book

Bayesian Analysis with Python

Name: Bayesian Analysis with Python
Author: Osvaldo Martin
ISBN: 9781785883804

by Osvaldo Martin

November 2016

Beginner to intermediate

282 pages

6h 58m

English

Packt Publishing

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Bayesian Analysis with Python
Table of Contents
Bayesian Analysis with Python
Credits
About the Author
About the Reviewer
www.PacktPub.com
eBooks, discount offers, and moreWhy subscribe?
Preface
What this book covers
What you need for this book
Who this book is for

Conventions
Reader feedback
Customer support
Downloading the example codeDownloading the color images of this bookErrataPiracyQuestions
1. Thinking Probabilistically - A Bayesian Inference Primer
Statistics as a form of modelingExploratory data analysisInferential statistics
Probabilities and uncertainty
Probability distributionsBayes' theorem and statistical inference
Single parameter inference
The coin-flipping problemThe general modelChoosing the likelihoodChoosing the priorGetting the posteriorComputing and plotting the posteriorInfluence of the prior and how to choose oneCommunicating a Bayesian analysisModel notation and visualizationSummarizing the posteriorHighest posterior density
Posterior predictive checks
Installing the necessary Python packages
Summary
Exercises
2. Programming Probabilistically – A PyMC3 Primer
Probabilistic programmingInference enginesNon-Markovian methodsGrid computingQuadratic methodVariational methodsMarkovian methodsMonte CarloMarkov chainMetropolis-HastingsHamiltonian Monte Carlo/NUTSOther MCMC methods
PyMC3 introduction
Coin-flipping, the computational approachModel specificationPushing the inference buttonDiagnosing the sampling processConvergenceAutocorrelationEffective size
Summarizing the posterior
Posterior-based decisionsROPELoss functions
Summary
Keep reading
Exercises
3. Juggling with Multi-Parametric and Hierarchical Models
Nuisance parameters and marginalized distributions
Gaussians, Gaussians, Gaussians everywhere
Gaussian inferencesRobust inferencesStudent's t-distribution
Comparing groups
The tips datasetCohen's dProbability of superiority
Hierarchical models
Shrinkage
Summary
Keep reading
Exercises
4. Understanding and Predicting Data with Linear Regression Models
Simple linear regressionThe machine learning connectionThe core of linear regression modelsLinear models and high autocorrelationModifying the data before runningChanging the sampling methodInterpreting and visualizing the posteriorPearson correlation coefficientPearson coefficient from a multivariate Gaussian
Robust linear regression
Hierarchical linear regression
Correlation, causation, and the messiness of life
Polynomial regression
Interpreting the parameters of a polynomial regressionPolynomial regression – the ultimate model?
Multiple linear regression
Confounding variables and redundant variablesMulticollinearity or when the correlation is too highMasking effect variablesAdding interactions
The GLM module
Summary
Keep reading
Exercises
5. Classifying Outcomes with Logistic Regression
Logistic regressionThe logistic modelThe iris datasetThe logistic model applied to the iris datasetMaking predictions
Multiple logistic regression
The boundary decisionImplementing the modelDealing with correlated variablesDealing with unbalanced classesHow do we solve this problem?Interpreting the coefficients of a logistic regressionGeneralized linear modelsSoftmax regression or multinomial logistic regression
Discriminative and generative models
Summary
Keep reading
Exercises
6. Model Comparison
Occam's razor – simplicity and accuracyToo many parameters leads to overfittingToo few parameters leads to underfittingThe balance between simplicity and accuracy
Regularizing priors
Regularizing priors and hierarchical models
Predictive accuracy measures
Cross-validationInformation criteriaThe log-likelihood and the devianceAkaike information criterionDeviance information criterionWidely available information criterionPareto smoothed importance sampling leave-one-out cross-validationBayesian information criterionComputing information criteria with PyMC3A note on the reliability of WAIC and LOO computationsInterpreting and using information criteria measuresPosterior predictive checks
Bayes factors
Analogy with information criteriaComputing Bayes factorsCommon problems computing Bayes factors
Bayes factors and information criteria
Summary
Keep reading
Exercises
7. Mixture Models
Mixture modelsHow to build mixture modelsMarginalized Gaussian mixture modelMixture models and count dataThe Poisson distributionThe Zero-Inflated Poisson modelPoisson regression and ZIP regressionRobust logistic regression
Model-based clustering
Fixed component clusteringNon-fixed component clustering
Continuous mixtures
Beta-binomial and negative binomialThe Student's t-distribution
Summary
Keep reading
Exercises
8. Gaussian Processes
Non-parametric statistics
Kernel-based models
The Gaussian kernelKernelized linear regressionOverfitting and priors
Gaussian processes
Building the covariance matrixSampling from a GP priorUsing a parameterized kernelMaking predictions from a GPImplementing a GP using PyMC3Posterior predictive checksPeriodic kernel
Summary
Keep reading
Exercises
Index

Content preview from Bayesian Analysis with Python

Exercises

In the kernelized regression example, try changing the number of knots and the bandwidth (one at a time). What is the effect of those changes? Try also using a single knot; what do you observe?
Experiment with fitting other functions using kernelized regression. For example y = np.sin(x) + x**0.7 or y = x. Using these functions changes the number of data points and parameters like in Exercise 1
In the example where we sample from the GP prior increase the number or realizations, by replacing:
```
plt.plot(test_points, stats.multivariate_normal.rvs(cov=cov, size=6).T)
```
with
```
plt.plot(test_points, stats.multivariate_normal.rvs(cov=cov, size=1000).T, alpha=0.05, color='b')
```
How does the GP prior look? Do you see that f(x) is distributed as a Gaussian ...

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Bayesian Analysis with Python

by Osvaldo Martin

Exercises

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