3
$\begingroup$

What kind of integral is this?

It pops up in Pattern Recognition and Machine Learning, Chapter 1.2, when the author is talking about multivariate probability densities.

$$ \int p(\vec{x}) d\vec{x} = 1$$

The author states "the integral is taken over the whole of $\vec{x}$ space."

I've heard of line integrals of vector fields and surface integrals of vector fields, are there other types of weird integrals that pop up where you are taking the integral with respect to a vector or matrix for example?

Another integral (1.45) pops up again in Chapter 1.2 when the author is talking about Bayes theorem, the posterior, likelihood and prior.

$$p(D) = \int p(D|\vec{w})p(\vec{w}) d\vec{w} $$

Is this different to the previous integral?

$\endgroup$
1
  • $\begingroup$ These are the Lebesgue integrals on $\mathbb{R}^d$. $\endgroup$ Commented Sep 26, 2020 at 8:39

1 Answer 1

Reset to default
2
$\begingroup$

If $\vec{x}\in\Bbb R^n$, it's an $n$-tuple integral. We often write $d\vec{x}$ as $d^n\vec{x}$. In fact, such an exponent makes it so obvious $x$ is a vector we can, as I just did, drop the $\vec{}$, viz. $\int_Sp(x)d^nx=1$. (I've also added an integration range $S\subseteq\Bbb R^n$, typically $S=\Bbb R^n$; unitarity is specified with a definite integral.) The second integral is also a definite multiple integral.

$\endgroup$
4
  • 1
    $\begingroup$ When you say they are 'multiple integrals' do you mean that $$ \int p(\vec{x}) d\vec{x} = \int \int ... \int p(\vec{x}) dx_{1} dx_{2} ... dx_{n} $$ $\endgroup$ Commented Sep 26, 2020 at 11:47
  • 1
    $\begingroup$ @tail_recursion Exactly: $d^nx=\prod_{i=1}^ndx_i$. $\endgroup$ Commented Sep 26, 2020 at 11:49
  • $\begingroup$ @J.G. It is thus a volume integral. However $p(\vec{x})d\vec{x}$ might also mean a line integral. So how do we understand when it is a line integral and when it is volume integral? Do we determine this from the domain of integration (like in this case it was all of $\Bbb R^n$)? $\endgroup$ Commented Jan 11, 2022 at 2:01
  • 1
    $\begingroup$ @AnirbanChakraborty Apart from the author having provided that context, even one so lazy as to omit a definite integral's limits should denote line integrals with $\oint$. $\endgroup$ Commented Jan 11, 2022 at 7:24

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.