Publications by Ken Wood
Hierarchical Mixture Models & Likelihoods
We have the mixture model written as \[f(x) = \sum_{k=1}^{K}w_kg_k(x)\] We will introduce an indicator \(C\), which is a (discrete) random variable where \(C \in 1,2,,,,K\). Then, \(X|C \sim g_c(x)\) and \(C\sim Pr(C=k) = w_k\), and \[Pr(X)=\sum_{k=1}^{K}f(x|C=k)Pr(C=k)=\sum_{k=1}^{K}w_kg_k(x)\] Setting up the hierarchical problem: \(X|C \sim g_c(x...
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Hierarchical Mixture Models & Likelihoods
We have the mixture model written as \[f(x) = \sum_{k=1}^{K}\omega_kg_k(x)\] We will introduce an indicator \(C\), which is a (discrete) random variable where \(C \in 1,2,,,,K\). Then, \(X|C \sim g_c(x)\) and \(C\sim Pr(C=k) = \omega_k\), and \[Pr(X)=\sum_{k=1}^{K}f(x|C=k)Pr(C=k)=\sum_{k=1}^{K}\omega_kg_k(x)\] Setting up the hierarchical problem: \...
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Bayesian Zero-Inflated Mixture Modes
Zero inflated negative binomial distribution x = seq(0, 15) y = dnbinom(x, 8, 0.6) z = 0.2*c(1,rep(0,length(x)-1)) + (1-0.2)*y par(mfrow=c(2,1)) par(mar=c(4,4,2,2)+0.1) barplot(y, names.arg=x, las=1, xlab = "x", ylab="Probability", border=NA, main="Negative Binomial") par(mar=c(4,4,1,1)+0.1) barplot(z, names.arg=x, las=1, xlab = "x", ylab=...
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Bayesian Mixture Model Examples
Mixture of univariate Gaussians, bimodal x = seq(-5, 12, length=100) y = 0.6*dnorm(x, 0, 1) + 0.4*dnorm(x, 5, 2) par(mar=c(4,4,1,1)+0.1) plot(x, y, type="l", ylab="Density", las=1, lwd=2) Mixture of univariate Gaussians, unimodal skewed x = seq(-5, 12, length=100) y = 0.55*dnorm(x, 0, sqrt(2)) + 0.45*dnorm(x, 3, 4) par(mar=c(4,4,1,1)+0.1) plot(x, ...
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Data Analysis Project - Bayesian Statistics V2
Executive Summary The purpose of this project was to fit a series of regression models to a dataset containing housing features and a corresponding sale price as the response variable. Three models were constructed using both R and JAGS. One of the JAGS models used 3 features to predict sale prices while the final iteration used 4 features. A numbe...
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Bayesian Mixture Models - Basic Definitions
Suppose we have a variable \(X\) with probability density function of \(f(x)\) (continuous) or probability mass function (discrete). Then \(F(x)\) is the probability distribution where \(F(x) = P(X<x)\). The Mixture model takes the form: \[F(x) = \sum_{k=1}^{K}w_kG_k(x),\] where \(w_k\) are the weights and \(G_k(x)\) are the components such that \(...
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Data Analysis Project - Bayesian Statistics
Introduction The Ames Housing dataset, which is available on Kaggle.com, was compiled by Dean De Cock for use in data science education. It’s an incredible dataset resource for data scientists and statisticians looking for a modernized and expanded version of the often-cited Boston Housing dataset. The subject dataset contains 79 explanatory vari...
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Hierarchical Modeling using Bayesian Methods
Data Input Let’s fit our hierarhical model for counts of chocolate chips. The data can be found in cookies.dat. dat = read.table(file="cookies.dat", header=TRUE) head(dat) ## chips location ## 1 12 1 ## 2 12 1 ## 3 6 1 ## 4 13 1 ## 5 12 1 ## 6 12 1 table(dat$location) ## ## 1 2 3 ...
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Poisson Regression using Bayesian Methods
For an example of Poisson regression, we’ll use the badhealth data set from the COUNT package in R. library("COUNT") ## Loading required package: msme ## Loading required package: MASS ## Loading required package: lattice ## Loading required package: sandwich data("badhealth") ?badhealth head(badhealth) ## numvisit badh age ## 1 30 0 ...
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Logistic Regression using Bayesian Methods
Data For an example of logistic regression, we’ll use the urine data set from the boot package in R. The response variable is r, which takes on values of \(0\) or \(1\). We will remove some rows from the data set which contain missing values. library("boot") data("urine") ?urine head(urine) ## r gravity ph osmo cond urea calc ## 1 0 1.021 4...
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