2 S. MUKHERJEE, PROBABILISTIC MACHINE LEARNING
LECTURE 1
Course preliminaries
The term machine learning goes back to Arthur Samuels and his computer
checker playing algoriths. In 1959 Samuels described machine learning as: ”Field
of study that gives computers the ability to learn without being explicitly pro-
grammed.”
Machine learning is considered a subfield of artificial intelligence and the idea
of a learning machine is given in ”Computing Machinery and Intelligence,” by Alan
Turing in 1950 in Mind: A Quarterly Review of Psychology and Philosophy. The
question posed in the fist sentence of this paper was “Can machines think ?”.
For this class by ML we are going to consider algorithms and probabilistic
methods to ”learn from data.” The material is at the interface of statistics and
computer science and one caricature of ML is computer scientists doing statistics.
ML is often also associated with the term ”big data” which is often meant to be
statistical analysis with very large data sets, here the computational challenge is as
serious as the inference problem.
Broadly speaking the methods we will discuss can be placed into two categories:
proceduralists: This will cover both frequentist statistics, as well as algorithmic
approaches to ML. This approach is based upon coming up with good procedures
to apply to data. What is meant by good is some long run probability of the
procedure, for example the long run probability of errors made in classification is
small.
Bayesian: A coherent axiomatic approach to inference based on inference of the
posterior probability of parameters or models given data. Bayesian inference may
not be feasible or practical in certain situations.
1.1. Review
We’ll start with a basic review of statistics. We will examine a statistical question
using both Bayesian and frequentist analysis. The following formalism will be
quantified in both models
P(M | D) =
P(D | M )P(M)
P(D)
∝ P(D | M )P(M),
where P(M | D) is evidence for model M given data D, P(D | M) is evidence for
D given model M, P(M ) is the probability of model M, and P(D) the probability
of data, The standard statistical terms for these objects are
P(D | M ) ≡ Lik(D; M), Likelihood of data given model M
P(M | D) ≡ Post(D; M), Posterior evidence of model M given data
P(M) ≡ π(M), prior probability (before seeing data) for model M.
Example 1: Motif estimation
We consider a random variable X that is drawn from a alphabet of k = 4 letters
{A, C, T, G} where we represent A ≡ 1, C ≡ 2, T ≡ 3, and G ≡ 4. We set the
probability distribution on X as the following multinomial distribution, note we
