I have many z_n and g_n (perhaps up to N=50,000,000) and I want to find
the value of 0 < x < 1 that maximises
argmax_x \sum_n (z_n log x) + (1 - z_n) log (1 - x g_n)
0 <= z_n <= 1 for all n and most are near 0.
The g_n are all positive and most are near 1.
Obviously I can do Newton's method but this scales badly in N. Is there
some way to get an approximate answer or to use the fact that most z_n
are near 0 or that most g_n are near 1?
If g_n = 1 for all n, clearly the problem is trivial and
x = \sum_n x_n / N
Sorry if I'm asking the wrong forum, pointers to the right place
appreciated.
Thanks,
John.
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