Last updated: 2024-07-02

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File Version Author Date Message
Rmd a1e004a Matthew Stephens 2024-07-02 workflowr::wflow_publish("poisson_log1p.Rmd")

Introduction

I wanted to look at normal approximation to the Poisson log-likelihood with the log1p link.

The model is: \[Y \sim Poisson(\lambda)\]

\[\log(1+\lambda) = b, (b>0)\] So \[Y | b \sim Poisson(\exp(b)-1)\] The log-likelihood is \[l(b) = Y \log(\exp(b)-1) - (\exp(b)-1)\] The derivative of the log-likelihood is \[l'(b) = Y \frac{\exp(b)}{\exp(b)-1} - \exp(b)\] The second derivative of the log-likelihood is \[l''(b) = Y \frac{-exp(b)}{(\exp(b)-1)^2} - \exp(b) = -exp(b)(Y/(exp(b)-1)^2 + 1)\]

Writing the RHS in terms of \(\lambda = exp(b)-1\) we have \[l''(b) = -(\lambda+1)[Y/\lambda^2 + 1]\] Also the mle for b is \(\hat{b} = log(1+Y)\)

This suggests the normal approximation to the likelihood of \(l(b) \approx N(b; \hat{b}, s^2)\) where \[s^2 = -1/l''(\hat{b}) = 1/(\hat{\lambda}+1)[Y/\hat{\lambda}^2 + 1] = 1/(Y+1)(1/Y+1)=Y/(1+Y)^2 \quad (Y>0)\] and \(s^2 = 1\) for \(Y=0\).

Plots

First we plot the log-likelihood for \(Y=1\) as a function of \(b\). Then we overlay the normal approximation to the log-likelihood for \(Y=1\).

b = seq(0,3,length=100)
loglik = function(b,Y) {
  if(Y==0){
    return(-(exp(b)-1))} 
  else 
    return(Y*log(exp(b)-1) - (exp(b)-1))
}

y = 1
ll = loglik(b,y)
plot(b,ll-max(ll),type="l",xlab="b",ylab="log-likelihood",main="log-likelihood for Y=1")
s2 = y/(1+y)^2
bhat = log(y+1)
ll_approx = dnorm(b,bhat,sqrt(s2),log=TRUE)
lines(b,ll_approx-max(ll_approx),col="red")

Do the same for \(y=2\):

y = 2
ll = loglik(b,y)
plot(b,ll-max(ll),type="l",xlab="b",ylab="log-likelihood",main="log-likelihood for Y=2")
s2 = y/(1+y)^2
bhat = log(y+1)
ll_approx = dnorm(b,bhat,sqrt(s2),log=TRUE)
lines(b,ll_approx-max(ll_approx),col="red")

For \(Y=0\) we see a problem: the derivative of the log-likelihood does not vanish at \(b=0\) (it is -1). So we need to modify the normal approximation to be \(N(-1,1)\) rather than \(N(0,1)\).

b = seq(0,1,length=100)
y = 0
ll = loglik(b,y)
plot(b,ll-max(ll),type="l",xlab="b",ylab="log-likelihood",main="log-likelihood for Y=0")
s2 = 1
bhat = -1
ll_approx = dnorm(b,bhat,sqrt(s2),log=TRUE)

lines(b,ll_approx-max(ll_approx),col="red")


sessionInfo()
R version 4.2.1 (2022-06-23)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur ... 10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.2/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):
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 [9] fastmap_1.1.1     fansi_1.0.6       highr_0.10        stringr_1.5.1    
[13] tools_4.2.1       xfun_0.41         utf8_1.2.4        cli_3.6.2        
[17] git2r_0.33.0      jquerylib_0.1.4   htmltools_0.5.7   rprojroot_2.0.4  
[21] yaml_2.3.8        digest_0.6.33     tibble_3.2.1      lifecycle_1.0.4  
[25] later_1.3.2       sass_0.4.8        vctrs_0.6.5       fs_1.6.3         
[29] promises_1.2.1    cachem_1.0.8      glue_1.6.2        evaluate_0.23    
[33] rmarkdown_2.25    stringi_1.8.3     bslib_0.6.1       compiler_4.2.1   
[37] pillar_1.9.0      jsonlite_1.8.8    httpuv_1.6.13     pkgconfig_2.0.3