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I would like to draw samples from the following conditional distribution of the following (don't ask why):

fRandomFunction[x_, y_] := 
 PDF[MultinormalDistribution[{3, 3}, {{0.5, 0}, {0, 0.5}}], {x, y}] + 
 PDF[MultinormalDistribution[{6, 6}, {{0.6, -0.5}, {-0.5, 0.6}}], {x,
   y}] + PDF[MultinormalDistribution[{6, 6}, {{1, 0.5}, {0.5, 1}}], {x, y}] + 
  PDF[MultinormalDistribution[{6, 4}, {{0.6, 0.5}, {0.5, 0.6}}], {x, 
   y}] + PDF[MultinormalDistribution[{2, 8}, {{0.2, 0}, {0, 0.2}}], {x, y}]

At the moment I am using Mathematica's inbuilt ProbabilityDistribution function to draw individual independent samples:

fGenerateSample[aX_] :=
 RandomVariate[
   ProbabilityDistribution[fRandomFunction[aX, y], {y, 0, Infinity}], {1}
 ][[1]]

If I map this onto a random list of X inputs, this takes quite a while to output an independent sample:

lInputs = RandomVariate[NormalDistribution[5, 1], {1}];
lSamples = fGenerateSample /@ lInputs // Timing;

This takes about 1/20 second to execute on my computer. I would like a way of speeding this process up. Does anyone know of a way I could generate a single sample at a faster rate from the conditional density here?

Note this problem is quite different from my actual problem, where I need to essentially do a NestList with my fGenerateSample function, up to 100,000 times. That is why speed is of the essence.

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  • 4
    $\begingroup$ You said not to ask "why" but why do you call fRandomFunction a conditional distribution when it is the sum of 5 multivariate normal densities. Also, fRandomFunction is not a proper density until you divide by 5. And finally why does y go from 0 to positive infinity when fRandomFunction deals with values of y between minus and plus infinity? Or am I totally misunderstanding what you've written. (My running of your code takes about a minute rather than 1/20 of a second so I also need a new machine.) $\endgroup$ – JimB Apr 4 '16 at 20:56
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    $\begingroup$ I must agree with @JimBaldwin: "...don't ask why.", "... this problem is quite different from my actual problem..." - then why aren't you posting the actual problem, and for that matter, what is so Double-Naught Spy secret about the actual goal? There are many here that are not only Mathematica experts, but mathematicians & probabilists as well. Knowing what the end goal is for those readers may well result in a far superior way to meet the needs. $\endgroup$ – ciao Apr 5 '16 at 6:03
  • $\begingroup$ @ciao Well, at least the "actual" problem here is acknowledged. $\endgroup$ – LLlAMnYP Apr 5 '16 at 13:50
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I agree with the comments. I am uncertain what the aim is, so I apologize if this is unhelpful. I only post to motivate clarification.

If the aim is a marginal of mixture of binormals:

m1 = MultinormalDistribution[{3, 3}, {{0.5, 0}, {0, 0.5}}];
m2 = MultinormalDistribution[{6, 6}, {{0.6, -0.5}, {-0.5, 0.6}}];
m3 = MultinormalDistribution[{6, 6}, {{1, 0.5}, {0.5, 1}}];
m4 = MultinormalDistribution[{6, 4}, {{0.6, 0.5}, {0.5, 0.6}}];
m5 = MultinormalDistribution[{2, 8}, {{0.2, 0}, {0, 0.2}}];
mix = MixtureDistribution[{1, 1, 1, 1, 1}, {m1, m2, m3, m4, m5}];
marg = MarginalDistribution[mix, 2];
Histogram[RandomVariate[marg, 100000]]
Histogram3D[RandomVariate[mix, 10000]]
Histogram3D[RandomVariate[#, 1000] & /@ {m1, m2, m3, m4, m5}]

enter image description here

If the aim is sum of binormals:

td = TransformedDistribution[{x1, y1} + {x2, y2} + {x3, y3} + {x4, 
     y4} + {x5, y5}, {{x1, y1} \[Distributed] 
     m1, {x2, y2} \[Distributed] m2, {x3, y3} \[Distributed] 
     m3, {x4, y4} \[Distributed] m4, {x5, y5} \[Distributed] m5}];
ListPlot[RandomVariate[td, 100000]]

enter image description here

Again apologies if this is unhelpful.

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  • $\begingroup$ I'd take a wild guess and imagine, that the 5 distributions can be arranged into one for a vector of length 10, possibly leading to a performance boost. Unless of course MMA does that under the hood. $\endgroup$ – LLlAMnYP Apr 5 '16 at 13:54

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