[HN Gopher] Monte Carlo Crash Course: Sampling
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       Monte Carlo Crash Course: Sampling
        
       Author : ibobev
       Score  : 49 points
       Date   : 2025-04-14 14:54 UTC (8 hours ago)
        
 (HTM) web link (thenumb.at)
 (TXT) w3m dump (thenumb.at)
        
       | vitus wrote:
       | Perhaps the most commonly-cited (but not actually all that
       | practical) example for rejection sampling is estimating the value
       | of pi (as hinted by the first example under uniform random
       | sampling): generate two random numbers between 0 and 1, then
       | check the fraction of points that satisfy x^2 + y^2 <= 1. As
       | number of samples tends to infinity, this will converge to pi/4.
       | (You can instead take x,y from (-1,1) if you want to generate
       | points on the full circle.)
       | 
       | > However, rejection sampling is only efficient when f_Omega can
       | make use of a significant proportion of the probability density
       | in f_D.
       | 
       | Perhaps a more relevant example: the unit n-sphere encloses a
       | vanishing amount of volume as the number of dimensions increases.
       | 
       | https://en.wikipedia.org/wiki/Volume_of_an_n-ball
       | 
       | This is one of those weird consequences that gets labeled as the
       | curse of dimensionality, especially in ML contexts.
       | 
       | "As the dimension d of the space increases, the hypersphere
       | becomes an insignificant volume relative to that of the
       | hypercube."
       | 
       | https://en.wikipedia.org/wiki/Curse_of_dimensionality#Distan...
        
       | atum47 wrote:
       | Slightly related, I once created an app that calculate the volume
       | of a random shape using Monte Carlo
       | https://github.com/victorqribeiro/monteCarlo
        
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       (page generated 2025-04-14 23:00 UTC)