**. A parameter's true value is uknown and fixed, and so we *condition* on them.\n", 91 | "\n", 92 | "In mathematical notation, this implies a (very) general model of the following form:\n", 93 | "\n", 94 | "
\n", 95 | "\\\\[f(y | \\theta)\\\\]\n", 96 | "
\n", 97 | "\n", 98 | "Here, the model \\\\(f\\\\) accepts data values \\\\(y\\\\) as an argument, conditional on particular values of \\\\(\\theta\\\\).\n", 99 | "\n", 100 | "Frequentist inference typically involves deriving **estimators** for the unknown parameters. Estimators are formulae that return estimates for particular estimands, as a function of data. They are selected based on some chosen optimality criterion, such as *unbiasedness*, *variance minimization*, or *efficiency*.\n", 101 | "\n", 102 | "> For example, lets say that we have collected some data on the prevalence of autism spectrum disorder (ASD) in some defined population. Our sample includes \\\\(n\\\\) sampled children, \\\\(y\\\\) of them having been diagnosed with autism. A frequentist estimator of the prevalence \\\\(p\\\\) is:\n", 103 | "\n", 104 | ">
\n", 105 | "> $$\\hat{p} = \\frac{y}{n}$$\n", 106 | ">
\n", 107 | "\n", 108 | "> Why this particular function? Because it can be shown to be unbiased and minimum-variance.\n", 109 | "\n", 110 | "It is important to note that, in a frequentist world, new estimators need to be derived for every estimand that is introduced." 111 | ] 112 | }, 113 | { 114 | "cell_type": "markdown", 115 | "metadata": {}, 116 | "source": [ 117 | "### The Bayesian World View\n", 118 | "\n", 119 | "![Bayes](images/bayes.png)\n", 120 | "\n", 121 | "- Data are considered **fixed**. They used to be random, but once they were written into your lab notebook/spreadsheet/IPython notebook they do not change.\n", 122 | "- Model parameters themselves may not be random, but Bayesians use probability distribtutions to describe their uncertainty in parameter values, and are therefore treated as **random**. In some cases, it is useful to consider parameters as having been sampled from probability distributions.\n", 123 | "\n", 124 | "This implies the following form:\n", 125 | "\n", 126 | "## $$p(\\theta | y)$$\n", 127 | "\n", 128 | "This formulation used to be referred to as ***inverse probability***, because it infers from observations to parameters, or from effects to causes.\n", 129 | "\n", 130 | "Bayesians do not seek new estimators for every estimation problem they encounter. There is only one estimator for Bayesian inference: **Bayes' Formula**." 131 | ] 132 | }, 133 | { 134 | "cell_type": "markdown", 135 | "metadata": {}, 136 | "source": [ 137 | "## Bayes' Formula\n", 138 | "\n", 139 | "Now that we have some probability under our belt, we turn to Bayes' formula. While frequentist statistics uses different estimators for different problems, Bayes formula is the **only estimator** that Bayesians need to obtain estimates of unknown quantities that we care about. \n", 140 | "\n", 141 | "![bayes formula](images/bayes_formula.png)\n", 142 | "\n", 143 | "The equation expresses how our belief about the value of \\\\(\\theta\\\\), as expressed by the **prior distribution** \\\\(P(\\theta)\\\\) is reallocated following the observation of the data \\\\(y\\\\).\n", 144 | "\n", 145 | "The innocuous denominator \\\\(P(y)\\\\) usuallt cannot be computed directly, and is actually the expression in the numerator, integrated over all \\\\(\\theta\\\\):\n", 146 | "\n", 147 | "
\n", 148 | "\\\\[Pr(\\theta|y) = \\frac{Pr(y|\\theta)Pr(\\theta)}{\\int Pr(y|\\theta)Pr(\\theta) d\\theta}\\\\]\n", 149 | "
\n", 150 | "\n", 151 | "The intractability of this integral is one of the factors that has contributed to the under-utilization of Bayesian methods by statisticians.\n", 152 | "\n", 153 | "### Priors\n", 154 | "\n", 155 | "Once considered a controversial aspect of Bayesian analysis, the prior distribution characterizes what is known about an unknown quantity before observing the data from the present study. Thus, it represents the information state of that parameter. It can be used to reflect the information obtained in previous studies, to constrain the parameter to plausible values, or to represent the population of possible parameter values, of which the current study's parameter value can be considered a sample.\n", 156 | "\n", 157 | "### Likelihood functions\n", 158 | "\n", 159 | "The likelihood represents the information in the observed data, and is used to update prior distributions to posterior distributions. This updating of belief is justified becuase of the **likelihood principle**, which states:\n", 160 | "\n", 161 | "> Following observation of \\\\(y\\\\), the likelihood \\\\(L(\\theta|y)\\\\) contains all experimental information from \\\\(y\\\\) about the unknown \\\\(\\theta\\\\).\n", 162 | "\n", 163 | "Bayesian analysis satisfies the likelihood principle because the posterior distribution's dependence on the data is **only through the likelihood**. In comparison, most frequentist inference procedures violate the likelihood principle, because inference will depend on the design of the trial or experiment.\n", 164 | "\n", 165 | "Remember from the density estimation section that the likelihood is closely related to the probability density (or mass) function. The difference is that the likelihood varies the parameter while holding the observations constant, rather than *vice versa*." 166 | ] 167 | }, 168 | { 169 | "cell_type": "markdown", 170 | "metadata": {}, 171 | "source": [ 172 | "## Bayesian Inference, in 3 Easy Steps\n", 173 | "\n", 174 | "![123](images/123.png)\n", 175 | "\n", 176 | "Gelman et al. (2013) describe the process of conducting Bayesian statistical analysis in 3 steps.\n", 177 | "\n", 178 | "### Step 1: Specify a probability model\n", 179 | "\n", 180 | "As was noted above, Bayesian statistics involves using probability models to solve problems. So, the first task is to *completely specify* the model in terms of probability distributions. This includes everything: unknown parameters, data, covariates, missing data, predictions. All must be assigned some probability density.\n", 181 | "\n", 182 | "This step involves making choices.\n", 183 | "\n", 184 | "- what is the form of the sampling distribution of the data?\n", 185 | "- what form best describes our uncertainty in the unknown parameters?" 186 | ] 187 | }, 188 | { 189 | "cell_type": "markdown", 190 | "metadata": {}, 191 | "source": [ 192 | "### Discrete Random Variables\n", 193 | "\n", 194 | "$$X = \\{0,1\\}$$\n", 195 | "\n", 196 | "$$Y = \\{\\ldots,-2,-1,0,1,2,\\ldots\\}$$\n", 197 | "\n", 198 | "**Probability Mass Function**: \n", 199 | "\n", 200 | "For discrete $X$,\n", 201 | "\n", 202 | "$$Pr(X=x) = f(x|\\theta)$$\n", 203 | "\n", 204 | "![Discrete variable](http://upload.wikimedia.org/wikipedia/commons/1/16/Poisson_pmf.svg)\n", 205 | "\n", 206 | "***e.g. Poisson distribution***\n", 207 | "\n", 208 | "The Poisson distribution models unbounded counts:\n", 209 | "\n", 210 | "
\n", 211 | "$$Pr(X=x)=\\frac{e^{-\\lambda}\\lambda^x}{x!}$$\n", 212 | "
\n", 213 | "\n", 214 | "* $X=\\{0,1,2,\\ldots\\}$\n", 215 | "* $\\lambda > 0$\n", 216 | "\n", 217 | "$$E(X) = \\text{Var}(X) = \\lambda$$" 218 | ] 219 | }, 220 | { 221 | "cell_type": "code", 222 | "execution_count": 1, 223 | "metadata": { 224 | "collapsed": true 225 | }, 226 | "outputs": [], 227 | "source": [ 228 | "%matplotlib inline\n", 229 | "import numpy as np\n", 230 | "import pandas as pd\n", 231 | "import matplotlib.pylab as plt\n", 232 | "import seaborn as sns\n", 233 | "sns.set_context('notebook')\n", 234 | "\n", 235 | "RANDOM_SEED = 20090425" 236 | ] 237 | }, 238 | { 239 | "cell_type": "code", 240 | "execution_count": 2, 241 | "metadata": { 242 | "collapsed": true 243 | }, 244 | "outputs": [], 245 | "source": [ 246 | "from pymc3 import Poisson\n", 247 | "\n", 248 | "x = Poisson.dist(mu=1)\n", 249 | "samples = x.random(size=10000)" 250 | ] 251 | }, 252 | { 253 | "cell_type": "code", 254 | "execution_count": 3, 255 | "metadata": {}, 256 | "outputs": [ 257 | { 258 | "data": { 259 | "text/plain": [ 260 | "0.9859" 261 | ] 262 | }, 263 | "execution_count": 3, 264 | "metadata": {}, 265 | "output_type": "execute_result" 266 | } 267 | ], 268 | "source": [ 269 | "samples.mean()" 270 | ] 271 | }, 272 | { 273 | "cell_type": "code", 274 | "execution_count": 4, 275 | "metadata": {}, 276 | "outputs": [ 277 | { 278 | "data": { 279 | "image/png": "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\n", 280 | "text/plain": [ 281 | "
" 282 | ] 283 | }, 284 | "metadata": {}, 285 | "output_type": "display_data" 286 | } 287 | ], 288 | "source": [ 289 | "plt.hist(samples, bins=len(set(samples)));" 290 | ] 291 | }, 292 | { 293 | "cell_type": "markdown", 294 | "metadata": {}, 295 | "source": [ 296 | "### Continuous Random Variables\n", 297 | "\n", 298 | "$$X \\in [0,1]$$\n", 299 | "\n", 300 | "$$Y \\in (-\\infty, \\infty)$$\n", 301 | "\n", 302 | "**Probability Density Function**: \n", 303 | "\n", 304 | "For continuous $X$,\n", 305 | "\n", 306 | "$$Pr(x \\le X \\le x + dx) = f(x|\\theta)dx \\, \\text{ as } \\, dx \\rightarrow 0$$\n", 307 | "\n", 308 | "![Continuous variable](https://upload.wikimedia.org/wikipedia/commons/7/74/Normal_Distribution_PDF.svg)" 309 | ] 310 | }, 311 | { 312 | "cell_type": "markdown", 313 | "metadata": {}, 314 | "source": [ 315 | "***e.g. normal distribution***\n", 316 | "\n", 317 | "
\n", 318 | "$$f(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right]$$\n", 319 | "
\n", 320 | "\n", 321 | "* $X \\in \\mathbf{R}$\n", 322 | "* $\\mu \\in \\mathbf{R}$\n", 323 | "* $\\sigma>0$\n", 324 | "\n", 325 | "$$\\begin{align}\n", 326 | "E(X) &= \\mu \\cr\n", 327 | "\\text{Var}(X) &= \\sigma^2 \n", 328 | "\\end{align}$$" 329 | ] 330 | }, 331 | { 332 | "cell_type": "code", 333 | "execution_count": 5, 334 | "metadata": { 335 | "collapsed": true 336 | }, 337 | "outputs": [], 338 | "source": [ 339 | "from pymc3 import Normal\n", 340 | "\n", 341 | "y = Normal.dist(mu=-2, sd=4)\n", 342 | "samples = y.random(size=10000)" 343 | ] 344 | }, 345 | { 346 | "cell_type": "code", 347 | "execution_count": 6, 348 | "metadata": {}, 349 | "outputs": [ 350 | { 351 | "data": { 352 | "text/plain": [ 353 | "-2.0303670533348503" 354 | ] 355 | }, 356 | "execution_count": 6, 357 | "metadata": {}, 358 | "output_type": "execute_result" 359 | } 360 | ], 361 | "source": [ 362 | "samples.mean()" 363 | ] 364 | }, 365 | { 366 | "cell_type": "code", 367 | "execution_count": 7, 368 | "metadata": {}, 369 | "outputs": [ 370 | { 371 | "data": { 372 | "text/plain": [ 373 | "3.972885354039681" 374 | ] 375 | }, 376 | "execution_count": 7, 377 | "metadata": {}, 378 | "output_type": "execute_result" 379 | } 380 | ], 381 | "source": [ 382 | "samples.std()" 383 | ] 384 | }, 385 | { 386 | "cell_type": "code", 387 | "execution_count": 8, 388 | "metadata": {}, 389 | "outputs": [ 390 | { 391 | "data": { 392 | "image/png": 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393 | "text/plain": [ 394 | "
" 395 | ] 396 | }, 397 | "metadata": {}, 398 | "output_type": "display_data" 399 | } 400 | ], 401 | "source": [ 402 | "plt.hist(samples);" 403 | ] 404 | }, 405 | { 406 | "cell_type": "markdown", 407 | "metadata": {}, 408 | "source": [ 409 | "### Step 2: Calculate a posterior distribution\n", 410 | "\n", 411 | "The mathematical form \\\\(p(\\theta | y)\\\\) that we associated with the Bayesian approach is referred to as a **posterior distribution**.\n", 412 | "\n", 413 | "> posterior /pos·ter·i·or/ (pos-tēr´e-er) later in time; subsequent.\n", 414 | "\n", 415 | "Why posterior? Because it tells us what we know about the unknown \\\\(\\theta\\\\) *after* having observed \\\\(y\\\\).\n", 416 | "\n", 417 | "This posterior distribution is formulated as a function of the probability model that was specified in Step 1. Usually, we can write it down but we cannot calculate it analytically. In fact, the difficulty inherent in calculating the posterior distribution for most models of interest is perhaps the major contributing factor for the lack of widespread adoption of Bayesian methods for data analysis. Various strategies for doing so comprise this tutorial.\n", 418 | "\n", 419 | "**But**, once the posterior distribution is calculated, you get a lot for free:\n", 420 | "\n", 421 | "- point estimates\n", 422 | "- credible intervals\n", 423 | "- quantiles\n", 424 | "- predictions\n", 425 | "\n", 426 | "### Step 3: Check your model\n", 427 | "\n", 428 | "Though frequently ignored in practice, it is critical that the model and its outputs be assessed before using the outputs for inference. Models are specified based on assumptions that are largely unverifiable, so the least we can do is examine the output in detail, relative to the specified model and the data that were used to fit the model.\n", 429 | "\n", 430 | "Specifically, we must ask:\n", 431 | "\n", 432 | "- does the model fit data?\n", 433 | "- are the conclusions reasonable?\n", 434 | "- are the outputs sensitive to changes in model structure?\n", 435 | "\n" 436 | ] 437 | }, 438 | { 439 | "cell_type": "markdown", 440 | "metadata": {}, 441 | "source": [ 442 | "## References and Resources\n", 443 | "\n", 444 | "- Goodman, S. N. (1999). Toward evidence-based medical statistics. 1: The P value fallacy. Annals of Internal Medicine, 130(12), 995–1004. http://doi.org/10.7326/0003-4819-130-12-199906150-00008\n", 445 | "- Johnson, D. (1999). The insignificance of statistical significance testing. Journal of Wildlife Management, 63(3), 763–772.\n", 446 | "- Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis, Third Edition. CRC Press." 447 | ] 448 | } 449 | ], 450 | "metadata": { 451 | "kernelspec": { 452 | "display_name": "Python 3", 453 | "language": "python", 454 | "name": "python3" 455 | }, 456 | "language_info": { 457 | "codemirror_mode": { 458 | "name": "ipython", 459 | "version": 3 460 | }, 461 | "file_extension": ".py", 462 | "mimetype": "text/x-python", 463 | "name": "python", 464 | "nbconvert_exporter": "python", 465 | "pygments_lexer": "ipython3", 466 | "version": "3.6.5" 467 | } 468 | }, 469 | "nbformat": 4, 470 | "nbformat_minor": 2 471 | } 472 | -------------------------------------------------------------------------------- /notebooks/2. Markov Chain Monte Carlo.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Markov chain Monte Carlo\n", 8 | "\n", 9 | "Let's briefly cover some theory regarding Bayesian analysis using Markov chain Monte Carlo (MCMC) methods. You might wonder why a numerical simulation method like MCMC is the standard approach for fitting Bayesian models. \n", 10 | "\n", 11 | "Gelman et al. (2013) break down the business of Bayesian analysis into three primary steps:\n", 12 | "\n", 13 | "1. Specify a full probability model, including all parameters, data, transformations, missing values and predictions that are of interest.\n", 14 | "2. Calculate the posterior distribution of the unknown quantities in the model, conditional on the data.\n", 15 | "3. Perform model checking to evaluate the quality and suitablility of the model.\n", 16 | "\n", 17 | "While each of these steps is challenging, it is the second step that is the most difficult for non-trivial models, and was a bottleneck for the adoption of Bayesian methods for decades. \n", 18 | "\n", 19 | "### Bayesian Inference\n", 20 | "\n", 21 | "At this point, we should all be familiar with **Bayes Formula**:\n", 22 | "\n", 23 | "![bayes formula](images/bayes_formula.png)\n", 24 | "\n", 25 | "The equation expresses how our belief about the value of $\\theta$, as expressed by the **prior distribution** $P(\\theta)$ is reallocated following the observation of the data $y$, as expressed by the posterior distribution the posterior distribution.\n", 26 | "\n", 27 | "Computing the posterior distribution is called the **inference problem**, and is usually the goal of Bayesian analysis.\n", 28 | "\n", 29 | "The innocuous denominator $P(y)$ (the model **evidence**, or **marginal likelihood**) cannot be calculated directly, and is actually the expression in the numerator, integrated over all $\\theta$:\n", 30 | "\n", 31 | "
\n", 32 | "$$Pr(\\theta|y) = \\frac{Pr(y|\\theta)Pr(\\theta)}{\\int Pr(y|\\theta)Pr(\\theta) d\\theta}$$\n", 33 | "
\n", 34 | "\n", 35 | "Computing this integral, which may involve many variables, is generally intractible with analytic methods. This is the major compuational hurdle for Bayesian analysis.\n", 36 | "\n", 37 | "### Simulation Approaches for Bayesian Computation\n", 38 | "\n", 39 | "Since analysis is off the table, a reasonable alternative is to attempt to estimate the integral using numerical methods. For example, consider the expected value of a random variable $\\mathbf{x}$:\n", 40 | "\n", 41 | "$$E[{\\bf x}] = \\int {\\bf x} f({\\bf x}) d{\\bf x}, \\qquad\n", 42 | "{\\bf x} = \\{x_1,...,x_k\\}$$\n", 43 | "\n", 44 | "where $k$ (the dimension of vector $x$) is perhaps very large. If we can produce a reasonable number of random vectors $\\{{\\bf x_i}\\}$, we can use these values to approximate the unknown integral. This process is known as *Monte Carlo integration*. In general, MC integration allows integrals against probability density functions:\n", 45 | "\n", 46 | "$$I = \\int h(\\mathbf{x}) f(\\mathbf{x}) \\mathbf{dx}$$\n", 47 | "\n", 48 | "to be estimated by finite sums:\n", 49 | "\n", 50 | "$$\\hat{I} = \\frac{1}{n}\\sum_{i=1}^n h(\\mathbf{x}_i),$$\n", 51 | "\n", 52 | "where $\\mathbf{x}_i$ is a sample from $f$. This estimate is valid and useful because:\n", 53 | "\n", 54 | "- By the strong law of large numbers:\n", 55 | "\n", 56 | "$$\\hat{I} \\rightarrow I \\,\\, \\text{with probability 1}$$\n", 57 | "\n", 58 | "- Simulation error can be measured and controlled:\n", 59 | "\n", 60 | "$$Var(\\hat{I}) = \\frac{1}{n(n-1)}\\sum_{i=1}^n (h(\\mathbf{x}_i)-\\hat{I})^2$$" 61 | ] 62 | }, 63 | { 64 | "cell_type": "markdown", 65 | "metadata": {}, 66 | "source": [ 67 | "### How is this relevant to Bayesian analysis? \n", 68 | "\n", 69 | "When we observe data $y$ that we hypothesize as being obtained from a sampling model $f(y|\\theta)$, where $\\theta$ is a vector of (unknown) model parameters, a Bayesian places a *prior* distribution $p(\\theta)$ on the parameters to describe the uncertainty in the true values of the parameters. Bayesian inference, then, is obtained by calculating the *posterior* distribution, which is proportional to the product of these quantities:\n", 70 | "\n", 71 | "$$p(\\theta | y) \\propto f(y|\\theta) p(\\theta)$$\n", 72 | "\n", 73 | "unfortunately, for most problems of interest, the normalizing constant cannot be calculated because it involves mutli-dimensional integration over $\\theta$.\n", 74 | "\n", 75 | "Returning to our integral for MC sampling, if we replace $f(\\mathbf{x})$\n", 76 | "with a posterior, $p(\\theta|y)$ and make $h(\\theta)$ an interesting function of the unknown parameter, the resulting expectation is that of the posterior of $h(\\theta)$:\n", 77 | "\n", 78 | "$$E[h(\\theta)|y] = \\int h(\\theta) p(\\theta|y) d\\theta \\approx \\frac{1}{n}\\sum_{i=1}^n h(\\theta)$$\n", 79 | "\n", 80 | "We also require integrals to obtain marginal estimates from a joint model. If $\\theta$ is of length $K$, then inference about any particular parameter is obtained by:\n", 81 | "\n", 82 | "$$p(\\theta_i|y) \\propto \\int p(\\theta|y) d\\theta_{-i}$$\n", 83 | "\n", 84 | "where the `-i` subscript indicates all elements except the $i^{th}$." 85 | ] 86 | }, 87 | { 88 | "cell_type": "markdown", 89 | "metadata": {}, 90 | "source": [ 91 | "### Sampling Markov Chains\n", 92 | "\n", 93 | "The expectation above assumes that the draws of $\\theta$ are **independent**. The limitation in using Monte Carlo sampling for Bayesian inference is that it is not usually feasible to make independent draws from the posterior distribution. \n", 94 | "\n", 95 | "The first \"MC\" in MCMC stands for **Markov chain**. A Markov chain is a **stochastic process**, an indexed set of random variables, where the value of a particular random variable in the set is dependent only on the random variable corresponding to the prevous index. This is a Markovian dependence structure:\n", 96 | "\n", 97 | "$$Pr(X_{t+1}=x_{t+1} | X_t=x_t, X_{t-1}=x_{t-1},\\ldots,X_0=x_0) = Pr(X_{t+1}=x_{t+1} | X_t=x_t)$$\n", 98 | "\n", 99 | "This conditioning specifies that the future depends on the current state, but not past states. Thus, the Markov chain wanders about the state space, remembering only where it has just been in the last time step. The collection of transition probabilities is sometimes called a *transition matrix* when dealing with discrete states, or more generally, a *transition kernel*.\n", 100 | "\n", 101 | "MCMC allows us to generate samples from a particular posterior distribution as a Markov chain. The magic is that the resulting sample, even though it is dependent in this way, is indistinguishable from an independent sample from the true posterior." 102 | ] 103 | }, 104 | { 105 | "cell_type": "markdown", 106 | "metadata": {}, 107 | "source": [ 108 | "## Why MCMC Works: Reversible Markov Chains\n", 109 | "\n", 110 | "Markov chain Monte Carlo simulates a Markov chain for which some function of interest\n", 111 | "(*e.g.* the joint distribution of the parameters of some model) is the unique, invariant limiting distribution. An invariant distribution with respect to some Markov chain with transition kernel $Pr(y \\mid x)$ implies that:\n", 112 | "\n", 113 | "$$\\int_x Pr(y \\mid x) \\pi(x) dx = \\pi(y).$$\n", 114 | "\n", 115 | "Invariance is guaranteed for any *reversible* Markov chain. Consider a Markov chain in reverse sequence:\n", 116 | "$\\{\\theta^{(n)},\\theta^{(n-1)},...,\\theta^{(0)}\\}$. This sequence is still Markovian, because:\n", 117 | "\n", 118 | "$$Pr(\\theta^{(k)}=y \\mid \\theta^{(k+1)}=x,\\theta^{(k+2)}=x_1,\\ldots ) = Pr(\\theta^{(k)}=y \\mid \\theta^{(k+1)}=x)$$\n", 119 | "\n", 120 | "Forward and reverse transition probabilities may be related through Bayes theorem:\n", 121 | "\n", 122 | "$$\\frac{Pr(\\theta^{(k+1)}=x \\mid \\theta^{(k)}=y) \\pi^{(k)}(y)}{\\pi^{(k+1)}(x)}$$\n", 123 | "\n", 124 | "Though not homogeneous in general, $\\pi$ becomes homogeneous if:\n", 125 | "\n", 126 | "- $n \\rightarrow \\infty$\n", 127 | "\n", 128 | "- $\\pi^{(i)}=\\pi$ for some $i < k$\n", 129 | "\n", 130 | "If this chain is homogeneous it is called reversible, because it satisfies the ***detailed balance equation***:\n", 131 | "\n", 132 | "$$\\pi(x)Pr(y \\mid x) = \\pi(y) Pr(x \\mid y)$$\n", 133 | "\n", 134 | "Reversibility is important because it has the effect of balancing movement through the entire state space. When a Markov chain is reversible, $\\pi$ is the unique, invariant, stationary distribution of that chain. Hence, if $\\pi$ is of interest, we need only find the reversible Markov chain for which $\\pi$ is the limiting distribution.\n", 135 | "This is what MCMC does!" 136 | ] 137 | }, 138 | { 139 | "cell_type": "markdown", 140 | "metadata": {}, 141 | "source": [ 142 | "## The Metropolis-Hastings Algorithm\n", 143 | "\n", 144 | "One of the simplest and most flexible MCMC algorithms is the Metropolis-Hastings sampler. This algorithm generates candidate state transitions from an auxilliary distribution, and accepts or rejects each candidate probabilistically, according to the posterior distribution of the model.\n", 145 | "\n", 146 | "Let us first consider a simple Metropolis-Hastings algorithm for a single parameter, $\\theta$. We will use a standard sampling distribution, referred to as the *proposal distribution*, to produce candidate variables $q_t(\\theta^{\\prime} | \\theta)$. That is, the generated value, $\\theta^{\\prime}$, is a *possible* next value for\n", 147 | "$\\theta$ at step $t+1$. We also need to be able to calculate the probability of moving back to the original value from the candidate, or\n", 148 | "$q_t(\\theta | \\theta^{\\prime})$. These probabilistic ingredients are used to define an *acceptance ratio*:\n", 149 | "\n", 150 | "$$a(\\theta^{\\prime},\\theta) = \\frac{q_t(\\theta^{\\prime} | \\theta) \\pi(\\theta^{\\prime})}{q_t(\\theta | \\theta^{\\prime}) \\pi(\\theta)}$$\n", 151 | "\n", 152 | "The value of $\\theta^{(t+1)}$ is then determined by:\n", 153 | "\n", 154 | "$$\\theta^{(t+1)} = \\left\\{\\begin{array}{l@{\\quad \\mbox{with prob.} \\quad}l}\\theta^{\\prime} & \\min(a(\\theta^{\\prime},\\theta^{(t)}),1) \\\\ \\theta^{(t)} & 1 - \\min(a(\\theta^{\\prime},\\theta^{(t)}),1) \\end{array}\\right.$$\n", 155 | "\n", 156 | "This transition kernel implies that movement is not guaranteed at every step. It only occurs if the suggested transition is likely based on the acceptance ratio.\n", 157 | "\n", 158 | "A single iteration of the Metropolis-Hastings algorithm proceeds as follows:\n", 159 | "\n", 160 | "The original form of the algorithm specified by Metropolis required that\n", 161 | "$q_t(\\theta^{\\prime} | \\theta) = q_t(\\theta | \\theta^{\\prime})$, which reduces $a(\\theta^{\\prime},\\theta)$ to\n", 162 | "$\\pi(\\theta^{\\prime})/\\pi(\\theta)$, but this is not necessary. In either case, the state moves to high-density points in the distribution with high probability, and to low-density points with low probability. After convergence, the Metropolis-Hastings algorithm describes the full target posterior density, so all points are recurrent.\n", 163 | "\n", 164 | "1. Sample $\\theta^{\\prime}$ from $q(\\theta^{\\prime} | \\theta^{(t)})$.\n", 165 | "\n", 166 | "2. Generate a Uniform[0,1] random variate $u$.\n", 167 | "\n", 168 | "3. If $a(\\theta^{\\prime},\\theta) > u$ then\n", 169 | " $\\theta^{(t+1)} = \\theta^{\\prime}$, otherwise\n", 170 | " $\\theta^{(t+1)} = \\theta^{(t)}$.\n", 171 | "\n", 172 | "### Random-walk Metropolis-Hastings\n", 173 | "\n", 174 | "A practical implementation of the Metropolis-Hastings algorithm makes use of a random-walk proposal.\n", 175 | "Recall that a random walk is a Markov chain that evolves according to:\n", 176 | "\n", 177 | "$$\n", 178 | "\\theta^{(t+1)} = \\theta^{(t)} + \\epsilon_t \\\\\n", 179 | "\\epsilon_t \\sim f(\\phi)\n", 180 | "$$\n", 181 | "\n", 182 | "As applied to the MCMC sampling, the random walk is used as a proposal distribution, whereby dependent proposals are generated according to:\n", 183 | "\n", 184 | "$$q(\\theta^{\\prime} | \\theta^{(t)}) = f(\\theta^{\\prime} - \\theta^{(t)}) = \\theta^{(t)} + \\epsilon_t$$\n", 185 | "\n", 186 | "Generally, the density generating $\\epsilon_t$ is symmetric about zero,\n", 187 | "resulting in a symmetric chain. Chain symmetry implies that\n", 188 | "$q(\\theta^{\\prime} | \\theta^{(t)}) = q(\\theta^{(t)} | \\theta^{\\prime})$,\n", 189 | "which reduces the Metropolis-Hastings acceptance ratio to:\n", 190 | "\n", 191 | "$$a(\\theta^{\\prime},\\theta) = \\frac{\\pi(\\theta^{\\prime})}{\\pi(\\theta)}$$\n", 192 | "\n", 193 | "The choice of the random walk distribution for $\\epsilon_t$ is frequently a normal or Student’s $t$ density, but it may be any distribution that generates an irreducible proposal chain.\n", 194 | "\n", 195 | "An important consideration is the specification of the scale parameter for the random walk error distribution. Large values produce random walk steps that are highly exploratory, but tend to produce proposal values in the tails of the target distribution, potentially resulting in very small acceptance rates. Conversely, small values tend to be accepted more frequently, since they tend to produce proposals close to the current parameter value, but may result in chains that mix very slowly.\n", 196 | "Some simulation studies suggest optimal acceptance rates in the range of 20-50%. It is often worthwhile to optimize the proposal variance by iteratively adjusting its value, according to observed acceptance rates early in the MCMC simulation ." 197 | ] 198 | }, 199 | { 200 | "cell_type": "markdown", 201 | "metadata": {}, 202 | "source": [ 203 | "# Hamiltonian Monte Carlo" 204 | ] 205 | }, 206 | { 207 | "cell_type": "markdown", 208 | "metadata": {}, 209 | "source": [ 210 | "While flexible and easy to implement, Metropolis-Hastings sampling is a random walk\n", 211 | "sampler that might not be statistically efficient for many models. In\n", 212 | "this context, and when sampling from continuous variables, Hamiltonian (or Hybrid) Monte\n", 213 | "Carlo (HMC) can prove to be a powerful tool. It avoids\n", 214 | "random walk behavior by simulating a physical system governed by\n", 215 | "Hamiltonian dynamics, potentially avoiding tricky conditional\n", 216 | "distributions in the process.\n", 217 | "\n", 218 | "![hmc comparison](images/hmc.png)\n", 219 | "\n", 220 | "In HMC, model samples are obtained by simulating a physical system,\n", 221 | "where particles move about a high-dimensional landscape, subject to\n", 222 | "potential and kinetic energies. Adapting the notation from [Neal (1993)](http://www.cs.toronto.edu/~radford/review.abstract.html),\n", 223 | "particles are characterized by a position vector or state\n", 224 | "$s \\in \\mathcal{R}^D$ and velocity vector $\\phi \\in \\mathcal{R}^D$. The\n", 225 | "combined state of a particle is denoted as $\\chi=(s,\\phi)$. The\n", 226 | "Hamiltonian is then defined as the sum of potential energy $E(s)$ and kinetic energy\n", 227 | "$K(\\phi)$, as follows:\n", 228 | "\n", 229 | "$$\\mathcal{H}(s,\\phi) = E(s) + K(\\phi)\n", 230 | "= E(s) + \\frac{1}{2} \\sum_i \\phi_i^2$$\n", 231 | "\n", 232 | "Instead of sampling $p(s)$ directly, HMC operates by sampling from the\n", 233 | "canonical distribution\n", 234 | "$p(s,\\phi) = \\frac{1}{Z} \\exp(-\\mathcal{H}(s,\\phi))=p(s)p(\\phi)$.\n", 235 | "Because the two variables are independent, marginalizing over $\\phi$ is\n", 236 | "trivial and recovers the original distribution of interest.\n", 237 | "\n", 238 | "**Hamiltonian Dynamics**\n", 239 | "\n", 240 | "State $s$ and velocity $\\phi$ are modified such that\n", 241 | "$\\mathcal{H}(s,\\phi)$ remains constant throughout the simulation. The\n", 242 | "differential equations are given by:\n", 243 | "\n", 244 | "$$\\begin{aligned}\\frac{ds_i}{dt} &= \\frac{\\partial \\mathcal{H}}{\\partial \\phi_i} = \\phi_i \\\\\n", 245 | "\\frac{d\\phi_i}{dt} &= - \\frac{\\partial \\mathcal{H}}{\\partial s_i}\n", 246 | "= - \\frac{\\partial E}{\\partial s_i}\n", 247 | "\\end{aligned}$$\n", 248 | "\n", 249 | "As shown in [Neal (1993)](http://www.cs.toronto.edu/~radford/review.abstract.html), \n", 250 | "the above transformation preserves volume and is\n", 251 | "reversible. The above dynamics can thus be used as transition operators\n", 252 | "of a Markov chain and will leave $p(s,\\phi)$ invariant. That chain by\n", 253 | "itself is not ergodic however, since simulating the dynamics maintains a\n", 254 | "fixed Hamiltonian $\\mathcal{H}(s,\\phi)$. HMC thus alternates Hamiltonian\n", 255 | "dynamic steps, with Gibbs sampling of the velocity. Because $p(s)$ and\n", 256 | "$p(\\phi)$ are independent, sampling $\\phi_{new} \\sim p(\\phi|s)$ is\n", 257 | "trivial since $p(\\phi|s)=p(\\phi)$, where $p(\\phi)$ is often taken to be\n", 258 | "the univariate Gaussian.\n", 259 | "\n", 260 | "**The Leap-Frog Algorithm**\n", 261 | "\n", 262 | "In practice, we cannot simulate Hamiltonian dynamics exactly because of\n", 263 | "the problem of time discretization. There are several ways one can do\n", 264 | "this. To maintain invariance of the Markov chain however, care must be\n", 265 | "taken to preserve the properties of *volume conservation* and *time\n", 266 | "reversibility*. The **leap-frog algorithm** maintains these properties\n", 267 | "and operates in 3 steps:\n", 268 | "\n", 269 | "$$\\begin{aligned}\n", 270 | "\\phi_i(t + \\epsilon/2) &= \\phi_i(t) - \\frac{\\epsilon}{2} \\frac{\\partial{}}{\\partial s_i} E(s(t)) \\\\\n", 271 | "s_i(t + \\epsilon) &= s_i(t) + \\epsilon \\phi_i(t + \\epsilon/2) \\\\\n", 272 | "\\phi_i(t + \\epsilon) &= \\phi_i(t + \\epsilon/2) - \\frac{\\epsilon}{2} \\frac{\\partial{}}{\\partial s_i} E(s(t + \\epsilon)) \n", 273 | "\\end{aligned}$$\n", 274 | "\n", 275 | "We thus perform a half-step update of the velocity at time\n", 276 | "$t+\\epsilon/2$, which is then used to compute $s(t + \\epsilon)$ and\n", 277 | "$\\phi(t + \\epsilon)$.\n", 278 | "\n", 279 | "**Accept / Reject**\n", 280 | "\n", 281 | "In practice, using finite stepsizes $\\epsilon$ will not preserve\n", 282 | "$\\mathcal{H}(s,\\phi)$ exactly and will introduce bias in the simulation.\n", 283 | "Also, rounding errors due to the use of floating point numbers means\n", 284 | "that the above transformation will not be perfectly reversible.\n", 285 | "\n", 286 | "HMC cancels these effects **exactly** by adding a Metropolis\n", 287 | "accept/reject stage, after $n$ leapfrog steps. The new state\n", 288 | "$\\chi' = (s',\\phi')$ is accepted with probability $p_{acc}(\\chi,\\chi')$,\n", 289 | "defined as:\n", 290 | "\n", 291 | "$$p_{acc}(\\chi,\\chi') = min \\left( 1, \\frac{\\exp(-\\mathcal{H}(s',\\phi')}{\\exp(-\\mathcal{H}(s,\\phi)} \\right)$$\n", 292 | "\n", 293 | "**HMC Algorithm**\n", 294 | "\n", 295 | "We obtain a new HMC sample as follows:\n", 296 | "\n", 297 | "1. sample a new velocity from a univariate Gaussian distribution\n", 298 | "2. perform $n$ leapfrog steps to obtain the new state $\\chi'$\n", 299 | "3. perform accept/reject move of $\\chi'$" 300 | ] 301 | }, 302 | { 303 | "cell_type": "markdown", 304 | "metadata": {}, 305 | "source": [ 306 | "## No U-Turn Sampling\n", 307 | "\n", 308 | "The major drawback of the HMC algorithm is the extensive tuning required to make it sample efficiency. There are a handful of parameters that require specification by the user:\n", 309 | "\n", 310 | "- the scaling of the momentum distribution\n", 311 | "- the step size forthe leapfrog algorithm\n", 312 | "- the number of steps to be taken for the leapfrog algorithm\n", 313 | "\n", 314 | "When these parameters are poorly-chosen, the HMC algorithm can suffer severe losses in efficiency. For example, if we take steps that are too short, the simulation becomes a random walk, while steps that are too long end up retracing paths already taken.\n", 315 | "\n", 316 | "An efficient MCMC algorithm seeks to optimize mixing, while maintaining detailed balance. While HMC can be tuned on-the-fly, it requires costly burn-in runs to do so.\n", 317 | "\n", 318 | "![nuts](images/nuts.png)\n", 319 | "\n", 320 | "The No U-turn Sampling (NUTS) algorithm automatically tunes the step size and step number parameters, without any intervention from the user. To do so, NUTS constructs a binary tree of leapfrog steps by repeated doubling. When the trajectory of steps creates an angle of more than 90 degrees (*i.e.* a u-turn), the doubling stops, and a point is proposed.\n", 321 | "\n", 322 | "![binary doubling](images/binary_doubling.png)\n", 323 | "\n", 324 | "NUTS provides the efficiency of gradient-based MCMC sampling without extensive user intervention required to tune Hamiltonian Monte Carlo. As the result, NUTS is the default sampling algorithm for continuous variables in PyMC3." 325 | ] 326 | }, 327 | { 328 | "cell_type": "markdown", 329 | "metadata": {}, 330 | "source": [ 331 | "## References\n", 332 | "\n", 333 | "1. [Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013)](http://www.stat.columbia.edu/~gelman/book/). Bayesian Data Analysis. Chapman &Hall/CRC Press, London, third edition.\n", 334 | "2. [Geyer, C. (2013)](http://www.mcmchandbook.net/HandbookChapter1.pdf) Introduction to Markov Chain Monte Carlo. In *Handbook of Markov Chain Monte Carlo*, S. Brooks, A. Gelman, G. Jones, X.L. Meng, eds. CRC Press.\n", 335 | "3. [Neal, R.M. (1993)](http://www.cs.toronto.edu/~radford/review.abstract.html) Probabilistic Inference Using Markov Chain Monte Carlo Methods, Technical Report CRG-TR-93-1, Dept. of Computer Science, University of Toronto, 144 pages.\n" 336 | ] 337 | } 338 | ], 339 | "metadata": { 340 | "anaconda-cloud": {}, 341 | "kernelspec": { 342 | "display_name": "Python 3", 343 | "language": "python", 344 | "name": "python3" 345 | }, 346 | "language_info": { 347 | "codemirror_mode": { 348 | "name": "ipython", 349 | "version": 3 350 | }, 351 | "file_extension": ".py", 352 | "mimetype": "text/x-python", 353 | "name": "python", 354 | "nbconvert_exporter": "python", 355 | "pygments_lexer": "ipython3", 356 | "version": "3.6.5" 357 | }, 358 | "latex_envs": { 359 | "bibliofile": "biblio.bib", 360 | "cite_by": "apalike", 361 | "current_citInitial": 1, 362 | "eqLabelWithNumbers": true, 363 | "eqNumInitial": 0 364 | } 365 | }, 366 | "nbformat": 4, 367 | "nbformat_minor": 2 368 | } 369 | -------------------------------------------------------------------------------- /notebooks/images/123.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/123.png -------------------------------------------------------------------------------- /notebooks/images/bayes.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/bayes.png -------------------------------------------------------------------------------- /notebooks/images/bayes_formula.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/bayes_formula.png -------------------------------------------------------------------------------- /notebooks/images/binary_doubling.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/binary_doubling.png -------------------------------------------------------------------------------- /notebooks/images/f.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/f.png -------------------------------------------------------------------------------- /notebooks/images/fisher.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/fisher.png -------------------------------------------------------------------------------- /notebooks/images/hmc.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/hmc.png -------------------------------------------------------------------------------- /notebooks/images/ivh.gif: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/ivh.gif -------------------------------------------------------------------------------- /notebooks/images/nuts.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/nuts.png -------------------------------------------------------------------------------- /notebooks/images/prob_model.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/prob_model.png -------------------------------------------------------------------------------- /notebooks/images/radon_entry.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/radon_entry.jpg -------------------------------------------------------------------------------- /notebooks/images/test_stats.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/fonnesbeck/PyMC3_EUSS/09d519ca61d969c43aa2166ce19249f819a26e84/notebooks/images/test_stats.png --------------------------------------------------------------------------------