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Probability Theory Review for Machine Learning
Samuel Ieong
November 6, 2006
1 Basic Concepts
Broadly speaking, probability theory is the mathematical study of uncertainty. It plays a
central role in machine learning, as the design of learning algorithms often relies on proba-
bilistic assumption of the data. This set of notes attempts to cover some basic probability
theory that serves as a background for the class.
1.1 Probability Space
When we speak about probability, we often refer to the probability of an event of uncertain
nature taking place. For example, we speak about the probability of rain next Tuesday.
Therefore, in order to discuss probability theory formally, we must first clarify what the
possible events are to which we would like to attach probability.
Formally, a probability space is defined by the triple (Ω, F, P ), where
• Ω is the space of possible outcomes (or outcome space),
• F ⊆ 2
Ω
(the power set of Ω) is the space of (measurable) events (or event space),
• P is the probability measure (or probability distribution) that maps an event E ∈ F to
a real value between 0 and 1 (think of P as a function).
Given the outcome space Ω, there is some restrictions as to what subset of 2
Ω
can be
considered an event space F:
• The trivial event Ω and the empty event ∅ is in F.
• The event space F is closed under (countable) union, i.e., if α, β ∈ F, then α ∪β ∈ F.
• The even space F is closed under complement, i.e., if α ∈ F, then (Ω \ α) ∈ F.
Example 1. Suppose we throw a (six-sided) dice. The space of possible outcomes Ω =
{1, 2, 3, 4, 5, 6}. We may decide that the events of interest is whether the dice throw is odd
or even. This event space will be given by F = {∅, {1, 3, 5}, {2, 4, 6}, Ω}.
1
Note that when the outcome space Ω is finite, as in the previous example, we often take
the event space F to be 2
Ω
. This treatment is not fully general, but it is often sufficient
for practical purposes. However, when the outcome space is infinite, we must be careful to
define what the event space is.
Given an event space F, the probability measure P must satisfy certain axioms.
• (non-negativity) For all α ∈ F, P (α) ≥ 0.
• (trivial event) P (Ω) = 1.
• (additivity) For all α, β ∈ F and α ∩ β = ∅, P (α ∪ β) = P (α) + P (β).
Example 2. Returning to our dice example, suppose we now take the event space F to be
2
Ω
. Further, we define a probability distribution P over F such that
P ({1}) = P ({2}) = ··· = P ({6}) = 1/6
then this distribution P completely specifies the probability of any given event happening
(through the additivity axiom). For example, the probability of an even dice throw will be
P ({2, 4, 6}) = P ({2}) + P ({4}) + P({6}) = 1/6 + 1/6 + 1/6 = 1/2
since each of these events are disjoint.
1.2 Random Variables
Random variables play an important role in probability theory. The most important fact
about random variables is that they are not variables. They are actually functions that
map outcomes (in the outcome space) to real values. In terms of notation, we usually denote
random variables by a capital letter. Let’s see an example.
Example 3. Again, consider the process of throwing a dice. Let X be a random variable that
depends on the outcome of the throw. A natural choice for X would be to map the outcome
i to the value i, i.e., mapping the event of throwing an “one” to the value of 1. Note that
we could have chosen some strange mappings too. For example, we could have a random
variable Y that maps all outcomes to 0, which would be a very boring function, or a random
variable Z that maps the outcome i to the value of 2
i
if i is odd and the value of −i if i is
even, which would be quite strange indeed.
In a sense, random variables allow us to abstract away from the formal notion of event
space, as we can define random variables that capture the appropriate events. For example,
consider the event space of odd or even dice throw in Example 1. We could have defined a
random variable that takes on value 1 if outcome i is odd and 0 otherwise. These type of
binary random variables are very common in practice, and are known as indicator variables,
taking its name from its use to indicate whether a certain event has happened. So why
did we introduce event space? That is because when one studies probability theory (more
2
rigorously) using measure theory, the distinction between outcome space and event space
will be very important. This topic is too advanced to be covered in this short review note.
In any case, it is good to keep in mind that event space is not always simply the power set
of the outcome space.
From here onwards, we will talk mostly about probability with respect to random vari-
ables. While some probability concepts can be defined meaningfully without using them,
random variables allow us to provide a more uniform treatment of probability theory. For
notations, the probability of a random variable X taking on the value of a will be denoted
by either
P (X = a) or P
X
(a)
We will also denote the range of a random variable X by V al(X).
1.3 Distributions, Joint Distributions, and Marginal Distributions
We often speak about the distribution of a variable. This formally refers to the probability
of a random variable taking on certain values. For example,
Example 4. Let random variable X be defined on the outcome space Ω of a dice throw
(again!). If the dice is fair, then the distribution of X would be
P
X
(1) = P
X
(2) = ··· = P
X
(6) = 1/6
Note that while this example resembles that of Example 2, they have different semantic
meaning. The probability distribution defined in Example 2 is over events, whereas the one
here is defined over random variables.
For notation, we will use
P
(
X
) to denote the distribution of the random variable
X
.
Sometimes, we speak about the distribution of more than one variables at a time. We
call these distributions joint distributions, as the probability is determined jointly by all the
variables involved. This is best clarified by an example.
Example 5. Let X be a random variable defined on the outcome space of a dice throw. Let
Y be an indicator variable that takes on value 1 if a coin flip turns up head and 0 if tail.
Assuming both the dice and the coin are fair, the joint distribution of X and Y is given by
P X = 1 X = 2 X = 3 X = 4 X = 5 X = 6
Y = 0 1/12 1/12 1/12 1/12 1/12 1/12
Y = 1 1/12 1/12 1/12 1/12 1/12 1/12
As before, we will denote the probability of X taking value a and Y taking value b by
either the long hand of P (X = a, Y = b), or the short hand of P
X,Y
(a, b). We refer to their
joint distribution by P (X, Y ).
Given a joint distribution, say over random variables X and Y , we can talk about the
marginal distribution of X or that of Y . The marginal distribution refers to the probability
distribution of a random variable on its own. To find out the marginal distribution of a
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