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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.
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