ch 20.1 - a-mixture-model-of-the-speed-accuracy-trade-off-the-fast-guess-model-account.html#the-global-motion-detection-task

This is nothing crucial, but I have a small suggestion for clarity. In the part that introduces the implementation of the mixture model (using log_sum_exp()), I think the exposition could be made a bit more uncompressed:

we have a problem of computing log(A+B)
for numerical stability we cannot just add that, we need a hack: log(A+B) = log_sum_exp(log(A), log(B))
log(A) = log(p_task*pdf1) = log(p) + llpdf1 (and remind that we were always dealing with lpdfs not pdfs)
log(B) = log((1-p_task)*pdf2) = log(1-p_task) + llpdf2
and then by the way we also need to deal with log(1-p) in a stabile way, --> log1m()
so we end up with: log_sum_exp(llog(p) + llpdf1, log1m(p) + llpdf2)

even though I was familiar with LSE and log1m, it took me a while to reconstruct this and parse why this works in this way. I assume a typical reader won't know LSE, be unfamiliar with transitions between log and lin scales, and be math-anxious, so being super explicit might help.

One part I found confusing was the mention that LSE = log(exp(x) + exp(y)), even though there were no exp()s in the formulas above that part of the text.

More generally, whenever I don't understand something in math, it makes me less motivated to understand anything that follows it, so I always appreciate it that whenever I'm exposed to math (because there is no other way around), it is presented in an explicit way.

vasishth / bayescogsci

ch 20.1 - a-mixture-model-of-the-speed-accuracy-trade-off-the-fast-guess-model-account.html#the-global-motion-detection-task #16