1

On page 4 of arXiv:1701.02434v2 ("A Conceptual Introduction to Hamiltonian Monte Carlo"), we have the following text and expectation:

We begin by assuming that the target sample space, $Q$, can be parameterized by the real numbers such that every point $q\in Q$ can be specified with $D$ real numbers. Given a parameter space, $\mathcal{Q}$, we can then specify the target distribution as a smooth probability density function, $\pi(q)$, while expectations reduce to integrals over parameter space,

$$\mathbb{E}_\pi[f] = \int_\mathcal{Q} \text{d}q\pi(q)f(q)$$

The way this integral is written does not make sense to me. I would have expected it to be written as,

$$\mathbb{E}_\pi[f] = \int_\mathcal{Q} \pi(q)f(q) \text{d}q$$

i.e., we are integrating over the target distribution $\pi$ and the function $f$ with respect to $q$, in line with how we write e.g. mean for some random (continuous) variable $X$ and density $p$:

$$\mathbb{E}[X] = \int xp(x)\text{d}x$$

Are these somehow equivalent? Am I misunderstanding either the integral or what this expectation represents?

an4s
  • 3,716
Surculus
  • 143
  • 3
    It's a notational thing; while most folks tend to use the differential to 'bound' the integral by demarcating the end of the quantity being integrated, others put it right next to the integral sign presumably to make it clear what the variable (or measure) of integration is. Since $dq$ binds the variable $q$ as being the integration variable, the two are really equivalent. – Steven Stadnicki May 21 '20 at 16:29
  • Ohh, I was under the impression that the differential, by convention, must be at the end. Thank you, that makes sense. – Surculus May 21 '20 at 16:30
  • 3
    Putting the differential at the beginning, or elsewhere in the expression, seems a little more common in physics or other sciences, where they like to think of the differential as really being an infinitesimal quantity that's multiplied by the integrand. From that standpoint, since multiplication is commutative, it doesn't matter where it goes. Mathematicians tend to think of it instead as just part of the notation of an integral, and so they don't tend to move it around. – Nate Eldredge May 21 '20 at 16:57
  • @Surculus: You can also sometimes see notation such as

    $$ \int\frac{\mathrm dx}x;, $$

    so the differential is not only moved around but even treated as a factor, confirming what Nate Eldredge wrote about it being conceived of as a quantity. I agree that this is more common in physics than in mathematics. Why did you delete your answer? I thought it was a good idea to post an answer so you can mark the question as solved.

     – joriki May 21 '20 at 17:21
  • 1
    @joriki Thanks, that makes sense. Honestly I deleted it because I wasn't allowed to accept it until 2 days after asking the question, so I wasn't really sure what to do about it. Going to undelete and accept in another day.. – Surculus May 22 '20 at 13:57

1 Answers1

1

Steven Stadnicki answered the question in a comment: This is a notation I had never seen, i.e. not keeping the differential at the end of the integral.

Creating this answer so that I can mark the question as solved.

Surculus
  • 143