Understanding SMC as approximations to a sequence of measures

Question

I am used to thinking about Sequential Monte Carlo (SMC) methods as a method that discretely approximates a sequence of probability distributions. For example, in a state-space framework, SMC can yield an approximation to the state distribution at each iteration. This is also how it is explained in much applied work.

More formal sources refer to SMC approximations as a sequence of measures. I am vaguely familiar with the idea of measures and sigma algebras, but I don't fully understand their role in SMC. For example, this article by Jasra and Del Moral contains the passage:

A process is written $\{X_t\}_{t\in[0,T]}$. A measurable space is denoted $(E,\mathscr{E})$. Given a sequence of spaces $E_0,\dots,E_n$ (resp. $\sigma$-algebras $\mathscr{E}_0,\dots,\mathscr{E}_n$) the product space is written as $E_{[0,n]}$ (resp. product $\sigma$-algebra $\mathscr{E}_{[0,n]}$). For a probability $\pi$ and $\pi$-integrable function $h$, the notation $\pi(h):=\int h(x)\pi(x)\mathrm{d}x$ is sometimes used. The Dirac measure on $\{x\}$ is written $\delta_x(\mathrm{d}x')$. Probability densities are often assigned a standard notation $p$.

The following things are not clear to me:

• What is the relationship between the process $\{X_t\}_{t\in[0,T]}$ and the sequence of spaces $\{E_i\}_{i\in[0,n]}$ (or $\sigma$-algebras)
• What is the difference between saying that SMC approximates a sequence of distributions and saying that it approximates a sequence of measures. Which one is formally more correct?
• The Dirac measure is also often used without the differential operator. What is the difference in interpretation and which is more appropriate?

Any help is much appreciated. Note that I am posting this question on stats.stackexchange.com because I want to understand the concepts in order to do empirical work and a purely formal answer may not help me tie the concepts back to their applications.

Answer 1

Anyone feel free to add to these three answers. I tried to balance informative-ness and relevance.

1. They are just different ways of talking about the same thing. Recall that a random variable is a function that's "measurable." For any $t$, $X_t$ is just a label for the random variable's output, while $E_t$ could be the set of inputs for that random variable. The sigma field for one of these random variables, $\mathcal{E}_i$, is a bunch of collections of the inputs; this is useful for defining probabilities.

2. The word "distribution" is just more general. I, personally, take it to mean anything that defines a random variable. That can be just a label or a name (e.g. normal distribution), a probability density function or probability mass function (in the case of purely discrete or continuous rvs), a cumulative distribution function, a moment generating function if it exists, a characteristic function, or just a more general measure. The latter is more formal and more general. It might be required, for example, if you're in a situation where a density or mass function does not exist.

3. The dirac measure is a function that takes as inputs sets. You can integrate stuff with respect to this measure. The differential thing you're seeing is to remind you of this. If you see another resource that doesn't have this notation, perhaps it's the dirac function, which is something different.

Another theorem I'd look up is the Radon-Nikodym theorem. This tells you a) when you can talk about densities instead of measures, which has to do with some of the above answers, and also b) a requirement that your proposal/instrumental distributions have to fulfill (i.e. they have dominate what you're trying to approximate).