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Computational
Molecular Biology, aka Algorithms for Computational Biology
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Module Guide: Probability
This module is intended to teach the basics of
probabilistic thinking and estimation of probabilities. We do not assume any
prior exposure to probability theory, but the review will be useful for those
who do know something about it.
The abilities you should come away from this
module with are:
- To create and reason about simple discrete probability models.
- To manipulate conditional probability expressions and to
demonstrate the validity of the rules for manipulating them.
- To estimate probabilities using maximum likelihood (ML) estimators.
You should also be able to articulate the difference between ML and
Bayesian estimators and how pseudo counts relate to ML and Bayesian
estimation approaches.
- To use the expectation maximization (EM) framework to estimate
probabilities in the presence of hidden variables.
In advance of each class, you should study
the assigned sections of the printed lecture notes and read the corresponding
sections of Probability and Statistics by Morris de Groot. This book is on
reserve at the libraries, and I have scanned in a few selected sections, which
are linked to the reading assignments below. The lecture notes are not meant as
a substitute for a textbook, but rather to highlight some of the most important
and relevant material from the textbook.
There are exercises to do at home in advance
of each class. These will not be collected. Instead, we will have similar
exercises for you to do in class and turn in for a grade. The process of doing
the exercises in class is intended to be a learning experience, but there won't
be enough time for you to get much out of it unless you have already studied
the text and struggled with the homework problems. In class, we will help you
like a workout partner would, by taking a few ounces of weight off you just
before you drop the barbell on your chest. If you come to class without having
looked at the material, the in-class experience may feel more like a quiz that
you aren't prepared for.
Day 1: Friday, August
28
In class
Expectation Maximization
Demo
Before the next class
- Read and think about the probability
course notes through Section 4, Combinations.
- Read at least Sections 1.6-1.8 of de Groot. If possible, read 1.3-1.5, as
well.
- Do these exercises, referring
to the course notes or text as needed.
Day 2: Monday, August
31
In class
- In class exercises, similar to the previous homework.
- Presentation/Discussion: Counting problems.
Before the next class
- Read the probability course notes, through Section 7, Random
Variables.
- Read at least Section 1.11 and 2.1 of de Groot. If possible, read 3.1 as well.
- Do these exercises, referring
to the course notes or text as needed.
Day 3: Wednesday,
September 2
In class
- In class exercises, similar to the previous homework.
- Discussion: Conditional probability, independence.
Before the next class
- Read the probability course notes, Section 9, Four Rules.
- Read 2.2 of de Groot.
- Do these exercises, referring
to the course notes or text as needed.
Day 4: Friday,
September 4
In class
- In class exercises, similar to the previous homework.
- Discussion: Random variables, Bayes’ rule.
Before the next class
- Read the probability course notes sections entitled Expectation and
Estimation.
- Read at least Section 4.1 and 4.2 of de Groot.
Before September 11
Labor Day: Monday,
September 7
Day 5: Wednesday,
September 9
In class
- Workshop: Problems and issues with EM Programming Assignment Part
1.
- Discussion: Expectation, Estimation.
Before the next class
- Read the probability course notes sections entitled Maximum
Likelihood and Expectation Maximization.
Day 6: Friday,
September 11
In class
- Due: EM programming assignment, Part 1
- Discussion: Maximum Likelihood estimation and its limitations;
Expectation Maximization; continuous random variables
Before the next class
- Start on EM programming assignment, Part2
Day 7: Monday, September
14
In class
- Workshop: Problems and issues with EM Programming Assignment Part 2
- Discussion: Bayes estimators, conjugate priors, and pseudocounts