Last updated: 2017-03-04

Code version: 5d0fa13282db4a97dc7d62e2d704e88a5afdb824

# Pre-requisites

include the most complex concepts required to understand the material.

# Overview

Suppose we have a logistic regression $$Y_i | X_i \sim Bern(p_i)$$ where $log(p_i/(1-p_i)) = \mu + \theta X_i.$

We will assume that $$X_i \in {-1,+1}$$, and assume priors for $$\mu$$ and $$\theta$$: $\mu \sim N(0,100)$ $\theta \sim N(0,1)$

For illustration we simulate data where $$\mu=\theta=0$$:

x = sample(c(-1,1),1000,replace=TRUE)
y = rbinom(1000,1,0.5)

#b is a vector b=(mu,theta)
#loglikelihood for logistic regression
loglik = function(b){
eta = b+b*x
p = exp(eta)/(1+exp(eta))
return(sum(log(y*p+(1-y)*(1-p))))
}

#b is a vector b=(mu,theta)
log_prior = function(b){
return(dnorm(b,0,10, log=TRUE)+dnorm(b,0,1,log=TRUE))
}

#b is a vector b=(mu,theta)
log_post = function(b){
return(loglik(b)+log_prior(b))
}

Let’s compute a 95% CI for $$\theta$$. First try a discrete grid

Note: This is still a work in progress.

m = seq(-10,10,length=100)
t = seq(-2,2,length=100)
df = expand.grid(m=m,t=t)
head(df)
#df = c(df,dplyr::ddply(df,log_post))

# Examples

## Session information

sessionInfo()
R version 3.3.0 (2016-05-03)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.10.5 (Yosemite)

locale:
 en_NZ.UTF-8/en_NZ.UTF-8/en_NZ.UTF-8/C/en_NZ.UTF-8/en_NZ.UTF-8

attached base packages:
 stats     graphics  grDevices utils     datasets  methods   base

other attached packages:
 tidyr_0.4.1     dplyr_0.5.0     ggplot2_2.1.0   knitr_1.15.1
 MASS_7.3-45     expm_0.999-0    Matrix_1.2-6    viridis_0.3.4
 workflowr_0.3.0 rmarkdown_1.3

loaded via a namespace (and not attached):
 Rcpp_0.12.5      git2r_0.18.0     plyr_1.8.4       tools_3.3.0
 digest_0.6.9     evaluate_0.10    tibble_1.1       gtable_0.2.0
 lattice_0.20-33  shiny_0.13.2     DBI_0.4-1        yaml_2.1.14
 gridExtra_2.2.1  stringr_1.2.0    gtools_3.5.0     rprojroot_1.2
 grid_3.3.0       R6_2.1.2         reshape2_1.4.1   magrittr_1.5
 backports_1.0.5  scales_0.4.0     htmltools_0.3.5  assertthat_0.1
 mime_0.5         colorspace_1.2-6 xtable_1.8-2     httpuv_1.3.3
 labeling_0.3     stringi_1.1.2    munsell_0.4.3   

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