[HN Gopher] Mathematical Foundations of Monte Carlo Methods
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       Mathematical Foundations of Monte Carlo Methods
        
       Author : poindontcare
       Score  : 68 points
       Date   : 2022-04-20 11:56 UTC (1 days ago)
        
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       | it wrote:
       | Here's an alternative for Bayesians:
       | http://georglsm.r-forge.r-project.org/site-projects/pdf/7113...
        
       | Spivakov wrote:
       | An informal perspective on some implication of Monte Carlo on
       | integral:
       | 
       | One of the most intuitive (and used in applications) definition
       | of being integrable is Riemann integral based on the geometric
       | idea that you can compute the area/volume by dividing region into
       | pieces and summing them all up. Now you can (mathematically)
       | prove that for any such integrable function, its integral can be
       | approximated by Monte Carlo and the results are consistent.
       | 
       | Now what about the other direction? You can theoretically run
       | Monte Carlo approx on wildly zigzag functions that does not make
       | any geometry sense (i.e. not Riemann integrable), if the
       | "probability" in the space is well-defined. The idea that uses
       | probability, instead of geometry, turns out to give a broader
       | class of integrable objects.
       | 
       | One interesting observation is that these ideas are intuitive and
       | meaningful if put informally. But when you formally look into
       | these ideas (integration/measure theory) it suddenly collapses
       | into lines of terse mathematical constructs.
        
         | l33t2328 wrote:
         | You're certainly aware of this, but for those that aren't:
         | asking that the "probability" in the space to be well defined
         | you are essentially invoking the idea that the function is
         | _measurable_. Measure is a way to generalize things like length
         | and volume to sets which have no length or volume in the
         | traditional sense. One good way to do this is say that lines
         | have "measure" equal to their length, and if you can combine
         | line segments(even infinitely many) to get some new line then
         | that line has measure as well. If you do make this precise, you
         | get the so-called "Borel measure" on the real line.
         | 
         | If you want all subsets of sets with 0 Borel measure to also
         | have 0 measure, then this leads to the notion of the lebesgue
         | measure, and it can be used to define the lebesgue integral.
        
           | pfortuny wrote:
           | Two insights more
           | 
           | a) But for probability you nees the measure of the whole
           | space to be 1, which forces points "far away" to have very
           | small "weight". Thus, there is no uniform distribution in the
           | whole R.
           | 
           | b) and then your mind blows up when you realize that discrete
           | probability is the very same thing, and that an integral in a
           | finite set is just summation.
        
         | ronald_raygun wrote:
         | Yeah to add to this when you study probability at a more
         | advanced level, you learn that really what a probability is
         | "just" an integral ie P(x) = integral{Indicator function(X)}.
         | So it's actually not a crazy object to pop up when you're
         | thinking about integrals and integral approximations.
        
       | arc-in-space wrote:
       | When I was beginning to study path tracing I've learned to avoid
       | this domain, the presentation always seems so friendly but the
       | actual material is unfortunately much too barebones
        
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