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I am trying to understand the delta function notation used to be express a monte carlo approximation of a probability distribution. The notation used in this (p10) is

$\pi(x_{1:n}) = \frac{1}{N}\sum^N_{i=1}\delta_{X^i_{1:n}}(x_{1:n})$

where $X^i_{1:n} \sim \pi(x_{1:n})$ for i=1,..,N

I'm wondering whether this is equivalent to the following

$\pi(x_{1:n}) = \frac{1}{N}\sum^N_{i=1}\delta(x_{1:n}-{X^i_{1:n}})$

for a specific value of $x_{1:n}$, say $x_{1:n} = y$. I understand the latter to mean that the probability of a particular $x_{1:n}$ is given by the number of samples that take this value, (hence $\delta(y - X^i_{1:n}) = \delta(0) = 1 )$ divided by the total number of samples, N.

Is this correct?

I realise my questions is similar to the question asked here: Delta Function in Monte Carlo Sampling and have read it and looked at linked answers.

Community
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user3235916
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  • Related: http://stats.stackexchange.com/questions/73623. – whuber Apr 05 '16 at 21:38
  • This Monte Carlo approximation consists in replacing the unknown distribution $\pi$ with a finite support distribution, which support is the simulated sample $X_{1:n}$. This is also the basis of bootstrap. – Xi'an Apr 08 '16 at 17:04
  • I wrote a blog post about it. I explain what the Dirac-delta function is and how it can be used for sampling, [here](https://maurocamaraescudero.netlify.app/post/towards-smc-using-the-dirac-delta-function-in-sampling-and-sequential-monte-carlo/) – Euler_Salter May 19 '20 at 15:38

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Yeah, that's correct. You can see from the wikipedia definition here how the generalization with subscript is defined, and your transformation, in the case at hand, renders it back into original delta notation.

Chris Bonnell
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