Re: and sampling a distribution

*To*: mathgroup at smc.vnet.net*Subject*: [mg110595] Re: and sampling a distribution*From*: Bill Rowe <readnews at sbcglobal.net>*Date*: Sun, 27 Jun 2010 04:55:43 -0400 (EDT)

On 6/26/10 at 3:09 AM, readnews at sbcglobal.net (Bill Rowe) wrote: >On 6/25/10 at 7:27 AM, stone at geology.washington.edu (John Stone) >wrote: > >>I am trying to use RandomReal[ ] to sample from bins of different >>widths that span the interval 0 - 1. The bin widths represent the >>weights I'm assigning to a family of trial solutions in an >>optimization problem. The aim is to sample the solutions in >>proportion to their weights using a uniform distribution of random >>numbers generated by RandomReal[ ]. >>For a simple example, however, suppose there are 10 equally >>weighted solutions. My selection process would use some code that >>looks like: >weights = Table[0.1, {10}]; bins = Accumulate[weights]; Select[bins, >(# >= RandomReal[] &)][[1]] >Rather than RandomReal you should be using RandomChoice. >Specifically, >RandomChoice[weights->bins,10] >will return a list of 10 values with the desired distribution. This >can be seen by doing: >Histogram[RandomChoice[weights -> bins, 1000]] >and note with equal weights and equally spaced bins of size 0.1, the >following is equivalent >RandomInteger[{1,10}]/10//N Up to this point my response was fine. RandomChoice is the thing to use when you want random selection from a pre-defined list of things with various weights. But the explanation I gave for why the code didn't work as expected is simply wrong. Peter Pain correctly pointed out something I should have immediately realized. RandomReal generate a new random value for each comparison made. And it is this characteristic that causes the distribution to differ from uniform. A simple demonstration that this is the case is to look at the length of the lists returned that start with 0.1. That is: In[12]:= Union[ Length /@ Cases[Table[Select[bins, (# >= RandomReal[] &)], {1000}], {0.1, __}]] Out[12]= {3,4,5,6,7,8,9} If there were only one random value selected whenever the selection was done, clearly the length of the lists with a given starting value would be constant. The idea of using Select to create the distribution can be made to work as follows: With[a = RandomReal[], Select[bins, (# >= a) &]][[1]] Repeating the demonstration above using this code yields: In[13]:= Union[ Length /@ Cases[Table[ With[{a = RandomReal[]}, Select[bins, (# >= a) &]], {1000}], {0.1, __}]] Out[13]= {10} showing every list returned that starts with the value 0.1 contains all ten values. But while this corrects the issue, this code will execute slower than code using RandomChoice will.