Following paper about Glicko rating have a expression below:

Parameter estimation in large dynamic paired comparison experiments (Equation 18, 19 on Appendix A)

$$ \int \frac{ (10^{(\theta-\theta_j)/400}) ^{s_{jk}}}{1+ 10^{(\theta-\theta_j)/400} }d \varphi(\theta_j|\mu, \sigma^2_j) d\theta_j \approx \frac{( 10^{g(\sigma^2)(\theta-\mu)/400}) ^{s_{jk}} }{1+ 10^{g(\sigma^2)(\theta-\mu)/400} } $$


$$ g(\sigma^2) = \frac{1}{\sqrt{1+3q^2\sigma^2/\pi^2}} $$

$$ q = log(10)/400 $$

It looks using the similarity between normal distribution and the integral of the logistic distribution.

But, I could not get how to get the expression. Is there any idea?


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