The goal of this section is to study a type of mathematical object that arises naturally in the context of conditional expected value and parametric probability distributions, and is of fundamental importance in the study of stochastic processes, particularly Markov processes. In a sense, the main object of study in this section is a generalization of a matrix, and the operations generalizations of matrix operations. If you keep this in mind, this section may seem less abstract.

Basic Theory

Definitions

More generally, \( S \) usually comes with a topology that is locally compact, Hausdorff, with a countable base (LCCB), and \( \mathscr S \) is the Borel \( \sigma \)-algebra, the \( \sigma \)-algebra generated by the topology (the collection of open subsets of \( S \)). The measure \( \mu \) is usually a Borel measure, and so satisfies \( \mu(C) \lt \infty \) if \( C \subseteq S \) is compact. A discrete measure space is of this type, corresponding to the discrete topology. A Euclidean measure space is also of this type, corresponding to the Euclidean topology, if \( S \) is open or closed (which is usually the case). In the discrete case, every function from \( S \) to another measurable space is measurable, and every from function from \( S \) to another topological space is continuous, so the measure theory is not really necessary.

Recall also that the measure space \((S, \mathscr S, \mu)\) is \(\sigma\)-finite if there exists a countable collection \(\{A_i: i \in I\} \subseteq \mathscr S\) such that \(\mu(A_i) \lt \infty\) for \(i \in I\) and \(S = \bigcup_{i \in I} A_i\). If \((S, \mathscr S, \mu)\) is a Borel measure space corresponding to an LCCB topology, then it is \(\sigma\)-finite.

If \(f: S \to \R\) is measurable, define \( \| f \| = \sup\{\left|f(x)\right|: x \in S\}\). Of course we may well have \(\|f\| = \infty\). Let \( \mathscr B(S) \) denote the collection of bounded measurable functions \( f: S \to \R \). Under the usual operations of pointwise addition and scalar multiplication, \( \mathscr B(S) \) is a vector space, and \(\| \cdot \|\) is the natural norm on this space, known as the supremum norm. This vector space plays an important role.

In this section, it is sometimes more natural to write integrals with respect to the positive measure \( \mu \) with the differential before the integrand, rather than after. However, rest assured that this is mere notation, the meaning of the integral is the same. So if \( f: S \to \R \) is measurable then we may write the integral of \( f \) with respect to \( \mu \) in operator notation as \[ \mu f = \int_S \mu(dx) f(x) \] assuming, as usual, that the integral exists. This will be the case if \( f \) is nonnegative, although \( \infty \) is a possible value. More generally, the integral exists in \( \R \cup \{-\infty, \infty\} \) if \( \mu f^+ \lt \infty \) or \( \mu f^- \lt \infty\) where \( f^+ \) and \( f^- \) are the positive and negative parts of \( f \). If both are finite, the integral exists in \( \R \) (and \( f \) is integrable with respect to \( \mu \)). If If \( \mu \) is a probability measure and we think of \( (S, \mathscr S) \) as the sample space of a random experiment, then we can think of \( f \) as a real-valued random variable, in which case our new notation is not too far from our traditional expected value \( \E(f) \). Our main definition comes next.

There are several classes of kernels that deserve special names.

So a probability kernel is bounded, a bounded kernel is finite, and a finite kernel is \(\sigma\)-finite. The terms stochastic kernel and Markov kernel are also used for probability kernels, and for a probability kernel \( \|K\| = 1 \) of course. The terms are consistent with terms used for measures: \( K \) is a finite kernel if and only if \( K(x, \cdot) \) is a finite measure for each \( x \in S \), and \( K \) is a probability kernel if and only if \( K(x, \cdot) \) is a probability measure for each \( x \in S \). Note that \( \|K\| \) is simply the supremum norm of the function \( x \mapsto K(x, T) \).

A kernel defines two natural integral operators, by operating on the left with measures, and by operating on the right with functions. As usual, we are often a bit casual witht the question of existence. Basically in this section, we assume that any integrals mentioned exist.

Thus, a kernel transforms measures on \( (S, \mathscr S) \) into measures on \( (T, \mathscr T) \), and transforms certain measurable functions from \( T \) to \( \R \) into measurable functions from \( S \) to \( \R \). Again, part (b) assumes that \( f \) is integrable with respect to the measure \( K(x, \cdot) \) for every \( x \in S \). In particular, the last statement will hold in the following important special case:

Thus, \( I(x, A) = 1 \) if \( x \in A \) and \( I(x, A) = 0 \) if \( x \notin A \). So \( x \mapsto I(x, A) \) is the indicator function of \( A \in \mathscr S \), while \( A \mapsto I(x, A) \) is point mass at \( x \in S \). Clearly the identity kernel is a probability kernel. If we need to indicate the dependence on the particular space, we will add a subscript. The following result justifies the name.

Constructions

We can create a new kernel from two given kernels, by the usual operations of addition and scalar multiplication.

A simple corollary of the last result is that if \(a, \, b \in [0, \infty)\) then \(a K + b L\) is a kerneal from \((S, \mathscr S)\) to \((T, \mathscr T)\). In particular, if \(K, \, L\) are probability kernels and \(p \in (0, 1)\) then \(p K + (1 - p) L\) is a probability kernel. A more interesting and important way to form a new kernel from two given kernels is via a multiplication operation.

Once again, the identity kernel lives up to its name:

The next several results show that the operations are associative whenever they make sense.

The next several results show that the distributive property holds whenever the operations makes sense.

In particular, note that if \( K \) is a kernel from \( (S, \mathscr S) \) to \( (T, \mathscr T) \), then the transformation \( \mu \mapsto \mu K \) defined for positive measures on \( (S, \mathscr S)\), and the transformation \( f \mapsto K f \) defined for measurable functions \( f: T \to \R \) (for which \( K f \) exists), are both linear operators. If \( \mu \) is a positive measure on \( (S, \mathscr S) \), then the integral operator \( f \mapsto \mu f \) defined for measurable \( f: S \to \R \) (for which \( \mu f \) exists) is also linear, but of course, we already knew that. Finally, note that the operator \( f \mapsto K f \) is positive: if \( f \ge 0 \) then \( K f \ge 0 \). Here is the important summary of our results when the kernel is bounded.

The commutative property for the product of kernels fails with a passion. If \( K \) and \( L \) are kernels, then depending on the measurable spaces, \( K L \) may be well defined, but not \( L K \). Even if both products are defined, they may be kernels from or to different measurable spaces. Even if both are defined from and to the same measurable spaces, it may well happen that \( K L \neq L K \). Exercise gives some examples.

If \( K \) is a kernel on \( (S, \mathscr S) \) and \( n \in \N \), we let \( K^n = K K \cdots K \), the \( n \)-fold power of \( K \). By convention, \( K^0 = I \), the identity kernel on \( S \).

Fixed points of the operators associated with a kernel turn out to be very important.

So in the language of linear algebra (or functional analysis), an invariant measure is a left eigenvector of the kernel, while an invariant function is a right eigenvector of the kernel, both corresponding to the eigenvalue 1. By our results above, if \( \mu \) and \( \nu \) are invariant measures and \( c \in [0, \infty) \), then \( \mu + \nu \) and \( c \mu \) are also invariant. Similarly, if \( f \) and \( g \) are invariant functions and \( c \in \R \), the \( f + g \) and \( c f \) are also invariant.

Of couse we are particularly interested in probability kernels.

As a corollary, it follows that if \( P \) is a probability kernel on \( (S, \mathscr S) \), then so is \( P^n \) for \( n \in \N \).

The operators associated with a kernel are of fundamental importance, and we can easily recover the kernel from the operators. Suppose that \( K \) is a kernel from \( (S, \mathscr S) \) to \( (T, \mathscr T) \), and let \( x \in S \) and \( A \in \mathscr T \). Then trivially, \(K \bs{1}_A(x) = K(x, A)\) where as usual, \( \bs{1}_A \) is the indicator function of \( A \). Trivially also \( \delta_x K(A) = K(x, A) \) where \( \delta_x \) is point mass at \( x \).

Kernel Functions

Usually our measurable spaces are in fact measure spaces, with natural measures associated with the spaces, as in the special cases described in . When we start with measure spaces, kernels are usually constructed from density functions in much the same way that positive measures are defined from density functions.

Clearly the kernel \( K \) depends on the positive measure \( \mu \) on \( (T, \mathscr T) \) as well as the function \( k \), while the measure \( \lambda \) on \( (S, \mathscr S) \) plays no role (and so is not even necessary). But again, our point of view is that the spaces have fixed, natural measures. Appropriately enough, the function \( k \) is called a kernel density function (with respect to \( \mu \)), or simply a kernel function.

A kernel function defines an operator on the left with functions on \( S \) in a completely analogous way to the operator on the right above with functions on \( T \).

The operator defined above depends on the measure \( \lambda \) on \( (S, \mathscr S) \) as well as the kernel function \( k \), while the measure \( \mu \) on \( (T, \mathscr T) \) playes no role (and so is not even necessary). But again, our point of view is that the spaces have fixed, natural measures. Here is how our new operation on the left with functions relates to our old operation on the left with measures.

As always, we are particularly interested in stochastic kernels. With a kernel function, we can have doubly stochastic kernels.

Of course, condition (a) simply means that the kernel associated with \( k \) is a stochastic kernel according to our original definition.

The most common and important special case is when the two spaces are the same. Thus, if \( (S, \mathscr S, \lambda) \) is a measure space and \( k : S \times S \to [0, \infty) \) is measurable, then we have an operator \( K \) that operates on the left and on the right with measurable functions \( f: S \to \R \): \begin{align*} f K(y) & = \int_S \lambda(dx) f(x) k(x, y), \quad y \in S \\ K f(x) & = \int_S k(x, y) f(y) \lambda(d y), \quad x \in S \end{align*} If \( f \) is nonnegative and \( \mu \) is the measure on with density function \( f \), then \( f K \) is the density function of the measure \( \mu K \) (both with respect to \( \lambda \)).

Of course, if \( k \) is a symmetric, stochastic kernel function on \( (S, \mathscr S, \lambda) \) then \( k \) is doubly stochastic, but the converse is not true.

Examples and Special Cases

The Discrete Case

In this subsection, we assume that the measure spaces are discrete, as described in (1). Since the \( \sigma \)-algebra (all subsets) and the measure (counting measure) are understood, we don't need to reference them. Recall that integrals with respect to counting measure are sums. Suppose now that \( K \) is a kernel from the discrete space \(S\) to the discrete space \(T\). For \( x \in S \) and \( y \in T \), let \( K(x, y) = K(x, \{y\}) \). Then more generally, \[ K(x, A) = \sum_{y \in A} K(x, y), \quad x \in S, \, A \subseteq T \] The function \( (x, y) \mapsto K(x, y) \) is simply the kernel function of the kernel \( K \), as defined in , but in this case we usually don't bother with using a different symbol for the function as opposed to the kernel. The function \( K \) can be thought of as a matrix, with rows indexed by \( S \) and columns indexed by \( T \) (and so an infinite matrix if \( S \) or \( T \) is countably infinite). With this interpretation, all of the operations defined above can be thought of as matrix operations. If \( f: T \to \R \) and \( f \) is thought of as a column vector indexed by \( T \), then \( K f \) is simply the ordinary product of the matrix \( K \) and the vector \( f \); the product is a column vector indexed by \( S \): \[K f(x) = \sum_{y \in S} K(x, y) f(y), \quad x \in S \] Similarly, if \( f: S \to \R \) and \( f \) is thought of as a row vector indexed by \( S \), then \( f K \) is simple the ordinary product of the vector \( f \) and the matrix \( K \); the product is a row vector indexed by \( T \): \[ f K(y) = \sum_{x \in S} f(x) K(x, y), \quad y \in T \] If \( L \) is another kernel from \( T \) to another discrete space \( U \), then as functions, \( K L \) is the simply the matrix product of \( K \) and \( L \): \[ K L(x, z) = \sum_{y \in T} K(x, y) L(x, z), \quad (x, z) \in S \times L \]

Conditional Probability

An important class of probability kernels arises from the distribution of one random variable, conditioned on the value of another random variable. In this subsection, suppose that \( (\Omega, \mathscr{F}, \P) \) is a probability space, and that \( (S, \mathscr S) \) and \( (T, \mathscr T) \) are measurable spaces. Further, suppose that \( X \) and \( Y \) are random variables defined on the probability space, with \( X \) taking values in \( S \) and \( Y \) taking values in \( T \). Informally, \( X \) and \( Y \) are random variables defined on the same underlying random experiment.

The operators associated with this kernel have natural interpretations.

As in the general discussion above, the measurable spaces \( (S, \mathscr S) \) and \( (T, \mathscr T) \) are usually measure spaces with natural measures attached. So the conditional probability distributions are often given via conditional probability density functions, which then play the role of kernel functions. The next two exercises give examples.

Parametric Distributions

A parametric probability distribution also defines a probability kernel in a natural way, with the parameter playing the role of the kernel variable, and the distribution playing the role of the measure. Such distributions are usually defined in terms of a parametric density function which then defines a kernel function, again with the parameter playing the role of the first argument and the variable the role of the second argument. If the parameter is thought of as a given value of another random variable, as in Bayesian analysis, then there is considerable overlap with the previous subsection. In most cases, (and in particular in the examples below), the spaces involved are either discrete or Euclidean, as described in .

Clearly the Poisson distribution has some very special and elegant properties. The next family of distributions also has some very special properties. Compare this exercise with .