Last updated: 2021-10-16
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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
\[f(\beta)\geq -\frac{\eta}{2}-\log(1+\exp(-\eta))+\frac{1}{4\eta}tanh(\frac{\eta}{2})(\beta^2-\eta^2).\]
This 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: