Probabilistic Machine Learning

Abstract

In this talk we review the probabilistic approach to machine learning. We start with a review of probability, and introduce the concepts of probabilistic modelling. We then apply the approach in practice to Naive Bayesian classification. In this session we review the Bayesian formalism in the context of linear models, reviewing initially maximum likelihood and introducing basis functions as a way of driving non-linearity in the model.

What is Machine Learning?

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

\[ \text{data} + \text{model} \xrightarrow{\text{compute}} \text{prediction}\]

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 blog post on "What is Machine Learning?"

Probabilities

We are now going to do some simple review of probabilities and use this review to explore some aspects of our data.

A probability distribution expresses uncertainty about the outcome of an event. We often encode this uncertainty in a variable. So if we are considering the outcome of an event, \(Y\), to be a coin toss, then we might consider \(Y=1\) to be heads and \(Y=0\) to be tails. We represent the probability of a given outcome with the notation: \[ P(Y=1) = 0.5 \] The first rule of probability is that the probability must normalize. The sum of the probability of all events must equal 1. So if the probability of heads (\(Y=1\)) is 0.5, then the probability of tails (the only other possible outcome) is given by \[ P(Y=0) = 1-P(Y=1) = 0.5 \]

Probabilities are often defined as the limit of the ratio between the number of positive outcomes (e.g. heads) given the number of trials. If the number of positive outcomes for event \(y\) is denoted by \(n\) and the number of trials is denoted by \(N\) then this gives the ratio \[ P(Y=y) = \lim_{N\rightarrow \infty}\frac{n_y}{N}. \] In practice we never get to observe an event infinite times, so rather than considering this we often use the following estimate \[ P(Y=y) \approx \frac{n_y}{N}. \]

Movie Body Count Example

There is a crisis in the movie industry, deaths are occuring on a massive scale. In every feature film the body count is tolling up. But what is the cause of all these deaths? Let's try and investigate.

For our first example of data science, we take inspiration from work by researchers at NJIT. They researchers were comparing the qualities of Python with R (my brief thoughts on the subject are available in a Google+ post here: https://plus.google.com/116220678599902155344/posts/5iKyqcrNN68). They put together a data base of results from the the "Internet Movie Database" and the Movie Body Count website which will allow us to do some preliminary investigation.

We will make use of data that has already been 'scraped' from the Movie Body Count website. Code and the data is available at a github repository. Git is a version control system and github is a website that hosts code that can be accessed through git. By sharing the code publicly through github, the authors are licensing the code publicly and allowing you to access and edit it. As well as accessing the code via github you can also download the zip file.

For ease of use we've packaged this data set in the pods library

pods

The pods library is a library for supporting open data science (python open data science). It allows you to load in various data sets and provides tools for helping teach in the notebook.

To install pods you can use pip:

pip install pods

The code is also available on github: https://github.com/sods/ods

Once pods is installed, it can be imported in the usual manner.

import pods

data = pods.datasets.movie_body_count()['Y']
data.head()

Once it is loaded in the data can be summarized using the describe method in pandas.

data.describe()

In jupyter and jupyter notebook it is possible to see a list of all possible functions and attributes by typing the name of the object followed by . for example in the above case if we type data. it show the columns available (these are attributes in pandas dataframes) such as Body_Count, and also functions, such as .describe().</p>

For functions we can also see the documentation about the function by following the name with a question mark. This will open a box with documentation at the bottom which can be closed with the x button.

data.describe?

The film deaths data is stored in an object known as a 'data frame'. Data frames come from the statistical family of programming languages based on S, the most widely used of which is R. The data frame gives us a convenient object for manipulating data. The describe method summarizes which columns there are in the data frame and gives us counts, means, standard deviations and percentiles for the values in those columns. To access a column directly we can write

print(data['Year'])
#print(data['Body_Count'])

This shows the number of deaths per film across the years. We can plot the data as follows.

# this ensures the plot appears in the web browser
%matplotlib inline 
import matplotlib.pyplot as plt # this imports the plotting library in python

plt.plot(data['Year'], data['Body_Count'], 'rx')

You may be curious what the arguments we give to plt.plot are for, now is the perfect time to look at the documentation

plt.plot?

We immediately note that some films have a lot of deaths, which prevent us seeing the detail of the main body of films. First lets identify the films with the most deaths.

data[data['Body_Count']>200]

Here we are using the command data['Kill_Count']>200 to index the films in the pandas data frame which have over 200 deaths. To sort them in order we can also use the sort command. The result of this command on its own is a data series of True and False values. However, when it is passed to the data data frame it returns a new data frame which contains only those values for which the data series is True. We can also sort the result. To sort the result by the values in the Kill_Count column in descending order we use the following command.

data[data['Body_Count']>200].sort_values(by='Body_Count', ascending=False)

We now see that the 'Lord of the Rings' is a large outlier with a very large number of kills. We can try and determine how much of an outlier by histograming the data.

Plotting the Data

data['Body_Count'].hist(bins=20) # histogram the data with 20 bins.
plt.title('Histogram of Film Kill Count')

Question 2

Read on the internet about the following python libraries: numpy, matplotlib, scipy and pandas. What functionality does each provide python. What is the pylab library and how does it relate to the other libraries?

10 marks

Write your answer to Question 2 here

We could try and remove these outliers, but another approach would be plot the logarithm of the counts against the year.

plt.plot(data['Year'], data['Body_Count'], 'rx')
ax = plt.gca() # obtain a handle to the current axis
ax.set_yscale('log') # use a logarithmic death scale
# give the plot some titles and labels
plt.title('Film Deaths against Year')
plt.ylabel('deaths')
plt.xlabel('year')

Note a few things. We are interacting with our data. In particular, we are replotting the data according to what we have learned so far. We are using the progamming language as a scripting language to give the computer one command or another, and then the next command we enter is dependent on the result of the previous. This is a very different paradigm to classical software engineering. In classical software engineering we normally write many lines of code (entire object classes or functions) before compiling the code and running it. Our approach is more similar to the approach we take whilst debugging. Historically, researchers interacted with data using a console. A command line window which allowed command entry. The notebook format we are using is slightly different. Each of the code entry boxes acts like a separate console window. We can move up and down the notebook and run each part in a different order. The state of the program is always as we left it after running the previous part.

Let's use the sum rule to compute the approximate probability that a film from the movie body count website has over 40 deaths.

deaths = (data.Body_Count>40).sum()  # number of positive outcomes (in sum True counts as 1, False counts as 0)
total_films = data.Body_Count.count()

prob_death = float(deaths)/float(total_films)
print("Probability of deaths being greather than 40 is:", prob_death)

Question 4

We now have an estimate of the probability a film has greater than 40 deaths. The estimate seems quite high. What could be wrong with the estimate? Do you think any film you go to in the cinema has this probability of having greater than 40 deaths?

Why did we have to use float around our counts of deaths and total films? What would the answer have been if we hadn't used the float command? If we were using Python 3 would we have this problem?

20 marks

Write your answer to Question 4 here

Conditioning

When predicting whether a coin turns up head or tails, we might think that this event is independent of the year or time of day. If we include an observation such as time, then in a probability this is known as condtioning. We use this notation, \(P(Y=y|T=t)\), to condition the outcome on a second variable (in this case time). Or, often, for a shorthand we use \(P(y|t)\) to represent this distribution (the \(Y=\) and \(T=\) being implicit). Because we don't believe a coin toss depends on time then we might write that \[ P(y|t) = p(y). \] However, we might believe that the number of deaths is dependent on the year. For this we can try estimating \(P(Y>40 | T=2000)\) and compare the result, for example to \(P(Y>40|2002)\) using our empirical estimate of the probability.

for year in [2000, 2002]:
    deaths = (data.Body_Count[data.Year==year]>40).sum()
    total_films = (data.Year==year).sum()

    prob_death = float(deaths)/float(total_films)
    print("Probability of deaths being greather than 40 in year", year, "is:", prob_death)

Question 5

Compute the probability for the number of deaths being over 40 for each year we have in our data data frame. Store the result in a numpy array and plot the probabilities against the years using the plot command from matplotlib. Do you think the estimate we have created of \(P(y|t)\) is a good estimate? Write your code and your written answers in the box below.

20 marks

# Write code for your answer to Question 5 in this box
# provide the answers so that the code runs correctly otherwise you will loose marks!

Notes for Question

Make sure the plot is included in this notebook file (the IPython magic command %matplotlib inline we ran above will do that for you, it only needs to be run once per file).

A Pictorial Definition of Probability

Inspired by lectures from Christopher Bishop

Definition of probability distributions.

Notational Details

Typically we should write out \(P\left(X=x,Y=y\right)\), but in practice we often shorten this to \(P\left(x,y\right)\). This looks very much like we might write a multivariate function, e.g. \[ f\left(x,y\right)=\frac{x}{y}, \] but for a multivariate function \[ f\left(x,y\right)\neq f\left(y,x\right). \] However, \[ P\left(x,y\right)=P\left(y,x\right) \] because \[ P\left(X=x,Y=y\right)=P\left(Y=y,X=x\right). \] Sometimes I think of this as akin to the way in Python we can write 'keyword arguments' in functions. If we use keyword arguments, the ordering of arguments doesn't matter.

We've now introduced conditioning and independence to the notion of probability and computed some conditional probabilities on a practical example The scatter plot of deaths vs year that we created above can be seen as a joint probability distribution. We represent a joint probability using the notation \(P(Y=y, T=t)\) or \(P(y, t)\) for short. Computing a joint probability is equivalent to answering the simultaneous questions, what's the probability that the number of deaths was over 40 and the year was 2002? Or any other question that may occur to us. Again we can easily use pandas to ask such questions.

year = 2000
deaths = (data.Body_Count[data.Year==year]>40).sum()
total_films = data.Body_Count.count() # this is total number of films
prob_death = float(deaths)/float(total_films)
print("Probability of deaths being greather than 40 and year being", year, "is:", prob_death)

The Product Rule

This number is the joint probability, \(P(Y, T)\) which is much smaller than the conditional probability. The number can never be bigger than the conditional probabililty because it is computed using the product rule. \[ p(Y=y, X=x) = p(Y=y|X=x)p(X=x) \] and \[p(X=x)\] is a probability distribution, which is equal or less than 1, ensuring the joint distribution is typically smaller than the conditional distribution.

The product rule is a fundamental rule of probability, and you must remember it! It gives the relationship between the two questions: 1) What's the probability that a film was made in 2002 and has over 40 deaths? and 2) What's the probability that a film has over 40 deaths given that it was made in 2002?

In our shorter notation we can write the product rule as \[ p(y, x) = p(y|x)p(x) \] We can see the relation working in practice for our data above by computing the different values for \(t=2000\).

p_x = float((data.Year==2002).sum())/float(data.Body_Count.count())
p_y_given_x = float((data.Body_Count[data.Year==2002]>40).sum())/float((data.Year==2002).sum())
p_y_and_x = float((data.Body_Count[data.Year==2002]>40).sum())/float(data.Body_Count.count())

print("P(x) is", p_x)
print("P(y|x) is", p_y_given_x)
print("P(y,x) is", p_y_and_x)

The Sum Rule

The other fundamental rule of probability is the sum rule this tells us how to get a marginal distribution from the joint distribution. Simply put it says that we need to sum across the value we'd like to remove. \[ P(Y=y) = \sum_{x} P(Y=y, X=x) \] Or in our shortened notation \[ P(y) = \sum_{x} P(y, x) \]

Question 6

Write code that computes \(P(y)\) by adding \(P(y, x)\) for all values of \(x\).

10 marks

# Write code for your answer to Question 6 in this box
# provide the answers so that the code runs correctly otherwise you will loose marks!

Bayes’ Rule

Bayes rule is a very simple rule, it's hardly worth the name of a rule at all. It follows directly from the product rule of probability. Because \(P(y, x) = P(y|x)P(x)\) and by symmetry \(P(y,x)=P(x,y)=P(x|y)P(y)\) then by equating these two equations and dividing through by \(P(y)\) we have \[ P(x|y) = \frac{P(y|x)P(x)}{P(y)} \] which is known as Bayes' rule (or Bayes's rule, it depends how you choose to pronounce it). It's not difficult to derive, and its importance is more to do with the semantic operation that it enables. Each of these probability distributions represents the answer to a question we have about the world. Bayes rule (via the product rule) tells us how to invert the probability.

Probabilities for Extracting Information from Data

What use is all this probability in data science? Let's think about how we might use the probabilities to do some decision making. Let's look at the information about the movies.

data.columns

Question 7

Now we see we have several additional features including the quality rating (IMDB_Rating). Let's assume we want to predict the rating given the other information in the data base. How would we go about doing it?

Using what you've learnt about joint, conditional and marginal probabilities, as well as the sum and product rule, how would you formulate the question you want to answer in terms of probabilities? Should you be using a joint or a conditional distribution? If it's conditional, what should the distribution be over, and what should it be conditioned on?

20 marks

Write your answer to Question 7 here

Probabilistic Modelling

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).

Graphical 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. \[p(\dataVector) = p(\dataScalar_\numData | \dataScalar_{\numData-1}) p(\dataScalar_{\numData-1}|\dataScalar_{\numData-2}) \dots p(\dataScalar_{2} | \dataScalar_{1})\]

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, 1995).

However, a challenge arise 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 et al., 2012).

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

Machine learning problems normally involve a prediction function and an objective function. So far in the course we've mainly focussed on the case where the prediction function was over the real numbers, so the codomain of the functions, \(\mappingFunction(\inputMatrix)\) was the real numbers or sometimes real vectors. The classification problem consists of predicting whether or not a particular example is a member of a particular class. So we may want to know if a particular image represents a digit 6 or if a particular user will click on a given advert. These are classification problems, and they require us to map to yes or no answers. That makes them naturally discrete mappings.

In classification we are given an input vector, \(\inputVector\), and an associated label, \(\dataScalar\) which either takes the value \(-1\) to represent no or \(1\) to represent yes.

• Classifiying hand written digits from binary images (automatic zip code reading)
• Detecting faces in images (e.g. digital cameras).
• Who a detected face belongs to (e.g. Picasa, Facebook, DeepFace, GaussianFace)
• Classifying type of cancer given gene expression data.
• Categorization of document types (different types of news article on the internet)

Our focus has been on models where the objective function is inspired by a probabilistic analysis of the problem. In particular we've argued that we answer questions about the data set by placing probability distributions over the various quantities of interest. For the case of binary classification this will normally involve introducing probability distributions for discrete variables. Such probability distributions, are in some senses easier than those for continuous variables, in particular we can represent a probability distribution over \(\dataScalar\), where \(\dataScalar\) is binary, with one value. If we specify the probability that \(\dataScalar=1\) with a number that is between 0 and 1, i.e. let's say that \(P(\dataScalar=1) = \pi\) (here we don't mean \(\pi\) the number, we are setting \(\pi\) to be a variable) then we can specify the probability distribution through a table.

Mathematically we can use a trick to implement this same table. We can use the value \(\dataScalar\) as a mathematical switch and write that \[ P(\dataScalar) = \pi^\dataScalar (1-\pi)^{(1-\dataScalar)} \] where our probability distribution is now written as a function of \(\dataScalar\). This probability distribution is known as the Bernoulli distribution. The Bernoulli distribution is a clever trick for mathematically switching between two probabilities if we were to write it as code it would be better described as

def bernoulli(y_i, pi):
    if y_i == 1:
        return pi
    else:
        return 1-pi

If we insert \(\dataScalar=1\) then the function is equal to \(\pi\), and if we insert \(\dataScalar=0\) then the function is equal to \(1-\pi\). So the function recreates the table for the distribution given above.

The probability distribution is named for Jacob Bernoulli, the swiss mathematician. In his book Ars Conjectandi he considered the distribution and the result of a number of 'trials' under the Bernoulli distribution to form the binomial distribution. Below is the page where he considers Pascal's triangle in forming combinations of the Bernoulli distribution to realise the binomial distribution for the outcome of positive trials.

Thomas Bayes also described the Bernoulli distribution, only he didn't refer to Jacob Bernoulli's work, so he didn't call it by that name. He described the distribution in terms of a table (think of a billiard table) and two balls. Bayes suggests that each ball can be rolled across the table such that it comes to rest at a position that is uniformly distributed between the sides of the table.

Let's assume that the first ball is rolled, and that it comes to reset at a position that is \(\pi\) times the width of the table from the left hand side.

Now, we roll the second ball. We are interested if the second ball ends up on the left side (+ve result) or the right side (-ve result) of the first ball. We use the Bernoulli distribution to determine this.

For this reason in Bayes's distribution there is considered to be aleatoric uncertainty about the distribution parameter.

Maximum Likelihood in the Bernoulli Distribution

Maximum likelihood in the Bernoulli distribution is straightforward. Let's assume we have data, \(\dataVector\) which consists of a vector of binary values of length \(n\). If we assume each value was sampled independently from the Bernoulli distribution, conditioned on the parameter \(\pi\) then our joint probability density has the form \[ p(\dataVector|\pi) = \prod_{i=1}^{\numData} \pi^{\dataScalar_i} (1-\pi)^{1-\dataScalar_i}. \] As normal in maximum likelihood we consider the negative log likelihood as our objective, \[\begin{align*} \errorFunction(\pi)& = -\log p(\dataVector|\pi)\\ & = -\sum_{i=1}^{\numData} \dataScalar_i \log \pi - \sum_{i=1}^{\numData} (1-\dataScalar_i) \log(1-\pi), \end{align*}\]

and we can derive the gradient with respect to the parameter \(\pi\). \[\frac{\text{d}\errorFunction(\pi)}{\text{d}\pi} = -\frac{\sum_{i=1}^{\numData} \dataScalar_i}{\pi} + \frac{\sum_{i=1}^{\numData} (1-\dataScalar_i)}{1-\pi},\]

and as normal we look for a stationary point for the log likelihood by setting this derivative to zero, \[0 = -\frac{\sum_{i=1}^{\numData} \dataScalar_i}{\pi} + \frac{\sum_{i=1}^{\numData} (1-\dataScalar_i)}{1-\pi},\] rearranging we form \[(1-\pi)\sum_{i=1}^{\numData} \dataScalar_i = \pi\sum_{i=1}^{\numData} (1-\dataScalar_i),\] which implies \[\sum_{i=1}^{\numData} \dataScalar_i = \pi\left(\sum_{i=1}^{\numData} (1-\dataScalar_i) + \sum_{i=1}^{\numData} \dataScalar_i\right),\]

and now we recognise that \(\sum_{i=1}^{\numData} (1-\dataScalar_i) + \sum_{i=1}^{\numData} \dataScalar_i = \numData\) so we have \[\pi = \frac{\sum_{i=1}^{\numData} \dataScalar_i}{\numData}\]

so in other words we estimate the probability associated with the Bernoulli by setting it to the number of observed positives, divided by the total length of \(\dataScalar\). This makes intiutive sense. If I asked you to estimate the probability of a coin being heads, and you tossed the coin 100 times, and recovered 47 heads, then the estimate of the probability of heads should be \(\frac{47}{100}\).

Exercise

Show that the maximum likelihood solution we have found is a minimum for our objective.

Write your answer to Exercise here

# Use this box for any code you need for the exercise

\[ \text{posterior} = \frac{\text{likelihood}\times\text{prior}}{\text{marginal likelihood}} \]

Four components:

1. Prior distribution
2. Likelihood
3. Posterior distribution
4. Marginal likelihood

Naive Bayes Classifiers

In probabilistic machine learning we place probability distributions (or densities) over all the variables of interest, our first classification algorithm will do just that. We will consider how to form a classification by making assumptions about the joint density of our observations. We need to make assumptions to reduce the number of parameters we need to optimise.

In the ideal world, given label data \(\dataVector\) and the inputs \(\inputMatrix\) we should be able to specify the joint density of all potential values of \(\dataVector\) and \(\inputMatrix\), \(p(\dataVector, \inputMatrix)\). If \(\inputMatrix\) and \(\dataVector\) are our training data, and we can somehow extend our density to incorporate future test data (by augmenting \(\dataVector\) with a new observation \(\dataScalar^*\) and \(\inputMatrix\) with the corresponding inputs, \(\inputVector^*\)), then we can answer any given question about a future test point \(\dataScalar^*\) given its covariates \(\inputVector^*\) by conditioning on the training variables to recover, \[ p(\dataScalar^*|\inputMatrix, \dataVector, \inputVector^*), \]

We can compute this distribution using the product and sum rules. However, to specify this density we must give the probability associated with all possible combinations of \(\dataVector\) and \(\inputMatrix\). There are \(2^{\numData}\) possible combinations for the vector \(\dataVector\) and the probability for each of these combinations must be jointly specified along with the joint density of the matrix \(\inputMatrix\), as well as being able to extend the density for any chosen test location \(\inputVector^*\).

In naive Bayes we make certain simplifying assumptions that allow us to perform all of the above in practice.

Data Conditional Independence

If we are given model parameters \(\paramVector\) we assume that conditioned on all these parameters that all data points in the model are independent. In other words we have, \[ p(\dataScalar^*, \inputVector^*, \dataVector, \inputMatrix|\paramVector) = p(\dataScalar^*, \inputVector^*|\paramVector)\prod_{i=1}^{\numData} p(\dataScalar_i, \inputVector_i | \paramVector). \] This is a conditional independence assumption because we are not assuming our data are purely independent. If we were to assume that, then there would be nothing to learn about our test data given our training data. We are assuming that they are independent given our parameters, \(\paramVector\). We made similar assumptions for regression, where our parameter set included \(\mappingVector\) and \(\dataStd^2\). Given those parameters we assumed that the density over \(\dataVector, \dataScalar^*\) was independent. Here we are going a little further with that assumption because we are assuming the joint density of \(\dataVector\) and \(\inputMatrix\) is independent across the data given the parameters.

Computing posterior distribution in this case becomes easier, this is known as the 'Bayes classifier'.

Feature Conditional Independence

\[ p(\inputVector_i | \dataScalar_i, \paramVector) = \prod_{j=1}^{\dataDim} p(\inputScalar_{i,j}|\dataScalar_i, \paramVector) \] where \(\dataDim\) is the dimensionality of our inputs.

The assumption that is particular to naive Bayes is to now consider that the features are also conditionally independent, but not only given the parameters. We assume that the features are independent given the parameters and the label. So for each data point we have \[p(\inputVector_i | \dataScalar_i, \paramVector) = \prod_{j=1}^{\dataDim} p(\inputScalar_{i,j}|\dataScalar_i,\paramVector)\] where \(\dataDim\) is the dimensionality of our inputs.

Marginal Density for \(\dataScalar_i\)

\[ p(\inputScalar_{i,j},\dataScalar_i| \paramVector) = p(\inputScalar_{i,j}|\dataScalar_i, \paramVector)p(\dataScalar_i). \]

We now have nearly all of the components we need to specify the full joint density. However, the feature conditional independence doesn't yet give us the joint density over \(p(\dataScalar_i, \inputVector_i)\) which is required to subsitute in to our data conditional independence to give us the full density. To recover the joint density given the conditional distribution of each feature, \(p(\inputScalar_{i,j}|\dataScalar_i, \paramVector)\), we need to make use of the product rule and combine it with a marginal density for \(\dataScalar_i\),

\[p(\inputScalar_{i,j},\dataScalar_i| \paramVector) = p(\inputScalar_{i,j}|\dataScalar_i, \paramVector)p(\dataScalar_i).\] Because \(\dataScalar_i\) is binary the Bernoulli density makes a suitable choice for our prior over \(\dataScalar_i\), \[p(\dataScalar_i|\pi) = \pi^{\dataScalar_i} (1-\pi)^{1-\dataScalar_i}\] where \(\pi\) now has the interpretation as being the prior probability that the classification should be positive.

Joint Density for Naive Bayes

This allows us to write down the full joint density of the training data, \[ p(\dataVector, \inputMatrix|\paramVector, \pi) = \prod_{i=1}^{\numData} \prod_{j=1}^{\dataDim} p(\inputScalar_{i,j}|\dataScalar_i, \paramVector)p(\dataScalar_i|\pi) \]

which can now be fit by maximum likelihood. As normal we form our objective as the negative log likelihood,

\[\begin{align*} \errorFunction(\paramVector, \pi)& = -\log p(\dataVector, \inputMatrix|\paramVector, \pi) \\ &= -\sum_{i=1}^{\numData} \sum_{j=1}^{\dataDim} \log p(\inputScalar_{i, j}|\dataScalar_i, \paramVector) - \sum_{i=1}^{\numData} \log p(\dataScalar_i|\pi), \end{align*}\] which we note decomposes into two objective functions, one which is dependent on \(\pi\) alone and one which is dependent on \(\paramVector\) alone so we have, \[ \errorFunction(\pi, \paramVector) = \errorFunction(\paramVector) + \errorFunction(\pi). \] Since the two objective functions are separately dependent on the parameters \(\pi\) and \(\paramVector\) we can minimize them independently. Firstly, minimizing the Bernoulli likelihood over the labels we have, \[ \errorFunction(\pi) = -\sum_{i=1}^{\numData}\log p(\dataScalar_i|\pi) = -\sum_{i=1}^{\numData} \dataScalar_i \log \pi - \sum_{i=1}^{\numData} (1-\dataScalar_i) \log (1-\pi) \] which we already minimized above recovering \[ \pi = \frac{\sum_{i=1}^{\numData} \dataScalar_i}{\numData}. \]

We now need to minimize the objective associated with the conditional distributions for the features, \[ \errorFunction(\paramVector) = -\sum_{i=1}^{\numData} \sum_{j=1}^{\dataDim} \log p(\inputScalar_{i, j} |\dataScalar_i, \paramVector), \] which necessarily implies making some assumptions about the form of the conditional distributions. The right assumption will depend on the nature of our input data. For example, if we have an input which is real valued, we could use a Gaussian density and we could allow the mean and variance of the Gaussian to be different according to whether the class was positive or negative and according to which feature we were measuring. That would give us the form, \[ p(\inputScalar_{i, j} | \dataScalar_i,\paramVector) = \frac{1}{\sqrt{2\pi \dataStd_{\dataScalar_i,j}^2}} \exp \left(-\frac{(\inputScalar_{i,j} - \mu_{\dataScalar_i, j})^2}{\dataStd_{\dataScalar_i,j}^2}\right), \] where \(\dataStd_{1, j}^2\) is the variance of the density for the \(j\)th output and the class \(\dataScalar_i=1\) and \(\dataStd_{0, j}^2\) is the variance if the class is 0. The means can vary similarly. Our parameters, \(\paramVector\) would consist of all the means and all the variances for the different dimensions.

As normal we form our objective as the negative log likelihood, \[ \errorFunction(\paramVector, \pi) = -\log p(\dataVector, \inputMatrix|\paramVector, \pi) = -\sum_{i=1}^{\numData} \sum_{j=1}^{\dataDim} \log p(\inputScalar_{i, j}|\dataScalar_i, \paramVector) - \sum_{i=1}^{\numData} \log p(\dataScalar_i|\pi), \] which we note decomposes into two objective functions, one which is dependent on \(\pi\) alone and one which is dependent on \(\paramVector\) alone so we have, \[ \errorFunction(\pi, \paramVector) = \errorFunction(\paramVector) + \errorFunction(\pi). \]

Movie Body Count Data

First we will load in the movie body count data. Our aim will be to predict whether a movie is rated R or not given the attributes in the data. We will predict on the basis of year, body count and movie genre. The genres in the CSV file are stored as a list in the following form:

Biography|Action|Sci-Fi

First we have to do a little work to extract this form and turn it into a vector of binary values. Let's first load in and remind ourselves of the data.

import pods

data = pods.datasets.movie_body_count()['Y']
data.head()

Now we will convert this data into a form which we can use as inputs X, and labels y.

import pandas as pd
import numpy as np

X = data[['Year', 'Body_Count']].copy()
y = data['MPAA_Rating']=='R' # set label to be positive for R rated films.

# Create series of movie genres with the relevant index
s = data['Genre'].apply(pd.Series, 1).stack() 
s.index = s.index.droplevel(-1) # to line up with df's index

# Extract from the series the unique list of genres.
genres = s.unique()

# For each genre extract the indices where it is present and add a column to X
for genre in genres:
    index = s[s==genre].index.tolist()
    X.loc[:, genre] = 0.0 
    X.loc[index, genre] = 1.0

This has given us a new data frame X which contains the different genres in different columns.

X.describe()

We can now specify the naive Bayes model. For the genres we want to model the data as Bernoulli distributed, and for the year and body count we want to model the data as Gaussian distributed. We set up two data frames to contain the parameters for the rows and the columns below.

# assume data is binary or real.
# this list encodes whether it is binary or real (1 for binary, 0 for real)
binary_columns = genres
real_columns = ['Year', 'Body_Count']
Bernoulli = pd.DataFrame(data=np.zeros((2,len(binary_columns))), columns=binary_columns, index=['theta_0', 'theta_1'])
Gaussian = pd.DataFrame(data=np.zeros((4,len(real_columns))), columns=real_columns, index=['mu_0', 'sigma2_0', 'mu_1', 'sigma2_1'])

Now we have the data in a form ready for analysis, let's construct our data matrix.

num_train = 200
indices = np.random.permutation(X.shape[0])
train_indices = indices[:num_train]
test_indices = indices[num_train:]
X_train = X.loc[train_indices]
y_train = y.loc[train_indices]
X_test = X.loc[test_indices]
y_test = y.loc[test_indices]

And we can now train the model. For each feature we can make the fit independently. The fit is given by either counting the number of positives (for binary data) which gives us the maximum likelihood solution for the Bernoulli. Or by computing the empirical mean and variance of the data for the Gaussian, which also gives us the maximum likelihood solution.

for column in X_train:
    if column in Gaussian:
        Gaussian[column]['mu_0'] = X_train[column][~y].mean()
        Gaussian[column]['mu_1'] = X_train[column][y].mean()
        Gaussian[column]['sigma2_0'] = X_train[column][~y].var(ddof=0)
        Gaussian[column]['sigma2_1'] = X_train[column][y].var(ddof=0)
    if column in Bernoulli:
        Bernoulli[column]['theta_0'] = X_train[column][~y].sum()/(~y).sum()
        Bernoulli[column]['theta_1'] = X_train[column][y].sum()/(y).sum()

We can examine the nature of the distributions we've fitted to the model by looking at the entries in these data frames.

Bernoulli

Gaussian

The final model parameter is the prior probability of the positive class, \(\pi\), which is computed by maximum likelihood.

prior = float(y_train.sum())/len(y_train)

Making Predictions

Naive Bayes has given us the class conditional densities: \(p(\inputVector_i | \dataScalar_i, \paramVector)\). To make predictions with these densities we need to form the distribution given by \[ P(\dataScalar^*| \dataVector, \inputMatrix, \inputVector^*, \paramVector) \] This can be computed by using the product rule. We know that \[ P(\dataScalar^*| \dataVector, \inputMatrix, \inputVector^*, \paramVector)p(\dataVector, \inputMatrix, \inputVector^*|\paramVector) = p(\dataScalar*, \dataVector, \inputMatrix, \inputVector^*| \paramVector) \] implying that \[ P(\dataScalar^*| \dataVector, \inputMatrix, \inputVector^*, \paramVector) = \frac{p(\dataScalar*, \dataVector, \inputMatrix, \inputVector^*| \paramVector)}{p(\dataVector, \inputMatrix, \inputVector^*|\paramVector)} \] and we've already defined \(p(\dataScalar^*, \dataVector, \inputMatrix, \inputVector^*| \paramVector)\) using our conditional independence assumptions above \[ p(\dataScalar^*, \dataVector, \inputMatrix, \inputVector^*| \paramVector) = \prod_{j=1}^{\dataDim} p(\inputScalar^*_{j}|\dataScalar^*, \paramVector)p(\dataScalar^*|\pi)\prod_{i=1}^{\numData} \prod_{j=1}^{\dataDim} p(\inputScalar_{i,j}|\dataScalar_i, \paramVector)p(\dataScalar_i|\pi) \] The other required density is \[ p(\dataVector, \inputMatrix, \inputVector^*|\paramVector) \] which can be found from \[p(\dataScalar^*, \dataVector, \inputMatrix, \inputVector^*| \paramVector)\] using the sum rule of probability, \[ p(\dataVector, \inputMatrix, \inputVector^*|\paramVector) = \sum_{\dataScalar^*=0}^1 p(\dataScalar^*, \dataVector, \inputMatrix, \inputVector^*| \paramVector). \] Because of our independence assumptions that is simply equal to \[ p(\dataVector, \inputMatrix, \inputVector^*| \paramVector) = \sum_{\dataScalar^*=0}^1 \prod_{j=1}^{\dataDim} p(\inputScalar^*_{j}|\dataScalar^*_i, \paramVector)p(\dataScalar^*|\pi)\prod_{i=1}^{\numData} \prod_{j=1}^{\dataDim} p(\inputScalar_{i,j}|\dataScalar_i, \paramVector)p(\dataScalar_i|\pi). \] Substituting both forms in to recover our distribution over the test label conditioned on the training data we have, \[ P(\dataScalar^*| \dataVector, \inputMatrix, \inputVector^*, \paramVector) = \frac{\prod_{j=1}^{\dataDim} p(\inputScalar^*_{j}|\dataScalar^*_i, \paramVector)p(\dataScalar^*|\pi)\prod_{i=1}^{\numData} \prod_{j=1}^{\dataDim} p(\inputScalar_{i,j}|\dataScalar_i, \paramVector)p(\dataScalar_i|\pi)}{\sum_{\dataScalar^*=0}^1 \prod_{j=1}^{\dataDim} p(\inputScalar^*_{j}|\dataScalar^*_i, \paramVector)p(\dataScalar^*|\pi)\prod_{i=1}^{\numData} \prod_{j=1}^{\dataDim} p(\inputScalar_{i,j}|\dataScalar_i, \paramVector)p(\dataScalar_i|\pi)} \] and we notice that all the terms associated with the training data actually cancel, the test prediction is conditionally independent of the training data given the parameters. This is a result of our conditional independence assumptions over the data points. \[ p(\dataScalar^*| \inputVector^*, \paramVector) = \frac{\prod_{j=1}^{\dataDim} p(\inputScalar^*_{j}|\dataScalar^*_i, \paramVector)p(\dataScalar^*|\pi)}{\sum_{\dataScalar^*=0}^1 \prod_{j=1}^{\dataDim} p(\inputScalar^*_{j}|\dataScalar^*_i, \paramVector)p(\dataScalar^*|\pi)} \] This formula is also fairly straightforward to implement. First we implement the log probabilities for the Gaussian density.

def log_gaussian(x, mu, sigma2):
    return -0.5* np.log(2*np.pi*sigma2)-((x-mu)**2)/(2*sigma2)

Now for any test point we compute the joint distribution of the Gaussian features by summing their log probabilities. Working in log space can be a considerable advantage over computing the probabilities directly: as the number of features we include goes up, because all the probabilities are less than 1, the joint probability will become smaller and smaller, and may be difficult to represent accurately (or even underflow). Working in log space can ameliorate this problem. We can also compute the log probability for the Bernoulli distribution.

def log_bernoulli(x, theta):
    return x*np.log(theta) + (1-x)*np.log(1-theta)

Laplace Smoothing

Before we proceed, let's just pause and think for a moment what will happen if theta here is either zero or one. This will result in \(\log 0 = -\infty\) and cause numerical problems. This definitely can happen in practice. If some of the features are rare or very common across the data set then the maximum likelihood solution could find values of zero or one respectively. Such values are problematic because they cause posterior probabilities of class membership of either one or zero. In practice we deal with this using Laplace smoothing (which actually has an interpretation as a Bayesian fit of the Bernoulli distribution. Laplace used an example of the sun rising each day, and a wish to predict the sun rise the following day to describe his idea of smoothing, which can be found at the bottom of following page from Laplace's 'Essai Philosophique ...'

Laplace suggests that when computing the probability of an event where a success or failure is rare (he uses an example of the sun rising across the last 5,000 years or 1,826,213 days) that even though only successes have been observed (in the sun rising case) that the odds for tomorrow shouldn't be given as \[ \frac{1,826,213}{1,826,213} = 1 \] but rather by adding one to the numerator and two to the denominator, \[ \frac{1,826,213 + 1}{1,826,213 + 2} = 0.99999945. \] This technique is sometimes called a 'pseudocount technique' because it has an intepretation of assuming some observations before you start, it's as if instead of observing \(\sum_{i}\dataScalar_i\) successes you have an additional success, \(\sum_{i}\dataScalar_i + 1\) and instead of having observed \(n\) events you've observed \(\numData + 2\). So we can think of Laplace's idea saying (before we start) that we have 'two observations worth of belief, that the odds are 50/50', because before we start (i.e. when \(\numData=0\)) our estimate is 0.5, yet because the effective \(n\) is only 2, this estimate is quickly overwhelmed by data. Laplace used ideas like this a lot, and it is known as his 'principle of insufficient reason'. His idea was that in the absence of knowledge (i.e. before we start) we should assume that all possible outcomes are equally likely. This idea has a modern counterpart, known as the principle of maximum entropy. A lot of the theory of this approach was developed by Ed Jaynes, who according to his erstwhile collaborator and friend, John Skilling, learnt French as an undergraduate by reading the works of Laplace. Although John also related that Jaynes's spoken French was not up to the standard of his scientific French. For me Ed Jaynes's work very much carries on the tradition of Laplace into the modern era, in particular his focus on Bayesian approaches. I'm very proud to have met those that knew and worked with him. It turns out that Laplace's idea also has a Bayesian interpretation (as Laplace understood), it comes from assuming a particular prior density for the parameter \(\pi\), but we won't explore that interpretation for the moment, and merely choose to estimate the probability as, \[ \pi = \frac{\sum_{i=1}^{\numData} \dataScalar_i + 1}{\numData + 2} \] to prevent problems with certainty causing numerical issues and misclassifications. Let's refit the Bernoulli features now.

# fit the Bernoulli with Laplace smoothing.
for column in X_train:
    if column in Bernoulli:
        Bernoulli[column]['theta_0'] = (X_train[column][~y].sum() + 1)/((~y).sum() + 2)
        Bernoulli[column]['theta_1'] = (X_train[column][y].sum() + 1)/((y).sum() + 2)

That places us in a position to write the prediction function.

def predict(X_test, Gaussian, Bernoulli, prior):
    log_positive = pd.Series(data = np.zeros(X_test.shape[0]), index=X_test.index)
    log_negative = pd.Series(data = np.zeros(X_test.shape[0]), index=X_test.index)
    for column in X_test.columns:
        if column in Gaussian:
            log_positive += log_gaussian(X_test[column], Gaussian[column]['mu_1'], Gaussian[column]['sigma2_1'])
            log_negative += log_gaussian(X_test[column], Gaussian[column]['mu_0'], Gaussian[column]['sigma2_0'])
        elif column in Bernoulli:
            log_positive += log_bernoulli(X_test[column], Bernoulli[column]['theta_1'])
            log_negative += log_bernoulli(X_test[column], Bernoulli[column]['theta_0'])
            
    return np.exp(log_positive + np.log(prior))/(np.exp(log_positive + np.log(prior)) + np.exp(log_negative + np.log(1-prior)))

Now we are in a position to make the predictions for the test data.

p_y = predict(X_test, Gaussian, Bernoulli, prior)

We can test the quality of the predictions in the following way. Firstly, we can threshold our probabilities at 0.5, allocating points with greater than 50% probability of membership of the positive class to the positive class. We can then compare to the true values, and see how many of these values we got correct. This is our total number correct.

correct = y_test.eq(p_y>0.5)
total_correct = sum(correct)
print("Total correct", total_correct, " out of ", len(y_test), "which is", float(total_correct)/len(y_test), "%")

We can also now plot the confusion matrix. A confusion matrix tells us where we are making mistakes. Along the diagonal it stores the true positives, the points that were positive class that we classified correctly, and the true negatives, the points that were negative class and that we classified correctly. The off diagonal terms contain the false positives and the false negatives. Along the rows of the matrix we place the actual class, and along the columns we place our predicted class.

confusion_matrix = pd.DataFrame(data=np.zeros((2,2)), 
                                columns=['predicted not R-rated', 'predicted R-rated'],
                                index =['actual not R-rated','actual R-rated'])
confusion_matrix['predicted R-rated']['actual R-rated'] = (y_test & (p_y>0.5)).sum()
confusion_matrix['predicted R-rated']['actual not R-rated'] = (~y_test & (p_y>0.5)).sum()
confusion_matrix['predicted not R-rated']['actual R-rated'] = (y_test & ~(p_y>0.5)).sum()
confusion_matrix['predicted not R-rated']['actual not R-rated'] = (~y_test & ~(p_y>0.5)).sum()
confusion_matrix

Exercise

How can you improve your classification, are all the features equally valid? Are some features more helpful than others? What happens if you remove features that appear to be less helpful. How might you select such features?

Write your answer to Exercise here

# Use this box for any code you need for the exercise

Exercise

We have decided to classify positive if probability of R rating is greater than 0.5. This has led us to accidentally classify some films as 'safe for children' when the aren't in actuallity. Imagine you wish to ensure that the film is safe for children. With your test set how low do you have to set the threshold to avoid all the false negatives (i.e. films where you said it wasn't R-rated, but in actuality it was?

Write your answer to Exercise here

# Use this box for any code you need for the exercise

Making Predictions

Naive Bayes has given us the class conditional densities: \(p(\inputVector_i | \dataScalar_i, \paramVector)\). To make predictions with these densities we need to form the distribution given by \[ P(\dataScalar^*| \dataVector, \inputMatrix, \inputVector^*, \paramVector) \]

Exercise

Write down the negative log likelihood of the Gaussian density over a vector of variables \(\inputVector\). Assume independence between each variable. Minimize this objective to obtain the maximum likelihood solution of the form. \[ \mu = \frac{\sum_{i=1}^{\numData} \inputScalar_i}{\numData} \] \[ \dataStd^2 = \frac{\sum_{i=1}^{\numData} (\inputScalar_i - \mu)^2}{\numData} \]

Write your answer to Exercise here

# Use this box for any code you need for the exercise

If the input data was binary then we could also make use of the Bernoulli distribution for the features. For that case we would have the form, \[ p(\inputScalar_{i, j} | \dataScalar_i,\paramVector) = \theta_{\dataScalar_i, j}^{\inputScalar_{i, j}}(1-\theta_{\dataScalar_i, j})^{(1-\inputScalar_{i,j})}, \] where \(\theta_{1, j}\) is the probability that the \(j\)th feature is on if \(\dataScalar_i\) is 1.

In either case, maximum likelihood fitting would proceed in the same way. The objective has the form, \[ \errorFunction(\paramVector) = -\sum_{j=1}^{\dataDim} \sum_{i=1}^{\numData} \log p(\inputScalar_{i,j} |\dataScalar_i, \paramVector), \] and if, as above, the parameters of the distributions are specific to each feature vector (we had means and variances for each continuous feature, and a probability for each binary feature) then we can use the fact that these parameters separate into disjoint subsets across the features to write, \[ \begin{align*} \errorFunction(\paramVector) &= -\sum_{j=1}^{\dataDim} \sum_{i=1}^{\numData} \log p(\inputScalar_{i,j} |\dataScalar_i, \paramVector_j)\\ & \sum_{j=1}^{\dataDim} \errorFunction(\paramVector_j), \end{align*} \] which means we can minimize our objective on each feature independently.

These characteristics mean that naive Bayes scales very well with big data. To fit the model we consider each feature in turn, we select the positive class and fit parameters for that class, then we select each negative class and fit features for that class. We have code below.

Naive Bayes Summary

Naive Bayes is making very simple assumptions about the data, in particular it is modeling the full joint probability of the data set, \(p(\dataVector, \inputMatrix | \paramVector, \pi)\) by very strong assumptions about factorizations that are unlikely to be true in practice. The data conditional independence assumption is common, and relies on a rich parameter vector to absorb all the information in the training data. The additional assumption of naive Bayes is that features are conditional independent given the class label \(\dataScalar_i\) (and the parameter vector, \(\paramVector\). This is quite a strong assumption. However, it causes the objective function to decompose into parts which can be independently fitted to the different feature vectors, meaning it is very easy to fit the model to large data. It is also clear how we should handle streaming data and missing data. This means that the model can be run 'live', adapting parameters and information as it arrives. Indeed, the model is even capable of dealing with new features that might arrive at run time. Such is the strength of the modeling the joint probability density. However, the factorization assumption that allows us to do this efficiently is very strong and may lead to poor decision boundaries in practice.

Other Reading

• Chapter 5 of Rogers and Girolami (2011) up to pg 179 (Section 5.1, and 5.2 up to 5.2.2).

References