Posts by Chris Fonnesbeck

Marginal Likelihood Implementation

The gp.Marginal class implements the more common case of GP regression: the observed data are the sum of a GP and Gaussian noise. gp.Marginal has a marginal_likelihood method, a conditional method, and a predict method. Given a mean and covariance function, the function \(f(x)\) is modeled as,

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Multivariate Gaussian Random Walk

This notebook shows how to fit a correlated time series using multivariate Gaussian random walks (GRWs). In particular, we perform a Bayesian regression of the time series data against a model dependent on GRWs.

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Reparameterizing the Weibull Accelerated Failure Time Model

The previous example notebook on Bayesian parametric survival analysis introduced two different accelerated failure time (AFT) models: Weibull and log-linear. In this notebook, we present three different parameterizations of the Weibull AFT model.

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Bayesian Survival Analysis

Survival analysis studies the distribution of the time to an event. Its applications span many fields across medicine, biology, engineering, and social science. This tutorial shows how to fit and analyze a Bayesian survival model in Python using PyMC.

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Introduction to Variational Inference with PyMC

The most common strategy for computing posterior quantities of Bayesian models is via sampling, particularly Markov chain Monte Carlo (MCMC) algorithms. While sampling algorithms and associated computing have continually improved in performance and efficiency, MCMC methods still scale poorly with data size, and become prohibitive for more than a few thousand observations. A more scalable alternative to sampling is variational inference (VI), which re-frames the problem of computing the posterior distribution as an optimization problem.

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Empirical Approximation overview

For most models we use sampling MCMC algorithms like Metropolis or NUTS. In PyMC we got used to store traces of MCMC samples and then do analysis using them. There is a similar concept for the variational inference submodule in PyMC: Empirical. This type of approximation stores particles for the SVGD sampler. There is no difference between independent SVGD particles and MCMC samples. Empirical acts as a bridge between MCMC sampling output and full-fledged VI utils like apply_replacements or sample_node. For the interface description, see variational_api_quickstart. Here we will just focus on Emprical and give an overview of specific things for the Empirical approximation.

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GLM: Robust Linear Regression

Duplicate implicit target name: “glm: robust linear regression”.

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Analysis of An AR(1) Model in PyMC

Consider the following AR(2) process, initialized in the infinite past: $\( y_t = \rho_0 + \rho_1 y_{t-1} + \rho_2 y_{t-2} + \epsilon_t, \)\( where \)\epsilon_t \overset{iid}{\sim} {\cal N}(0,1)\(. Suppose you'd like to learn about \)\rho\( from a a sample of observations \)Y^T = { y_0, y_1,\ldots, y_T }$.

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Multi-output Gaussian Processes: Coregionalization models using Hamadard product

This notebook shows how to implement the Intrinsic Coregionalization Model (ICM) and the Linear Coregionalization Model (LCM) using a Hamadard product between the Coregion kernel and input kernels. Multi-output Gaussian Process is discussed in this paper by Bonilla et al. [2007]. For further information about ICM and LCM, please check out the talk on Multi-output Gaussian Processes by Mauricio Alvarez, and his slides with more references at the last page.

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A Primer on Bayesian Methods for Multilevel Modeling

Hierarchical or multilevel modeling is a generalization of regression modeling.

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Gaussian Processes using numpy kernel

Example of simple Gaussian Process fit, adapted from Stan’s example-models repository.

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Modeling spatial point patterns with a marked log-Gaussian Cox process

The log-Gaussian Cox process (LGCP) is a probabilistic model of point patterns typically observed in space or time. It has two main components. First, an underlying intensity field \(\lambda(s)\) of positive real values is modeled over the entire domain \(X\) using an exponentially-transformed Gaussian process which constrains \(\lambda\) to be positive. Then, this intensity field is used to parameterize a Poisson point process which represents a stochastic mechanism for placing points in space. Some phenomena amenable to this representation include the incidence of cancer cases across a county, or the spatiotemporal locations of crime events in a city. Both spatial and temporal dimensions can be handled equivalently within this framework, though this tutorial only addresses data in two spatial dimensions.

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Gaussian Process for CO2 at Mauna Loa

This Gaussian Process (GP) example shows how to:

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Lasso regression with block updating

Sometimes, it is very useful to update a set of parameters together. For example, variables that are highly correlated are often good to update together. In PyMC block updating is simple. This will be demonstrated using the parameter step of pymc.sample.

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Bayesian Estimation Supersedes the T-Test

Non-consecutive header level increase; H1 to H3 [myst.header]

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