Last updated: 2019-03-31
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Knit directory: fiveMinuteStats/analysis/
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include the most complex concepts required to understand the material.
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)
head(df)
#df = c(df,dplyr::ddply(df,log_post))
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
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[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 This site was created with R Markdown