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# 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[1]+b[2]*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[1],0,10, log=TRUE)+dnorm(b[2],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)
#df = c(df,dplyr::ddply(df,log_post))

# Examples

sessionInfo()
R version 3.5.2 (2018-12-20)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.1

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base

loaded via a namespace (and not attached):
[1] workflowr_1.2.0 Rcpp_1.0.0      digest_0.6.18   rprojroot_1.3-2
[5] backports_1.1.3 git2r_0.24.0    magrittr_1.5    evaluate_0.12
[9] stringi_1.2.4   fs_1.2.6        whisker_0.3-2   rmarkdown_1.11
[13] tools_3.5.2     stringr_1.3.1   glue_1.3.0      xfun_0.4
[17] yaml_2.2.0      compiler_3.5.2  htmltools_0.3.6 knitr_1.21     

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