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More on Multivariate Gaussians
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For Standford CS229. Up to this point in class, you have seen multivariate Gaussians arise in a number of applications, such as the probabilistic interpretation of linear regression, Gaussian discriminant analysis, mixture of Gaussians clustering, and most recently, factor analysis. In these lecture notes, we attempt to demystify some of the fancier properties of multivariate Gaussians that were introduced in the recent factor analysis lecture. The goal of these notes is to give you some intuition into where these properties come from, so that you can use them with confidence on your homework (hint hint!) and beyond.
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More on Multivariate Gaussians
Chuong B. Do
November 21, 2008
Up to this point in class, you have seen multivariate Gaussians arise in a number of appli-
cations, such as the probabilistic interpretation of linear regression, Gaussian discriminant
analysis, mixture of Gaussians clustering, and most recently, factor analysis. In these lec-
ture notes, we attempt to demystify some of the fancier properties of multivariate Gaussians
that were introduced in the recent factor analysis lecture. The goal of these notes is to give
you some intuition into where these properties come from, so that you can use them with
confidence on your homework (hint hint!) and beyond.
1 Definition
A vector-valued random variable x ∈ R
n
is said to have a multivariate normal (or Gaus-
sian) distribution with mean µ ∈ R
n
and covariance matrix Σ ∈ S
n
++
1
if its probability
density function is given by
p(x; µ, Σ) =
1
(2π)
n/2
|Σ|
1/2
exp
−
1
2
(x − µ)
T
Σ
−1
(x − µ)
.
We write this as x ∼ N(µ, Σ).
2 Gaussian facts
Multivariate Gaussians turn out to be extremely handy in practice due to the following facts:
• Fact #1: If you know the mean µ and covariance matrix Σ of a Gaussian random
variable x, you can write down the probability density function for x directly.
1
Recall from the section notes on linear algebra that S
n
++
is the space of symmetric positive definite n ×n
matrices, defined as
S
n
++
=
A ∈ R
n×n
: A = A
T
and x
T
Ax > 0 for all x ∈ R
n
such that x 6= 0
.
1
• Fact #2: The following Gaussian integrals have closed-form solutions:
Z
x∈R
n
p(x; µ, Σ)dx =
Z
∞
−∞
···
Z
∞
−∞
p(x; µ, Σ)dx
1
. . . dx
n
= 1
Z
x∈R
n
x
i
p(x; µ, σ
2
)dx = µ
i
Z
x∈R
n
(x
i
− µ
i
)(x
j
− µ
j
)p(x; µ, σ
2
)dx = Σ
ij
.
• Fact #3: Gaussians obey a number of closure properties:
– The sum of independent Gaussian random variables is Gaussian.
– The marginal of a joint Gaussian distribution is Gaussian.
– The conditional of a joint Gaussian distribution is Gaussian.
At first glance, some of these facts, in particular facts #1 and #2, may seem either
intuitively obvious or at least plausible. What is probably not so clear, however, is why
these facts are so powerful. In this document, we’ll provide some intuition for how these facts
can be used when performing day-to-day manipulations dealing with multivariate Gaussian
random variables.
3 Closure properties
In this section, we’ll go through each of the closure properties described earlier, and we’ll
either prove the property using facts #1 and #2, or we’ll at least give some type of intuition
as to why the property is true.
The following is a quick roadmap of what we’ll cover:
sums marginals conditionals
why is it Gaussian? no yes yes
resulting density function
yes yes yes
3.1 Sum of independent Gaussians is Gaussian
The formal statement of this rule is:
Suppose that y ∼ N(µ, Σ) and z ∼ N(µ
′
, Σ
′
) are independent Gaussian dis-
tributed random variables, where µ , µ
′
∈ R
n
and Σ, Σ
′
∈ S
n
++
. Then, their sum
is also Gaussian:
y + z ∼ N(µ + µ
′
, Σ + Σ
′
).
Before we prove anything, here are some observations:
2
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