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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.

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  • $\begingroup$ Related: stats.stackexchange.com/questions/73623. $\endgroup$
    – whuber
    Apr 5, 2016 at 21:38
  • $\begingroup$ 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. $\endgroup$
    – Xi'an
    Apr 8, 2016 at 17:04
  • $\begingroup$ I wrote a blog post about it. I explain what the Dirac-delta function is and how it can be used for sampling, here $\endgroup$ May 19, 2020 at 15:38

1 Answer 1

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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.

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