# Multiplying two probability distributions represented by particles

Hello,

I have two probability distributions p1(x) and p2(x) given by (x_1i, w_1i) and (x_2i, w_2i) respectively, i.e. they are both represented by sets of particles. I need to create the pdf p(x) = p1(x) . p2(x), their product. I read that for functions supported on integers (these would be like the delta functions of the particles), multiplying is the same as convolution. But I still can't figure out how to create a new pdf represented by (x_i, w_i) from these two sets. Can somebody give me some ideas?

Thanks

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Usually when you multiply probabilities, they are probabilities for different events, not for the same event. I wonder if you really want p1(x)p2(x) or actually p1(x)p2(y). Multiplying delta functions depending on different arguments is no problem. I assume when you say p(x) is represented by $(x_i,w_i)$, you mean $x=x_i$ with probability $w_i$. With this interpretation $p1(x)p2(y)$ is represented by $(x_iy_j,w_{1i}w_{2j})$. – Michael Renardy May 23 '11 at 16:00
NBP has quite a definite algorithm. I do not quite understand which part could cause such a confusion whether to use a product of PDFs or a convolution. Anyway, if it helps, a product of PDFs is the same as a convolution of their characteristic functions. I'm just guessing that that might be what you were referring to. If you just multiply values of probabilities on the same outcome, do not forget to normalize the result. If you multiply outcomes too, be aware that some pairs of outcomes might give the same product. – Tunococ Jun 20 '11 at 10:28

You seem a bit confused about multiplication and convolution. Do you need the convolution of the two pdf's (this would give you the pdf of the sum of the random variables), or the product of the pdf's?

In either case, you could fit parametric distributions (e.g. normal, log normal, or whatever might be appropriate for your problem) to the two probability distributions and then multiply or convolve the pdf's.

You could also look at techniques that have been developed for nonparametric density estimation.

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I am doing non-parametric belief propagation :). Yes, I need to multiply the two pdfs. I will try to fit a single gaussian onto these particle pdfs but that kind of spoils the point of having a particle filter on the nodes of NBP. – Pratik May 10 '11 at 0:13

monte carlo simulation should do the trick: use a good random generator and fire at both functions, (this is for each function: randomly take a probability from the pdf, such that higher density probability has higher likelihood to be chosen) then take the mean of the probabilities chosen (i assume that both functions apply to the same random variable), do this e.g. a million times, the resulting pdf is the product

Crist-Jan Doedens

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