**Abstract**: In this talk we give an introduction to Gaussian processes for students
who are interested in working with GPs for the the R250 module.

Figure: *Pierre-Simon Laplace 1749-1827.*

In [ ]:

```
import pods
pods.notebook.display_google_book(id='1YQPAAAAQAAJ', page='PR17-IA2')
```

Famously, Laplace considered the idea of a deterministic Universe, one
in which the model is *known*, or as the below translation refers to it,
"an intelligence which could comprehend all the forces by which nature
is animated". He speculates on an "intelligence" that can submit this
vast data to analysis and propsoses that such an entity would be able to
predict the future.

Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it---an intelligence sufficiently vast to submit these data to analysis---it would embrace in the same formulate the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present in its eyes.

This notion is known as *Laplace's demon* or *Laplace's superman*.

Figure: *Laplace's determinsim in English translation.*

Unfortunately, most analyses of his ideas stop at that point, whereas
his real point is that such a notion is unreachable. Not so much
*superman* as *strawman*. Just three pages later in the "Philosophical
Essay on Probabilities" [@Laplace:essai14], Laplace goes on to observe:

The curve described by a simple molecule of air or vapor is regulated in a manner just as certain as the planetary orbits; the only difference between them is that which comes from our ignorance.

Probability is relative, in part to this ignorance, in part to our knowledge.

In [ ]:

```
import pods
pods.notebook.display_google_book(id='1YQPAAAAQAAJ', page='PR17-IA4')
```

Figure: *To Laplace, determinism is a strawman. Ignorance of mechanism
and data leads to uncertainty which should be dealt with through
probability.*

In other words, we can never make use of the idealistic deterministic Universe due to our ignorance about the world, Laplace's suggestion, and focus in this essay is that we turn to probability to deal with this uncertainty. This is also our inspiration for using probability in machine learning.

The "forces by which nature is animated" is our *model*, the "situation
of beings that compose it" is our *data* and the "intelligence
sufficiently vast enough to submit these data to analysis" is our
compute. The fly in the ointment is our *ignorance* about these aspects.
And *probability* is the tool we use to incorporate this ignorance
leading to uncertainty or *doubt* in our predictions.

One view of Bayesian inference is to assume we are given a mechanism for generating samples, where we assume that mechanism is representing on accurate view on the way we believe the world works.

This mechanism is known as our *prior* belief.

We combine our prior belief with our observations of the real world by
discarding all those samples that are inconsistent with our prior. The
*likelihood* defines mathematically what we mean by inconsistent with
the prior. The higher the noise level in the likelihood, the looser the
notion of consistent.

The samples that remain are considered to be samples from the
*posterior*.

This approach to Bayesian inference is closely related to two sampling
techniques known as *rejection sampling* and *importance sampling*. It
is realized in practice in an approach known as *approximate Bayesian
computation* (ABC) or likelihood-free inference.

In practice, the algorithm is often too slow to be practical, because most samples will be inconsistent with the data and as a result the mechanism has to be operated many times to obtain a few posterior samples.

However, in the Gaussian process case, when the likelihood also assumes
Gaussian noise, we can operate this mechanism mathematically, and obtain
the posterior density *analytically*. This is the benefit of Gaussian
processes.

In [ ]:

```
import pods
from ipywidgets import IntSlider
```

In [ ]:

```
pods.notebook.display_plots('gp_rejection_sample{sample:0>3}.png',
directory='../slides/diagrams/gp',
sample=IntSlider(1,1,5,1))
```

Figure: *One view of Bayesian inference is we have a machine for
generating samples (the prior), and we discard all samples
inconsistent with our data, leaving the samples of interest (the
posterior). The Gaussian process allows us to do this
analytically.*

What is machine learning? At its most basic level machine learning is a combination of

$$\text{data} + \text{model} \stackrel{\text{compute}}{\rightarrow} \text{prediction}$$where *data* is our observations. They can be actively or passively
acquired (meta-data). The *model* contains our assumptions, based on
previous experience. That experience can be other data, it can come from
transfer learning, or it can merely be our beliefs about the
regularities of the universe. In humans our models include our inductive
biases. The *prediction* is an action to be taken or a categorization or
a quality score. The reason that machine learning has become a mainstay
of artificial intelligence is the importance of predictions in
artificial intelligence. The data and the model are combined through
computation.

In practice we normally perform machine learning using two functions. To combine data with a model we typically make use of:

**a prediction function** a function which is used to make the
predictions. It includes our beliefs about the regularities of the
universe, our assumptions about how the world works, e.g. smoothness,
spatial similarities, temporal similarities.

**an objective function** a function which defines the cost of
misprediction. Typically it includes knowledge about the world's
generating processes (probabilistic objectives) or the costs we pay for
mispredictions (empiricial risk minimization).

The combination of data and model through the prediction function and
the objectie function leads to a *learning algorithm*. The class of
prediction functions and objective functions we can make use of is
restricted by the algorithms they lead to. If the prediction function or
the objective function are too complex, then it can be difficult to find
an appropriate learning algorithm. Much of the acdemic field of machine
learning is the quest for new learning algorithms that allow us to bring
different types of models and data together.

A useful reference for state of the art in machine learning is the UK Royal Society Report, Machine Learning: Power and Promise of Computers that Learn by Example.

You can also check my post blog post on What is Machine Learning?..

In practice, we normally also have uncertainty associated with these functions. Uncertainty in the prediction function arises from

- scarcity of training data and
- mismatch between the set of prediction functions we choose and all possible prediction functions.

There are also challenges around specification of the objective function, but for we will save those for another day. For the moment, let us focus on the prediction function.

Neural networks are adaptive non-linear function models. Originally, they were studied (by McCulloch and Pitts [@McCulloch:neuron43]) as simple models for neurons, but over the last decade they have become popular because they are a flexible approach to modelling complex data. A particular characteristic of neural network models is that they can be composed to form highly complex functions which encode many of our expectations of the real world. They allow us to encode our assumptions about how the world works.

We will return to composition later, but for the moment, let's focus on a one hidden layer neural network. We are interested in the prediction function, so we'll ignore the objective function (which is often called an error function) for the moment, and just describe the mathematical object of interest

$$ \mappingFunction(\inputVector) = \mappingMatrix^\top \activationVector(\mappingMatrixTwo, \inputVector) $$Where in this case $\mappingFunction(\cdot)$ is a scalar function with
vector inputs, and $\activationVector(\cdot)$ is a vector function with
vector inputs. The dimensionality of the vector function is known as the
number of hidden units, or the number of neurons. The elements of this
vector function are known as the *activation* function of the neural
network and $\mappingMatrixTwo$ are the parameters of the activation
functions.

In statistics activation functions are traditionally known as *basis
functions*. And we would think of this as a *linear model*. It's doesn't
make linear predictions, but it's linear because in statistics
estimation focuses on the parameters, $\mappingMatrix$, not the
parameters, $\mappingMatrixTwo$. The linear model terminology refers to
the fact that the model is *linear in the parameters*, but it is *not*
linear in the data unless the activation functions are chosen to be
linear.

The first difference in the (early) neural network literature to the classical statistical literature is the decision to optimize these parameters, $\mappingMatrixTwo$, as well as the parameters, $\mappingMatrix$ (which would normally be denoted in statistics by $\boldsymbol{\beta}$)[^1].

We're going to go revisit that decision, and follow the path of Radford Neal [@Neal:bayesian94] who, inspired by work of David MacKay [@MacKay:bayesian92] and others did his PhD thesis on Bayesian Neural Networks. If we take a Bayesian approach to parameter inference (note I am using inference here in the classical sense, not in the sense of prediction of test data, which seems to be a newer usage), then we don't wish to fit parameters at all, rather we wish to integrate them away and understand the family of functions that the model describes.

This Bayesian approach is designed to deal with uncertainty arising from fitting our prediction function to the data we have, a reduced data set.

The Bayesian approach can be derived from a broader understanding of what our objective is. If we accept that we can jointly represent all things that happen in the world with a probability distribution, then we can interogate that probability to make predictions. So, if we are interested in predictions, $\dataScalar_*$ at future points input locations of interest, $\inputVector_*$ given previously training data, $\dataVector$ and corresponding inputs, $\inputMatrix$, then we are really interogating the following probability density, $$ p(\dataScalar_*|\dataVector, \inputMatrix, \inputVector_*), $$ there is nothing controversial here, as long as you accept that you have a good joint model of the world around you that relates test data to training data, $p(\dataScalar_*, \dataVector, \inputMatrix, \inputVector_*)$ then this conditional distribution can be recovered through standard rules of probability ($\text{data} + \text{model} \rightarrow \text{prediction}$).

We can construct this joint density through the use of the following decomposition: $$ p(\dataScalar_*|\dataVector, \inputMatrix, \inputVector_*) = \int p(\dataScalar_*|\inputVector_*, \mappingMatrix) p(\mappingMatrix | \dataVector, \inputMatrix) \text{d} \mappingMatrix $$

where, for convenience, we are assuming *all* the parameters of the
model are now represented by $\parameterVector$ (which contains
$\mappingMatrix$ and $\mappingMatrixTwo$) and
$p(\parameterVector | \dataVector, \inputMatrix)$ is recognised as the
posterior density of the parameters given data and
$p(\dataScalar_*|\inputVector_*, \parameterVector)$ is the *likelihood*
of an individual test data point given the parameters.

The likelihood of the data is normally assumed to be independent across the parameters, $$ p(\dataVector|\inputMatrix, \mappingMatrix) = \prod_{i=1}^\numData p(\dataScalar_i|\inputVector_i, \mappingMatrix),$$

and if that is so, it is easy to extend our predictions across all future, potential, locations, $$ p(\dataVector_*|\dataVector, \inputMatrix, \inputMatrix_*) = \int p(\dataVector_*|\inputMatrix_*, \parameterVector) p(\parameterVector | \dataVector, \inputMatrix) \text{d} \parameterVector. $$

The likelihood is also where the *prediction function* is incorporated.
For example in the regression case, we consider an objective based
around the Gaussian density, $$
p(\dataScalar_i | \mappingFunction(\inputVector_i)) = \frac{1}{\sqrt{2\pi \dataStd^2}} \exp\left(-\frac{\left(\dataScalar_i - \mappingFunction(\inputVector_i)\right)^2}{2\dataStd^2}\right)
$$

In short, that is the classical approach to probabilistic inference, and
all approaches to Bayesian neural networks fall within this path. For a
deep probabilistic model, we can simply take this one stage further and
place a probability distribution over the input locations, $$
p(\dataVector_*|\dataVector) = \int p(\dataVector_*|\inputMatrix_*, \parameterVector) p(\parameterVector | \dataVector, \inputMatrix) p(\inputMatrix) p(\inputMatrix_*) \text{d} \parameterVector \text{d} \inputMatrix \text{d}\inputMatrix_*
$$ and we have *unsupervised learning* (from where we can get deep
generative models).

One way of representing a joint distribution is to consider conditional dependencies between data. Conditional dependencies allow us to factorize the distribution. For example, a Markov chain is a factorization of a distribution into components that represent the conditional relationships between points that are neighboring, often in time or space. It can be decomposed in the following form. $$p(\dataVector) = p(\dataScalar_\numData | \dataScalar_{\numData-1}) p(\dataScalar_{\numData-1}|\dataScalar_{\numData-2}) \dots p(\dataScalar_{2} | \dataScalar_{1})$$

Figure: *A Markov chain is a simple form of probabilistic graphical
model providing a particular decomposition of the joint density.*

By specifying conditional independencies we can reduce the parameterization required for our data, instead of directly specifying the parameters of the joint distribution, we can specify each set of parameters of the conditonal independently. This can also give an advantage in terms of interpretability. Understanding a conditional independence structure gives a structured understanding of data. If developed correctly, according to causal methodology, it can even inform how we should intervene in the system to drive a desired result [@Pearl:causality95].

However, a challenge arises when the data becomes more complex. Consider
the graphical model shown below, used to predict the perioperative risk
of *C Difficile* infection following colon surgery
[@Steele:predictive12].

Figure: *A probabilistic directed graph used to predict the
perioperative risk of C Difficile infection following colon surgery.
When these models have good predictive performance they are often
difficult to interpret. This may be due to the limited representation
capability of the conditional densities in the model.*

To capture the complexity in the interelationship between the data, the graph itself becomes more complex, and less interpretable.

As far as combining our data and our model to form our prediction, the devil is in the detail. While everything is easy to write in terms of probability densities, as we move from $\text{data}$ and $\text{model}$ to $\text{prediction}$ there is that simple $\stackrel{\text{compute}}{\rightarrow}$ sign, which is now burying a wealth of difficulties. Each integral sign above is a high dimensional integral which will typically need approximation. Approximations also come with computational demands. As we consider more complex classes of functions, the challenges around the integrals become harder and prediction of future test data given our model and the data becomes so involved as to be impractical or impossible.

Statisticians realized these challenges early on, indeed, so early that
they were actually physicists, both Laplace and Gauss worked on models
such as this, in Gauss's case he made his career on prediction of the
location of the lost planet (later reclassified as a asteroid, then
dwarf planet), Ceres. Gauss and Laplace made use of maximum a posteriori
estimates for simplifying their computations and Laplace developed
Laplace's method (and invented the Gaussian density) to expand around
that mode. But classical statistics needs better guarantees around model
performance and interpretation, and as a result has focussed more on the
*linear* model implied by $$
\mappingFunction(\inputVector) = \left.\mappingVector^{(2)}\right.^\top \activationVector(\mappingMatrix_1, \inputVector)
$$

The Gaussian likelihood given above implies that the data observation is related to the function by noise corruption so we have, $$ \dataScalar_i = \mappingFunction(\inputVector_i) + \noiseScalar_i, $$ where $$ \noiseScalar_i \sim \gaussianSamp{0}{\dataStd^2} $$

and while normally integrating over high dimensional parameter vectors
is highly complex, here it is *trivial*. That is because of a property
of the multivariate Gaussian.

Gaussian processes are initially of interest because

- linear Gaussian models are easier to deal with
- Even the parameters
*within*the process can be handled, by considering a particular limit.

Let's first of all review the properties of the multivariate Gaussian distribution that make linear Gaussian models easier to deal with. We'll return to the, perhaps surprising, result on the parameters within the nonlinearity, $\parameterVector$, shortly.

To work with linear Gaussian models, to find the marginal likelihood all you need to know is the following rules. If $$ \dataVector = \mappingMatrix \inputVector + \noiseVector, $$ where $\dataVector$, $\inputVector$ and $\noiseVector$ are vectors and we assume that $\inputVector$ and $\noiseVector$ are drawn from multivariate Gaussians, $$ \begin{align} \inputVector & \sim \gaussianSamp{\meanVector}{\covarianceMatrix}\\ \noiseVector & \sim \gaussianSamp{\zerosVector}{\covarianceMatrixTwo} \end{align} $$ then we know that $\dataVector$ is also drawn from a multivariate Gaussian with, $$ \dataVector \sim \gaussianSamp{\mappingMatrix\meanVector}{\mappingMatrix\covarianceMatrix\mappingMatrix^\top + \covarianceMatrixTwo}. $$

With apprioriately defined covariance, $\covarianceMatrixTwo$, this is
actually the marginal likelihood for Factor Analysis, or Probabilistic
Principal Component Analysis [@Tipping:probpca99], because we integrated
out the inputs (or *latent* variables they would be called in that
case).

However, we are focussing on what happens in models which are non-linear
in the inputs, whereas the above would be *linear* in the inputs. To
consider these, we introduce a matrix, called the design matrix. We set
each activation function computed at each data point to be $$
\activationScalar_{i,j} = \activationScalar(\mappingVector^{(1)}_{j}, \inputVector_{i})
$$ and define the matrix of activations (known as the *design matrix* in
statistics) to be, $$
\activationMatrix =
\begin{bmatrix}
\activationScalar_{1, 1} & \activationScalar_{1, 2} & \dots & \activationScalar_{1, \numHidden} \\
\activationScalar_{1, 2} & \activationScalar_{1, 2} & \dots & \activationScalar_{1, \numData} \\
\vdots & \vdots & \ddots & \vdots \\
\activationScalar_{\numData, 1} & \activationScalar_{\numData, 2} & \dots & \activationScalar_{\numData, \numHidden}
\end{bmatrix}.
$$ By convention this matrix always has $\numData$ rows and $\numHidden$
columns, now if we define the vector of all noise corruptions,
$\noiseVector = \left[\noiseScalar_1, \dots \noiseScalar_\numData\right]^\top$.

If we define the prior distribution over the vector $\mappingVector$ to be Gaussian, $$ \mappingVector \sim \gaussianSamp{\zerosVector}{\alpha\eye}, $$

then we can use rules of multivariate Gaussians to see that, $$ \dataVector \sim \gaussianSamp{\zerosVector}{\alpha \activationMatrix \activationMatrix^\top + \dataStd^2 \eye}. $$

In other words, our training data is distributed as a multivariate Gaussian, with zero mean and a covariance given by $$ \kernelMatrix = \alpha \activationMatrix \activationMatrix^\top + \dataStd^2 \eye. $$

This is an $\numData \times \numData$ size matrix. Its elements are in
the form of a function. The maths shows that any element, index by $i$
and $j$, is a function *only* of inputs associated with data points $i$
and $j$, $\dataVector_i$, $\dataVector_j$.
$\kernel_{i,j} = \kernel\left(\inputVector_i, \inputVector_j\right)$

If we look at the portion of this function associated only with
$\mappingFunction(\cdot)$, i.e. we remove the noise, then we can write
down the covariance associated with our neural network, $$
\kernel_\mappingFunction\left(\inputVector_i, \inputVector_j\right) = \alpha \activationVector\left(\mappingMatrix_1, \inputVector_i\right)^\top \activationVector\left(\mappingMatrix_1, \inputVector_j\right)
$$ so the elements of the covariance or *kernel* matrix are formed by
inner products of the rows of the *design matrix*.

This is the essence of a Gaussian process. Instead of making assumptions about our density over each data point, $\dataScalar_i$ as i.i.d. we make a joint Gaussian assumption over our data. The covariance matrix is now a function of both the parameters of the activation function, $\mappingMatrixTwo$, and the input variables, $\inputMatrix$. This comes about through integrating out the parameters of the model, $\mappingVector$.

We can basically put anything inside the basis functions, and many people do. These can be deep kernels [@Cho:deep09] or we can learn the parameters of a convolutional neural network inside there.

Viewing a neural network in this way is also what allows us to beform
sensible *batch* normalizations [@Ioffe:batch15].

The process described above is degenerate. The covariance function is of rank at most $\numHidden$ and since the theoretical amount of data could always increase $\numData \rightarrow \infty$, the covariance function is not full rank. This means as we increase the amount of data to infinity, there will come a point where we can't normalize the process because the multivariate Gaussian has the form, $$ \gaussianDist{\mappingFunctionVector}{\zerosVector}{\kernelMatrix} = \frac{1}{\left(2\pi\right)^{\frac{\numData}{2}}\det{\kernelMatrix}^\frac{1}{2}} \exp\left(-\frac{\mappingFunctionVector^\top\kernelMatrix \mappingFunctionVector}{2}\right) $$ and a non-degenerate kernel matrix leads to $\det{\kernelMatrix} = 0$ defeating the normalization (it's equivalent to finding a projection in the high dimensional Gaussian where the variance of the the resulting univariate Gaussian is zero, i.e. there is a null space on the covariance, or alternatively you can imagine there are one or more directions where the Gaussian has become the delta function).

In the machine learning field, it was Radford Neal [@Neal:bayesian94] that realized the potential of the next step. In his 1994 thesis, he was considering Bayesian neural networks, of the type we described above, and in considered what would happen if you took the number of hidden nodes, or neurons, to infinity, i.e. $\numHidden \rightarrow \infty$.

Figure: *Page 37 of Radford Neal's 1994
thesis*

In loose terms, what Radford considers is what happens to the elements
of the covariance function, $$
\begin{align*}
\kernel_\mappingFunction\left(\inputVector_i, \inputVector_j\right) & = \alpha \activationVector\left(\mappingMatrix_1, \inputVector_i\right)^\top \activationVector\left(\mappingMatrix_1, \inputVector_j\right)\\
& = \alpha \sum_k \activationScalar\left(\mappingVector^{(1)}_k, \inputVector_i\right) \activationScalar\left(\mappingVector^{(1)}_k, \inputVector_j\right)
\end{align*}
$$ if instead of considering a finite number you sample infinitely
many of these activation functions, sampling parameters from a prior
density, $p(\mappingVectorTwo)$, for each one, $$
\kernel_\mappingFunction\left(\inputVector_i, \inputVector_j\right) = \alpha \int \activationScalar\left(\mappingVector^{(1)}, \inputVector_i\right) \activationScalar\left(\mappingVector^{(1)}, \inputVector_j\right) p(\mappingVector^{(1)}) \text{d}\mappingVector^{(1)}
$$ And that's not *only* for Gaussian $p(\mappingVectorTwo)$. In fact
this result holds for a range of activations, and a range of prior
densities because of the *central limit theorem*.

To write it in the form of a probabilistic program, as long as the distribution for $\phi_i$ implied by this short probabilistic program, $$ \begin{align*} \mappingVectorTwo & \sim p(\cdot)\\ \phi_i & = \activationScalar\left(\mappingVectorTwo, \inputVector_i\right), \end{align*} $$ has finite variance, then the result of taking the number of hidden units to infinity, with appropriate scaling, is also a Gaussian process.

To understand this argument in more detail, I highly recommend reading chapter 2 of Neal's thesis [@Neal:bayesian94], which remains easy to read and clear today. Indeed, for readers interested in Bayesian neural networks, both Raford Neal's and David MacKay's PhD thesis [@MacKay:bayesian92] remain essential reading. Both theses embody a clarity of thought, and an ability to weave together threads from different fields that was the business of machine learning in the 1990s. Radford and David were also pioneers in making their software widely available and publishing material on the web.

We will consider a Gaussian distribution with a particular structure of
covariance matrix. We will generate *one* sample from a 25-dimensional
Gaussian density. $$
\mappingFunctionVector=\left[\mappingFunction_{1},\mappingFunction_{2}\dots \mappingFunction_{25}\right].
$$ in the figure below we plot these data on the $y$-axis against their
*indices* on the $x$-axis.

In [ ]:

```
%load -s Kernel mlai.py
```

In [ ]:

```
%load -s polynomial_cov mlai.py
```

In [ ]:

```
%load -s exponentiated_quadratic mlai.py
```

In [ ]:

```
import pods
from ipywidgets import IntSlider
```

In [ ]:

```
pods.notebook.display_plots('two_point_sample{sample:0>3}.svg', '../slides/diagrams/gp', sample=IntSlider(0, 0, 8, 1))
```

Figure: *A 25 dimensional correlated random variable (values ploted
against index)*

In [ ]:

```
import pods
from ipywidgets import IntSlider
```

In [ ]:

```
pods.notebook.display_plots('two_point_sample{sample:0>3}.svg',
'../slides/diagrams/gp',
sample=IntSlider(9, 9, 12, 1))
```

Figure: *The joint Gaussian over $\mappingFunction_1$ and
$\mappingFunction_2$ along with the conditional distribution of
$\mappingFunction_2$ given $\mappingFunction_1$*

Figure: *Uluru, the sacred rock in Australia. If we think of it as a
probability density, viewing it from this side gives us one marginal
from the density. Figuratively speaking, slicing through the rock would
give a conditional density.*

When viewing these contour plots, I sometimes find it helpful to think of Uluru, the prominent rock formation in Australia. The rock rises above the surface of the plane, just like a probability density rising above the zero line. The rock is three dimensional, but when we view Uluru from the classical position, we are looking at one side of it. This is equivalent to viewing the marginal density.

The joint density can be viewed from above, using contours. The
conditional density is equivalent to *slicing* the rock. Uluru is a holy
rock, so this has to be an imaginary slice. Imagine we cut down a
vertical plane orthogonal to our view point (e.g. coming across our view
point). This would give a profile of the rock, which when renormalized,
would give us the conditional distribution, the value of conditioning
would be the location of the slice in the direction we are facing.

Of course in practice, rather than manipulating mountains physically, the advantage of the Gaussian density is that we can perform these manipulations mathematically.

Prediction of $\mappingFunction_2$ given $\mappingFunction_1$ requires
the *conditional density*,
$p(\mappingFunction_2|\mappingFunction_1)$.Another remarkable property
of the Gaussian density is that this conditional distribution is *also*
guaranteed to be a Gaussian density. It has the form, $$
p(\mappingFunction_2|\mappingFunction_1) = \gaussianDist{\mappingFunction_2}{\frac{\kernelScalar_{1, 2}}{\kernelScalar_{1, 1}}\mappingFunction_1}{ \kernelScalar_{2, 2} - \frac{\kernelScalar_{1,2}^2}{\kernelScalar_{1,1}}}
$$where we have assumed that the covariance of the original joint
density was given by $$
\kernelMatrix = \begin{bmatrix} \kernelScalar_{1, 1} & \kernelScalar_{1, 2}\\ \kernelScalar_{2, 1} & \kernelScalar_{2, 2}.\end{bmatrix}
$$

Using these formulae we can determine the conditional density for any of the elements of our vector $\mappingFunctionVector$. For example, the variable $\mappingFunction_8$ is less correlated with $\mappingFunction_1$ than $\mappingFunction_2$. If we consider this variable we see the conditional density is more diffuse.

In [ ]:

```
import pods
from ipywidgets import IntSlider
```

In [ ]:

```
pods.notebook.display_plots('two_point_sample{sample:0>3}.svg',
'../slides/diagrams/gp',
sample=IntSlider(13, 13, 17, 1))
```

Figure: *Sample from the joint Gaussian model, points indexed by 1 and
8 highlighted.*

Figure: *The joint Gaussian over $\mappingFunction_1$ and
$\mappingFunction_8$ along with the conditional distribution of
$\mappingFunction_8$ given $\mappingFunction_1$*

Covariance function, $\kernelMatrix$

Determines properties of samples.

Function of $\inputMatrix$, $$\kernelScalar_{i,j} = \kernelScalar(\inputVector_i, \inputVector_j)$$

Posterior mean $$\mappingFunction_D(\inputVector_*) = \kernelVector(\inputVector_*, \inputMatrix) \kernelMatrix^{-1} \dataVector$$

Posterior covariance $$\mathbf{C}_* = \kernelMatrix_{*,*} - \kernelMatrix_{*,\mappingFunctionVector} \kernelMatrix^{-1} \kernelMatrix_{\mappingFunctionVector, *}$$

Posterior mean

In [ ]:

```
$$\mappingFunction_D(\inputVector_*) = \kernelVector(\inputVector_*, \inputMatrix) \boldsymbol{\alpha}$$
```

- Posterior covariance $$\covarianceMatrix_* = \kernelMatrix_{*,*} - \kernelMatrix_{*,\mappingFunctionVector} \kernelMatrix^{-1} \kernelMatrix_{\mappingFunctionVector, *}$$

The exponentiated quadratic covariance, also known as the Gaussian
covariance or the RBF covariance and the squared exponential. Covariance
between two points is related to the negative exponential of the squared
distnace between those points. This covariance function can be derived
in a few different ways: as the infinite limit of a radial basis
function neural network, as diffusion in the heat equation, as a
Gaussian filter in *Fourier space* or as the composition as a series of
linear filters applied to a base function.

The covariance takes the following form, $$
\kernelScalar(\inputVector, \inputVector^\prime) = \alpha \exp\left(-\frac{\ltwoNorm{\inputVector-\inputVector^\prime}^2}{2\lengthScale^2}\right)
$$ where $\ell$ is the *length scale* or *time scale* of the process and
$\alpha$ represents the overall process variance.

- Gold medal times for Olympic Marathon since 1896. - Marathons before 1924 didn't have a standardised distance. - Present results using pace per km. - In 1904 Marathon was badly organised leading to very slow times. |
Image from Wikimedia Commons |

In [ ]:

```
import numpy as np
import pods
```

In [ ]:

```
data = pods.datasets.olympic_marathon_men()
x = data['X']
y = data['Y']
offset = y.mean()
scale = np.sqrt(y.var())
```

In [ ]:

```
import matplotlib.pyplot as plt
import teaching_plots as plot
import mlai
```

In [ ]:

```
xlim = (1875,2030)
ylim = (2.5, 6.5)
yhat = (y-offset)/scale
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
_ = ax.plot(x, y, 'r.',markersize=10)
ax.set_xlabel('year', fontsize=20)
ax.set_ylabel('pace min/km', fontsize=20)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
mlai.write_figure(figure=fig,
filename='../slides/diagrams/datasets/olympic-marathon.svg',
transparent=True,
frameon=True)
```

Figure: *Olympic marathon pace times since 1892.*

Things to notice about the data include the outlier in 1904, in this year, the olympics was in St Louis, USA. Organizational problems and challenges with dust kicked up by the cars following the race meant that participants got lost, and only very few participants completed.

More recent years see more consistently quick marathons.

If we had to summarise the objectives of machine learning in one word, a
very good candidate for that word would be *generalization*. What is
generalization? From a human perspective it might be summarised as the
ability to take lessons learned in one domain and apply them to another
domain. If we accept the definition given in the first session for
machine learning, $$
\text{data} + \text{model} \stackrel{\text{compute}}{\rightarrow} \text{prediction}
$$ then we see that without a model we can't generalise: we only have
data. Data is fine for answering very specific questions, like "Who won
the Olympic Marathon in 2012?", because we have that answer stored,
however, we are not given the answer to many other questions. For
example, Alan Turing was a formidable marathon runner, in 1946 he ran a
time 2 hours 46 minutes (just under four minutes per kilometer, faster
than I and most of the other Endcliffe Park
Run runners can do 5 km).
What is the probability he would have won an Olympics if one had been
held in 1946?

To answer this question we need to generalize, but before we formalize the concept of generalization let's introduce some formal representation of what it means to generalize in machine learning.

Our first objective will be to perform a Gaussian process fit to the data, we'll do this using the GPy software.

In [ ]:

```
import GPy
```

In [ ]:

```
m_full = GPy.models.GPRegression(x,yhat)
_ = m_full.optimize() # Optimize parameters of covariance function
```

`m_full.optimize()`

optimizes
the parameters of the covariance function and the noise level of the
model. Once the fit is complete, we'll try creating some test points,
and computing the output of the GP model in terms of the mean and
standard deviation of the posterior functions between 1870 and 2030. We
plot the mean function and the standard deviation at 200 locations. We
can obtain the predictions using `y_mean, y_var = m_full.predict(xt)`

In [ ]:

```
xt = np.linspace(1870,2030,200)[:,np.newaxis]
yt_mean, yt_var = m_full.predict(xt)
yt_sd=np.sqrt(yt_var)
```

Now we plot the results using the helper function in `teaching_plots`

.

In [ ]:

```
import teaching_plots as plot
```

In [ ]:

```
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
plot.model_output(m_full, scale=scale, offset=offset, ax=ax, xlabel='year', ylabel='pace min/km', fontsize=20, portion=0.2)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
mlai.write_figure(figure=fig,
filename='../slides/diagrams/gp/olympic-marathon-gp.svg',
transparent=True, frameon=True)
```

Figure: *Gaussian process fit to the Olympic Marathon data. The error
bars are too large, perhaps due to the outlier from 1904.*

In the fit we see that the error bars (coming mainly from the noise
variance) are quite large. This is likely due to the outlier point in
1904, ignoring that point we can see that a tighter fit is obtained. To
see this making a version of the model, `m_clean`

, where that point is
removed.

In [ ]:

```
x_clean=np.vstack((x[0:2, :], x[3:, :]))
y_clean=np.vstack((y[0:2, :], y[3:, :]))
m_clean = GPy.models.GPRegression(x_clean,y_clean)
_ = m_clean.optimize()
```

Can we determine covariance parameters from the data?

$$ \gaussianDist{\dataVector}{\mathbf{0}}{\kernelMatrix}=\frac{1}{(2\pi)^\frac{\numData}{2}{\det{\kernelMatrix}^{\frac{1}{2}}}}{\exp\left(-\frac{\dataVector^{\top}\kernelMatrix^{-1}\dataVector}{2}\right)} $$$$ \begin{aligned} \gaussianDist{\dataVector}{\mathbf{0}}{\kernelMatrix}=\frac{1}{(2\pi)^\frac{\numData}{2}{\color{black} \det{\kernelMatrix}^{\frac{1}{2}}}}{\color{black}\exp\left(-\frac{\dataVector^{\top}\kernelMatrix^{-1}\dataVector}{2}\right)} \end{aligned} $$$$ \begin{aligned} \log \gaussianDist{\dataVector}{\mathbf{0}}{\kernelMatrix}=&{\color{black}-\frac{1}{2}\log\det{\kernelMatrix}}{\color{black}-\frac{\dataVector^{\top}\kernelMatrix^{-1}\dataVector}{2}} \\ &-\frac{\numData}{2}\log2\pi \end{aligned} $$$$ \errorFunction(\parameterVector) = {\color{black} \frac{1}{2}\log\det{\kernelMatrix}} + {\color{black} \frac{\dataVector^{\top}\kernelMatrix^{-1}\dataVector}{2}} $$The parameters are *inside* the covariance function (matrix).
\normalsize
$$\kernelScalar_{i, j} = \kernelScalar(\inputVals_i, \inputVals_j; \parameterVector)$$

\Large $$\kernelMatrix = \rotationMatrix \eigenvalueMatrix^2 \rotationMatrix^\top$$

In [ ]:

```
gpoptimizePlot1
```

$\eigenvalueMatrix$ represents distance on axes. $\rotationMatrix$ gives rotation. |

<!--```{.python} gpoptimizePlot3

In [ ]:

```
\only<1>{\input{../../../gp/tex/diagrams/gpOptimiseQuadratic1.svg}}\only<2>{\input{../../../gp/tex/diagrams/gpOptimiseQuadratic2.svg}}\only<3>{\input{../../../gp/tex/diagrams/gpOptimiseQuadratic3.svg}}-->
<table>
<tr>
<td width="50%">
<img src="http://inverseprobability.com/talks/slides/../slides/diagrams/gp/gp-optimise006.svg" class="" align="100%" style="vertical-align:middle;">
</td>
<td width="50%">
<img src="http://inverseprobability.com/talks/slides/../slides/diagrams/gp/gp-optimise010.svg" class="" align="100%" style="vertical-align:middle;">
</td>
</tr>
</table>
<table>
<tr>
<td width="50%">
<img src="http://inverseprobability.com/talks/slides/../slides/diagrams/gp/gp-optimise016.svg" class="" align="100%" style="vertical-align:middle;">
</td>
<td width="50%">
<img src="http://inverseprobability.com/talks/slides/../slides/diagrams/gp/gp-optimise021.svg" class="" align="100%" style="vertical-align:middle;">
</td>
</tr>
</table>
Figure: <i>Variation in the data fit term, the capacity term and the
negative log likelihood for different lengthscales.</i>
## Gene Expression Example \[<a href="https://github.com/lawrennd/talks/edit/gh-pages/_gp/includes/della-gatta-gene-gp.md" target="_blank" >edit</a>\]
We now consider an example in gene expression. Gene expression is the
measurement of mRNA levels expressed in cells. These mRNA levels show
which genes are 'switched on' and producing data. In the example we will
use a Gaussian process to determine whether a given gene is active, or
we are merely observing a noise response.
## Della Gatta Gene Data \[<a href="https://github.com/lawrennd/talks/edit/gh-pages/_ml/includes/della-gatta-gene-data.md" target="_blank" >edit</a>\]
- Given given expression levels in the form of a time series from
@DellaGatta:direct08.
``` {.python}
import numpy as np
import pods
```

In [ ]:

```
data = pods.datasets.della_gatta_TRP63_gene_expression(data_set='della_gatta',gene_number=937)
x = data['X']
y = data['Y']
offset = y.mean()
scale = np.sqrt(y.var())
```

In [ ]:

```
import matplotlib.pyplot as plt
import teaching_plots as plot
import mlai
```

In [ ]:

```
xlim = (-20,260)
ylim = (5, 7.5)
yhat = (y-offset)/scale
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
_ = ax.plot(x, y, 'r.',markersize=10)
ax.set_xlabel('time/min', fontsize=20)
ax.set_ylabel('expression', fontsize=20)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
mlai.write_figure(figure=fig,
filename='../slides/diagrams/datasets/della-gatta-gene.svg',
transparent=True,
frameon=True)
```

Figure: *Gene expression levels over time for a gene from data
provided by @DellaGatta:direct08. We would like to understand whethere
there is signal in the data, or we are only observing noise.*

- Want to detect if a gene is expressed or not, fit a GP to each gene @Kalaitzis:simple11.

Figure: *The example is taken from the paper "A Simple Approach to
Ranking Differentially Expressed Gene Expression Time Courses through
Gaussian Process Regression." @Kalaitzis:simple11.*

In [ ]:

```
import GPy
```

In [ ]:

```
m_full = GPy.models.GPRegression(x,yhat)
m_full.kern.lengthscale=50
_ = m_full.optimize() # Optimize parameters of covariance function
```

*time scale* of the covariance function) to a reasonable value. Default
would be 1, but here we set it to 50 minutes, given points are arriving
across zero to 250 minutes.

In [ ]:

```
xt = np.linspace(-20,260,200)[:,np.newaxis]
yt_mean, yt_var = m_full.predict(xt)
yt_sd=np.sqrt(yt_var)
```

Now we plot the results using the helper function in `teaching_plots`

.

In [ ]:

```
import teaching_plots as plot
```

In [ ]:

```
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
plot.model_output(m_full, scale=scale, offset=offset, ax=ax, xlabel='time/min', ylabel='expression', fontsize=20, portion=0.2)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
ax.set_title('log likelihood: {ll:.3}'.format(ll=m_full.log_likelihood()), fontsize=20)
mlai.write_figure(figure=fig,
filename='../slides/diagrams/gp/della-gatta-gene-gp.svg',
transparent=True, frameon=True)
```

Figure: *Result of the fit of the Gaussian process model with the time
scale parameter initialized to 50 minutes.*

Now we try a model initialized with a longer length scale.

In [ ]:

```
m_full2 = GPy.models.GPRegression(x,yhat)
m_full2.kern.lengthscale=2000
_ = m_full2.optimize() # Optimize parameters of covariance function
```

In [ ]:

```
import teaching_plots as plot
```

In [ ]:

```
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
plot.model_output(m_full2, scale=scale, offset=offset, ax=ax, xlabel='time/min', ylabel='expression', fontsize=20, portion=0.2)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
ax.set_title('log likelihood: {ll:.3}'.format(ll=m_full2.log_likelihood()), fontsize=20)
mlai.write_figure(figure=fig,
filename='../slides/diagrams/gp/della-gatta-gene-gp2.svg',
transparent=True, frameon=True)
```

Figure: *Result of the fit of the Gaussian process model with the time
scale parameter initialized to 2000 minutes.*

Now we try a model initialized with a lower noise.

In [ ]:

```
m_full3 = GPy.models.GPRegression(x,yhat)
m_full3.kern.lengthscale=20
m_full3.likelihood.variance=0.001
_ = m_full3.optimize() # Optimize parameters of covariance function
```

In [ ]:

```
import teaching_plots as plot
```

In [ ]:

```
fig, ax = plt.subplots(figsize=plot.big_wide_figsize)
plot.model_output(m_full3, scale=scale, offset=offset, ax=ax, xlabel='time/min', ylabel='expression', fontsize=20, portion=0.2)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
ax.set_title('log likelihood: {ll:.3}'.format(ll=m_full3.log_likelihood()), fontsize=20)
mlai.write_figure(figure=fig,
filename='../slides/diagrams/gp/della-gatta-gene-gp3.svg',
transparent=True, frameon=True)
```

Figure: *Result of the fit of the Gaussian process model with the
noise initialized low (standard deviation 0.1) and the time scale
parameter initialized to 20 minutes.*

Figure:

[]{style="text-align:right"}

As an example of using Gaussian process models within the full pipeline from data to decsion, we'll consider the prediction of Malaria incidence in Uganda. For the purposes of this study malaria reports come in two forms, HMIS reports from health centres and Sentinel data, which is curated by the WHO. There are limited sentinel sites and many HMIS sites.

The work is from Ricardo Andrade Pacheco's PhD thesis, completed in collaboration with John Quinn and Martin Mubangizi [@Andrade:consistent14; @Mubangizi:malaria14]. John and Martin were initally from the AI-DEV group from the University of Makerere in Kampala and more latterly they were based at UN Global Pulse in Kampala.

Malaria data is spatial data. Uganda is split into districts, and health reports can be found for each district. This suggests that models such as conditional random fields could be used for spatial modelling, but there are two complexities with this. First of all, occasionally districts split into two. Secondly, sentinel sites are a specific location within a district, such as Nagongera which is a sentinel site based in the Tororo district.

Figure: *Ugandan districs. Data SRTM/NASA from
https://dds.cr.usgs.gov/srtm/version2_1.*

[[@Andrade:consistent14; @Mubangizi:malaria14]]{style="text-align:right"}

Figure: *The Kapchorwa District, home district of Stephen
Kiprotich.*

Stephen Kiprotich, the 2012 gold medal winner from the London Olympics, comes from Kapchorwa district, in eastern Uganda, near the border with Kenya.

The common standard for collecting health data on the African continent is from the Health management information systems (HMIS). However, this data suffers from missing values [@Gething:hmis06] and diagnosis of diseases like typhoid and malaria may be confounded.

Figure: *The Tororo district, where the sentinel site, Nagongera, is
located.*

World Health Organization Sentinel Surveillance systems are set up "when high-quality data are needed about a particular disease that cannot be obtained through a passive system". Several sentinel sites give accurate assessment of malaria disease levels in Uganda, including a site in Nagongera.

Figure: *Sentinel and HMIS data along with rainfall and temperature
for the Nagongera sentinel station in the Tororo district.*

In collaboration with the AI Research Group at Makerere we chose to investigate whether Gaussian process models could be used to assimilate information from these two different sources of disease informaton. Further, we were interested in whether local information on rainfall and temperature could be used to improve malaria estimates.

The aim of the project was to use WHO Sentinel sites, alongside rainfall and temperature, to improve predictions from HMIS data of levels of malaria.

Figure: *The Mubende District.*

Figure: *Prediction of malaria incidence in Mubende.*

Figure: *The project arose out of the Gaussian process summer school
held at Makerere in Kampala in 2013. The school led, in turn, to the
Data Science Africa initiative.*

Figure: *The Kabarole district in Uganda.*

Figure: *Estimate of the current disease situation in the Kabarole
district over time. Estimate is constructed with a Gaussian process with
an additive covariance funciton.*

Health monitoring system for the Kabarole district. Here we have fitted the reports with a Gaussian process with an additive covariance function. It has two components, one is a long time scale component (in red above) the other is a short time scale component (in blue).

Monitoring proceeds by considering two aspects of the curve. Is the blue
line (the short term report signal) above the red (which represents the
long term trend? If so we have higher than expected reports. If this is
the case *and* the gradient is still positive (i.e. reports are going
up) we encode this with a *red* color. If it is the case and the
gradient of the blue line is negative (i.e. reports are going down) we
encode this with an *amber* color. Conversely, if the blue line is below
the red *and* decreasing, we color *green*. On the other hand if it is
below red but increasing, we color *yellow*.

This gives us an early warning system for disease. Red is a bad situation getting worse, amber is bad, but improving. Green is good and getting better and yellow good but degrading.

Finally, there is a gray region which represents when the scale of the effect is small.

Figure: *The map of Ugandan districts with an overview of the Malaria
situation in each district.*

These colors can now be observed directly on a spatial map of the districts to give an immediate impression of the current status of the disease across the country.

An additive covariance function is derived from considering the result of summing two Gaussian processes together. If the first Gaussian process is $g(\cdot)$, governed by covariance $\kernelScalar_g(\cdot, \cdot)$ and the second process is $h(\cdot)$, governed by covariance $\kernelScalar_h(\cdot, \cdot)$ then the combined process $f(\cdot) = g(\cdot) + h(\cdot)$ is govererned by a covariance function, $$ \kernelScalar_f(\inputVector, \inputVector^\prime) = \kernelScalar_g(\inputVector, \inputVector^\prime) + \kernelScalar_h(\inputVector, \inputVector^\prime) $$

Figure: *This is a retrospective analysis of US births by Aki Vehtari.
The challenges of forecasting. Even with seasonal and weekly effects
removed there are significant effects on holidays, weekends, etc.*

There's a nice analysis of US birth rates by Gaussian processes with additive covariances in @Gelman:bayesian13. A combination of covariance functions are used to take account of weekly and yearly trends. The analysis is summarized on the cover of the book.

The fixed basis function covariance just comes from the properties of a multivariate Gaussian, if we decide $$ \mappingFunctionVector=\basisMatrix\mappingVector $$ and then we assume $$ \mappingVector \sim \gaussianSamp{\zerosVector}{\alpha\eye} $$ then it follows from the properties of a multivariate Gaussian that $$ \mappingFunctionVector \sim \gaussianSamp{\zerosVector}{\alpha\basisMatrix\basisMatrix^\top} $$ meaning that the vector of observations from the function is jointly distributed as a Gaussian process and the covariance matrix is $\kernelMatrix = \alpha\basisMatrix \basisMatrix^\top$, each element of the covariance matrix can then be found as the inner product between two rows of the basis funciton matrix.

In [ ]:

```
%load -s basis_cov mlai.py
```

In [ ]:

```
%load -s radial mlai.py
```

In [ ]:

```
%load -s brownian_cov mlai.py
```

Brownian motion is also a Gaussian process. It follows a Gaussian random walk, with diffusion occuring at each time point driven by a Gaussian input. This implies it is both Markov and Gaussian. The covariance function for Brownian motion has the form $$ \kernelScalar(t, t^\prime)=\alpha \min(t, t^\prime) $$

In [ ]:

```
%load -s mlp_cov mlai.py
```

The multi-layer perceptron (MLP) covariance, also known as the neural network covariance or the arcsin covariance, is derived by considering the infinite limit of a neural network.

In [ ]:

```
%load -s relu_cov mlai.py
```

Another approach to developing covariance function exploits Bochner's
theorem @Bochner:book59. Bochner's theorem tells us that any positve
filter in Fourier space implies has an associated Gaussian process with
a stationary covariance function. The covariance function is the
*inverse Fourier transform* of the filter applied in Fourier space.

For example, in signal processing, *band limitations* are commonly
applied as an assumption. For example, we may believe that no frequency
above $w=2$ exists in the signal. This is equivalent to a rectangle
function being applied as a the filter in Fourier space.

The inverse Fourier transform of the rectangle function is the
$\text{sinc}(\cdot)$ function. So the sinc is a valid covariance
function, and it represents *band limited* signals.

Note that other covariance functions we've introduced can also be interpreted in this way. For example, the exponentiated quadratic covariance function can be Fourier transformed to see what the implied filter in Fourier space is. The Fourier transform of the exponentiated quadratic is an exponentiated quadratic, so the standard EQ-covariance implies a EQ filter in Fourier space.

In [ ]:

```
%load -s sinc_cov mlai.py
```

\includecovariane{sinc}{\kernelScalar(\inputVector, \inputVector^\prime) = \alpha \text{sinc}\left(\pi w r\right)}{Sinc covariance function.}

In [ ]:

```
%load -s lmc_cov mlai.py
```

In [ ]:

```
from IPython.core.display import HTML
```

In [ ]:

```
HTML(anim.to_jshtml())
```

In [ ]:

```
plot.save_animation(anim,
diagrams='../slides/diagrams/kern',
filename='lmc_covariance.html')
```

In [ ]:

```
%load -s icm_cov mlai.py
```

In [ ]:

```
from IPython.core.display import HTML
```

In [ ]:

```
HTML(anim.to_jshtml())
```

In [ ]:

```
plot.save_animation(anim,
diagrams='../slides/diagrams/kern',
filename='icm_covariance.html')
```

We'll cover extensions to Gaussian processes including approximate inference in non Gaussian models, large data [@Thang:unifying17; @Hensman:bigdata13], multiple output GPs [@Alvarez:vector12], Bayesian optimisation [@Snoek:practical12] and Deep GPs [@Damianou:deepgp13].

[^1]: In classical statistics we often interpret these parameters, $\beta$, whereas in machine learning we are normally more interested in the result of the prediction, and less in the prediction. Although this is changing with more need for accountability. In honour of this I normally use $\boldsymbol{\beta}$ when I care about the value of these parameters, and $\mappingVector$ when I care more about the quality of the prediction.