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Let \(y\) be a binary variable with \(p(y=1) = p\), where \(p = g(\beta)\) and \(g(x) = \frac{1}{1+\exp(-x)}\). The function \(g\) is usually called logistic function or sigmoid function.
The log likelihood is
\[\begin{equation} \begin{split} l(\beta) &= \log p(y|\beta) \\&= y\log g(\beta)+(1-y)\log (1-g(\beta)) \\&= (-\beta-\log(1+\exp(-\beta))) +y\beta \\&= (y-1)\beta - \log(1+\exp(-\beta)) \end{split} \end{equation}\]
It’s sometimes more convenient to write
\[l(\beta) = g((2y-1)\beta).\]
The summation inside \(\log\) function is a pain and it is hard to deal with. For example in variational inference of logistic regression. One method to bypass the log sum is to lower bound the function \(-\log(1+\exp(-\beta))\).
In the paper Jaakkola and Jordan, 2000, the following lower bound is suggested.
Notice that \[-\log(1+\exp(-\beta)) = \frac{\beta}{2}-\log(\exp(\beta/2)+\exp(-\beta/2)),\]
where \(f(\beta) = -\log(\exp(\beta/2)+\exp(-\beta/2))\) is a convex function in \(\beta^2\). So we can bound \(f(\beta)\) globally with a first order Taylor expansion in \(\beta^2\), leading to
\[f(\beta)\geq -\frac{\eta}{2}-\log(1+\exp(-\eta))-\frac{1}{4\eta}tanh(\frac{\eta}{2})(\beta^2-\eta^2).\]
(I find the derivative \(\frac{\partial f(\beta)}{\partial\beta^2}\) should be \(-\frac{1}{4\eta}tanh(\frac{\eta}{2})\). Let’s have a check. It suggests that it should be \(-\frac{1}{4\eta}tanh(\frac{\eta}{2})\))
g = function(x){
1/(1+exp(-x))
}
f0 = function(x){
-log(exp(x/2)+exp(-x/2))
}
f0_lb = function(x,eta){
-eta/2 + log(g(eta))-1/4/eta*tanh(eta/2)*(x^2-eta^2)
}
f0(0)[1] -0.6931472f0_lb(0,1)[1] -0.6977324f0_lb(0,-1)[1] -0.6977324This lower bound is exact if \(\beta^2 = \eta^2\). The lower bound is also a quadratic function in \(\beta^2\) which is a very useful property for doing Gaussian approximation.
The unknown parameter \(\eta\) is suggested to be optimized within the algorithm. It turns out that \(\eta^2 = E(\beta^2)\), the second moment of \(\beta\).
Another two useful reference:
Spike and slab variational Bayes for high dimensional logistic regression
VARIATIONAL BAYESIAN INFERENCE FOR LINEAR AND LOGISTIC REGRESSION
Now a question is how about \(f(x,y) = -\log(e^\frac{x+y}{2}+e^{-\frac{x+y}{2}})\).
library(plotly)Loading required package: ggplot2
Attaching package: 'plotly'The following object is masked from 'package:ggplot2':
last_plotThe following object is masked from 'package:stats':
filterThe following object is masked from 'package:graphics':
layoutf = function(x,y){
-log(exp((x+y)/2)+exp(-(x+y)/2))
}
x = y = seq(0,10,length.out = 30)
z = outer(x,y,f)
fig = plot_ly(x=x^2,y=y^2,z=z)
fig <- fig %>% add_surface()
figThe plot suggests it is a convex function in \((x^2,y^2)\). A first order Taylor series expansion of in the variable \((x^2,y^2)\) yields \[f(x,y)\geq f(a,b)-\frac{1}{4a}tanh(\frac{a+b}{2})(x^2-a^2) - \frac{1}{4b}tanh(\frac{a+b}{2})(y^2-b^2)\]
sessionInfo()R version 4.0.3 (2020-10-10)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS High Sierra 10.13.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/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
other attached packages:
[1] plotly_4.9.2.1 ggplot2_3.3.2 workflowr_1.6.2
loaded via a namespace (and not attached):
[1] Rcpp_1.0.5 pillar_1.4.6 compiler_4.0.3 later_1.1.0.1
[5] git2r_0.27.1 tools_4.0.3 digest_0.6.27 viridisLite_0.3.0
[9] jsonlite_1.7.1 evaluate_0.14 lifecycle_1.0.0 tibble_3.0.4
[13] gtable_0.3.0 pkgconfig_2.0.3 rlang_0.4.10 DBI_1.1.0
[17] rstudioapi_0.11 crosstalk_1.1.0.1 yaml_2.2.1 xfun_0.26
[21] httr_1.4.2 withr_2.3.0 stringr_1.4.0 dplyr_1.0.5
[25] knitr_1.36 htmlwidgets_1.5.2 generics_0.1.0 fs_1.5.0
[29] vctrs_0.3.7 tidyselect_1.1.0 rprojroot_1.3-2 grid_4.0.3
[33] data.table_1.13.2 glue_1.4.2 R6_2.4.1 rmarkdown_2.5
[37] farver_2.0.3 tidyr_1.1.2 purrr_0.3.4 magrittr_2.0.1
[41] whisker_0.4 backports_1.1.10 scales_1.1.1 promises_1.1.1
[45] ellipsis_0.3.1 htmltools_0.5.1.1 assertthat_0.2.1 colorspace_1.4-1
[49] httpuv_1.5.4 stringi_1.5.3 lazyeval_0.2.2 munsell_0.5.0
[53] crayon_1.3.4