Gaussian processes

case the used model is not in the conjugate-exponential family, there is a need for further approximations [55].

A drawback with VBEM is that, even for fairly simple models, the deriva-tions required for the analytical approximation of the posterior density can be troublesome. However, the resulting VBEM algorithm is computationally efficient compared to many alternative methods that rely on Markov Chain Monte Carlo sampling [56].

5.3 Gaussian processes

With the focus on learning unknown functions based on data, in this section we introduce the concept of Gaussian processes [57, 58]. This is a Bayesian non-parametric method that can be used to describe a distribution over func-tions. A major advantage with this approach is that Gaussian processes are very flexible and are capable to describe very complex functions. The follow-ing discussion is based on the definition of a Gaussian process:

Definition: “A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution.”

For illustration, we consider an unknown function f , with argument x ∈ Rⁿ and scalar output y. In the discussion on Bayesian filtering, many of the involved densities are assumed to be Gaussian, characterized by its mean and covariance. Similarly, a prior for the function f can be defined by a Gaussian process that is specified by its mean function and covariance function. This prior is denoted by

f (x)∼ GP m(x), k(x, x^′)

, (5.22)

where the mean and covariance functions are defined as m(x) = E

By choosing the mean and covariance functions, any knowledge regarding the function can be incorporated into the prior. Based on this prior and a set of training samplesD = {x⁽ⁱ⁾, y⁽ⁱ⁾}^Ni=1 that are related through f , the objective in this section is to find a description of f (x_∗), where x_∗is an arbitrary vector in Rⁿ.

As often in reality, the observations are affected by noise. This noise is modelled as Gaussian such that each training sample is

y⁽ⁱ⁾ = f (x⁽ⁱ⁾) + w⁽ⁱ⁾, (5.24)

Chapter 5. Parameter and function estimation

where w⁽ⁱ⁾ ∼ N (0, σn²). From the definition of a Gaussian process, the noisy samples from the process are jointly Gaussian, implying that

y ∼ N m(x), K + σn²IN×N

Given the training samples and the prior knowledge regarding the function, the output for an arbitrary input vector can be estimated. Denoting a set of M input vectors by x_∗ = [(x⁽¹⁾_∗ )^T, . . . , (x^{(M )}_∗ )^T]^T, we wish to find the corresponding outputs y_∗ = [y_∗⁽¹⁾, . . . , y^{(M )}_∗ ]^T. From the definition of the Gaussian process, the vector of training outputs, y, and the sought outputs y_∗ are jointly Gaussian. That is,

 y

In [58], it is shown that the conditional distribution p(y_∗|y) is given as p(y_∗|y) = N m(x∗) + K^T_∗K⁻¹(y− m(x)), K∗∗− K^T_∗K⁻¹K_∗

. (5.28) To illustrate the discussed theory, the section ends with an example.

Example 5.2 (Gaussian processes)

In this example we consider an unknown function f , such that y = f (x), where both x and y are scalars. A prior of f can be specified as

f ∼ GP(m(x), k(x, x^′)) (5.29)

where m(x) and k(x, x^′) shall be chosen to summarize our prior knowledge regarding the function. Assume that we know that the function is fairly smooth and varies around y = 0. These properties are captured by

m(x) = 0 (5.30)

where L = 3. Changing L affects the smoothness of the function.

46

5.3 Gaussian processes

For illustration, we study the function on a limited interval by construct-ing a vector x that consists of n equally spaced values on [−30, 30]. A sample from the process in (5.29) is a function [58]:

fs(x) = m(x) +√

Figure 5.2: Samples from the prior process.

Assume that we observe the function for 7 different values of x. For i = 1, 2, . . . , 7, these samples can be described as

y⁽ⁱ⁾ = f (x⁽ⁱ⁾) + w⁽ⁱ⁾, (5.33) where w(i) ∼ N (0, σn²) is the observation noise. Depending on the noise level, the information regarding f obtained from the samples will differ. For illustration, we consider two scenarios. First, the noise variance is σ_n² = 0, i.e., we observe the true function values, and second, σ_n² = 1.

The estimated functions, including uncertainties, are shown in Figure 5.3. Worth noting in 5.3(a) is that when the samples are noise-free, the true function is observed and consequently there are no uncertainties regarding the function at these points. This can be compared to the case where the samples are noisy. Then, as shown in Figure 5.3(b), the samples provide less information about the underlying function. Finally, in both figures it is clear that in unobserved intervals, such as x ∈ [20, 30], the posterior function is very similar to the prior. The reason is that the available samples do not provide much information regarding the function on these intervals.

Chapter 5. Parameter and function estimation

x

-30 -20 -10 0 10 20 30

y

-3 -2 -1 0 1 2 3

Samples

Posterior mean function

± 2 standard deviations

(a) Posterior mean function including uncertainties for σ_n² = 0.

x

-30 -20 -10 0 10 20 30

y

-3 -2 -1 0 1 2 3

Samples

Posterior mean function

± 2 standard deviations

(b) Posterior mean function including uncertainties for σ²_n= 1.

Figure 5.3: The posterior functions for training samples with two different noise levels.

48

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