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Notes on Probability

This document summarizes key concepts in probability theory.

The concepts in this note are introduced in Week 4.

Set Concepts and Notation

  • A set \(A\) is an unordered collection of elements.
  • \(\emptyset\) is the empty set.
  • \(A \cup B\) is the union: all elements in either \(A\) or \(B\) (or both).
  • \(A \cap B\) is the intersection: all elements in both \(A\) and \(B\).
  • \(A \setminus B\) is the set difference: all elements in \(A\) but not in \(B\).
  • If \(A\) is a subset of some larger set \(U\) that contains all the possible elements under consideration, then the complement \(A^c = U \setminus A\) is the set of elements not in \(A\).
  • \(|A|\) is the cardinality (or size) of \(A\). It may be infinite.
  • \(\mathcal{P}(A)\) is the power set of \(A\): the set of all subsets of \(A\).

Kinds of Sets

There are, broadly speaking, three kinds of sets in terms of their cardinality:

  • Finite sets have a finite number of elements. There is some natural number \(n\) such that \(|A| = n\).
  • Countable sets (or countably infinite sets) are infinite sets with the same cardinality as the set of natural numbers (\(|A| = \mathbb{N}\)). Formally, there exists an isomorphism (a 1:1 onto mapping) between members of \(A\) and \(\mathbb{N}\). Natural numbers, integers, rationals, and algebraics (rationals and roots) are all countable sets.
  • Uncountable sets are infinite sets whose cardinality is larger than that of the natural numbers. The real numbers (\(\mathbb{R}\)) and the power set of the natural numbers (\(\mathcal{P}(\mathbb{N})\)) are two frequently-encountered uncountable sets.

We also talk about discrete and continuous sets:

  • A continuous set \(A\) with an order \(<\) is a set where we can always find an element to fit between any two other elements: for any \(a, b \in A\) such that \(a < b\), there is a \(c \in A\) such that \(a < c < b\).
  • A discrete set is a set that is not continuous: there are irreducible gaps between elements.

All finite sets are discrete. The natural numbers and integers are also discrete. The real numbers are continuous. Rationals and algebraics are also continuous, but we won't be using them directly in this class.

Note

While \(\mathbb{R}\) is continuous, the floating-point numbers are actually discrete and, technically, finite (there are only \(2^{64}\) possible values of a 64-bit quantity). However, we are usually using floats to represent reals, so for logical, mathematical purposes, we usually treat quantities represented in floats as continuous. The representation is an approximation of the true quantities.

Events

A random process (or a process modeled as random) produces distinct individual outcomes, called elementary events. We use \(E\) to denote the set of such outcomes; for a coin flip, \(E = \{H, T\}\). For a random process that produces a count, \(E = \mathbb{N}\).

Probability is defined over events. An event \(A\) is a subset of \(E\) (\(A \subseteq E\)). If elementary events are events, they are represented as singletons: \(A = \{H\}\) means “coin is heads”. \(E\), the set of all elementary events, is the event “something happened”.

We use set operations to combine events:

  • \(A \cap B\) is the event “both \(A\) and \(B\) happened”.
  • \(A \cup B\) is the event “either \(A\) or \(B\) (or both) happened”.
  • \(A \setminus B\) is the event “\(A\) happened but not \(B\)”. If \(B \subseteq A\), then \(A \setminus B = \emptyset\).

With these definitions, we can now define the event space: \(\F\) is the set of all possible events (subsets of \(E\)). This is a set of sets. It does not necessarily contain every subset of \(E\), but it has the following properties:

  • \(E \in \F\).
  • If \(A \in \F\), then its complement \(A^c \in \F\). We say \(\F\) is closed under complement.
    • Since \(E \in \F\) and \(E^c = \emptyset\), \(\emptyset \in \F\).
  • If \(A_1, A_2, \dots, A_n \in \F\), then their union \(\bigcup_i A_i \in \F\). This applies also to unions of countably many sets. We say \(\F\) is closed under countable unions.

\(\F\) is called a sigma field (or sigma algebra). For a finite set \(E\), we usually use \(\F = \mathcal{P}(E)\), the power set of \(E\). This means that every possible subset of \(E\) is an event.

Here are some additional properties of sigma fields (these are listed separately from the previous properties because those are the definition of a sigma field and these are consequences — we can prove them from the definitions and axioms):

  • If \(A, B \in \F\), \(A \cap B \in \F\)

Probability

Now that we have a sigma field, we can define the concept of probability. A probability distribution (or measure) \(P\) over a sigma field \(\F\) obeys the following (Kolmogorov's axioms):

  • \(P(E) = 1\) — the probability of something happening is 1.
  • \(P(A) \ge 0\)non-negativity: probabilities are not negative.
  • If \(A_1, A_2, \dots, A_n\) are (countably many) disjoint events in \(\F\), then \(P(\bigcup_i A_i) = \sum_i P(A_i)\) (countable additivity).

A collection of disjoint sets is also called mutually exclusive. What it means is that for any \(A_i, A_j\) in the collection, \(A_i \cap A_j = \emptyset\) — the two events cannot both happen simultaneously.

We a field of events equipped with a probability measure \((E, \F, P)\) a probability space.

Some additional facts about probability:

  • \(P(A) \le 1\) (combined with non-negativity, we have \(0 \le P(A) \le 1\))
  • \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
  • \(P(A^c) = 1 - P(A)\)
  • \(P(A \setminus B) = P(A) - P(A \cap B)\)
  • If \(A \subseteq B\), then \(P(A) \le P(B)\)

Joint and Conditional Probability

We define the joint probability \(P(A, B) = P(A \cap B)\): the probability of both \(A\) and \(B\) happening in the same observation. This is sometimes also written \(P(A; B)\), and commas and semicolons are sometimes mixed. This is usually to separate different kinds of events in the probability statement.

The conditional probability \(P(B|A)\), read “the probability of \(A\) given \(B\)”, is the probability of \(B\) conditioned on the knowledge that \(A\) has happened.

Conditional and joint probabilities decompose as follows:

  • \(P(A,B) = P(A|B) P(B)\)
  • \(P(A,B) = P(B|A) P(A)\)

From this we can derive Bayes' theorem:

\[P(B|A) = \frac{P(A|B) P(B)}{P(A)}\]

We can marginalize a joint distribution by summing. If \(\mathcal{B} = {B_1, B_2, \dots, B_n}\) is a collection of mutually exclusive events that span \(E\), then:

\[P(A) = \sum_{B \in \mathcal{B}} P(A, B)\]

We call \(\mathcal{B}\) a partition of \(E\). By “span \(E\)”, we mean that for any \(e \in E\), there is some \(B_i \in \mathcal{B}\) such that \(e \in B_i\).

Independence

Two events are independent if knowing the outcome of one tells you nothing about the probability of the other. The following are true if and only if \(A\) and \(B\) are independent:

  • \(P(A|B) = P(A)\)
  • \(P(B|A) = P(B)\)
  • \(P(A, B) = P(A) P(B)\)

Continuous Probability & Random Variables

If \(E\) is continuous (typically \(E = \mathbb{R}\)), then we can't meaningfully talk about the probabilities of elementary events. The probability that an observation is exactly any particular value \(x \in \mathbb{R}\) is (typically) zero.

Instead, we define a sigma field where events are intervals:

  • \(E = \mathbb{R}\)
  • \(\F\) is the set of intervals, their complements, and their countable unions. It contains infinitesimally small intervals, but not singletons.

This is not the only way to define probabilities over continuous event spaces, but it is the common way of defining probabilities over real values. This particular sigma-field is called the Borel sigma algebra, and we will denote it \((\mathbb{R}, \mathcal{B})\).

We often talk about continuous distributions as the distribution of a random variable \(X\). A random variable is a variable that takes on random values. We can (often) observe or sample a random variable.

We define continuous probabilities in terms of a distribution function \(F\):

\[F(x) = P(X \le x)\]

This is also called the continuous distribution function (PDF).

We can use it to compute the probability for any interval:

\[P(x_1 < X \le x_2) = F(x_2) - F(x_1)\]

This probability is called the probability mass on a particular interval.

Distributions are often defined by a probability density function \(p\) such that

\[F(x) = \int_{-\infty}^x p(x_*) dx_*\]

Unlike probabilities or probability mass, densities can exceed 1. When you use sns.distplot and it shows the kernel density estimator (KDE), it is showing you an estimate of the density. That is why the \(y\) axis is weird.

We can also talk about joint and conditional continuous probabilities and densities. When marginalizing a continuous probability density, we replace the sum with an integral:

\[p(x) = \int p(x,y) dy\]

Note

Technically, a random variable for a probability space \((E, \mathcal{F}, P)\) is a function \(f_X: E \to \mathbb{R}\). For our purposes in this class, we can just treat random varaibles directly as the results of a probability space \((\mathbb{R}, \mathcal{B}, P)\).

Expectation

The expected value of a random variable \(X\), \(\E[X]\), is its mean. It is computed as the weighted sum over the possible values of \(x\), where the weight for each value is its probability (or density). For discrete \(X\) with probability measure \(P\), we have:

\[\E[X] = \sum_{x \in X} x P(x)\]

If \(X\) is continuous and has probability density \(p\), we have:

\[\E[X] = \int x p(x) dx\]

Note

If we use the technical definition of a random variable, then we denote:

\[\E_P[f_X] = \int f_X(e) P(e) de\]

We can also talk about the conditional expectation \(\E[X | A]\), the expected value of \(X\) given that we know event \(A\) happened. It is defined as \(\E[X|A] = \int x p(x|A) dx\).

Variance and Covariance

The variance of a random variable \(X\) is the expected value of its squared deviation from its mean:

\[\Var(X) = \E[(X - \E[X])^2]\]

The standard deviation is the square root of variance (\(\sigma_X = \sqrt{\Var(X)}\)).

The covariance of two random variables is the expected value of the product of their deviations from mean:

\[\Cov(X, Y) = \E[(X - \E[X]) (Y - \E[Y])]\]

The correlation \(r_{XY} = \frac{\Cov(X, Y)}{\sigma_X \sigma_Y}\).

We can also show that \(\Var(X) = \Cov(X, X)\).

Random variables can also be described as independent in the same way as events: knowing one tells you nothing about the other. If two random variables are independent then their covariance \(\Cov(X, Y) = 0\) (this implication is one-directional — there exist non-independent random variables whose covariance is 0).

Properties of Expected Values

Expected value obeys a number of useful properties (\(X\) and \(Y\) are random variables, and \(\alpha\), \(\beta\), etc. are real numbers):

  • Linearity of expectation:
    • \(\E[X + Y] = \E[X] + \E[Y]\)
    • \(\E[\alpha X] = \alpha \E[X]\)
  • If \(X\) and \(Y\) are independent, then \(\E[XY] = \E[X] \E[Y]\)
  • If \(\E[X] = 0\), then \(\Var(X) = \E[X^2]\)
  • If \(\E[X] = \E[Y] = 0\), then \(\Cov(X, Y) = \E[X Y]\)

Expectation of Indicator Functions

Sets can be described as an indicator function (or characteristic function) \(\IND_A: E \to {0,1}\). This function is defined as:

\[\IND_A(x) = \begin{cases} 1 & x \in A \\ 0 & x \not\in A \end{cases}\]

Then the expected value of this function is the same as the probability of \(A\):

\[\E[\IND_A(X)] = \P(A)\]

Odds

Another way of computing probability is to compute with odds: the ratio of probabilities for or against an event. This is given by:

\[\Odds(A) = \frac{P(A)}{P(A^c)} = \frac{P(A)}{1 - P(A)}\]

The log odds are often computationally convenient, and are the basis of logistic regression:

\[\log \Odds(a) = \log P(A) \log (1 - P(A))\]

The logit function converts probabilities to log-odds.

We can also compute an odds ratio of two outcomes:

\[\OR(A, B) = \frac{\Odds(A)}{\Odds(B)}\]

Further Reading

If you want to dive more deeply into probability theory, Michael Betancourt's case studies are rather mathematically dense but quite good:

For a book: