Theory of Inference 2012/13 (MATH 35600)
Reading week, Teaching Week 17 (week starting Mon
25 Feb). There is a revision sheet for weeks 1 to
3. Note that the final week of the course is Teaching Week 19 (ie the final
lecture is on Fri 15 Mar).
Navigation:
Course outline,
details,
homework and assessment.
The basic premise of inference is our judgement that the things we
would like to know are related to other things that we can
measure. This premise holds over the whole of the sciences. The
distinguishing features of statistical science are
- A probabilistic approach to quantifying uncertainty, and, within
that,
- A concern to assess the principles under which we make good
inferences, and
- The development of tools to facilitate the making of such
inferences.
In this course I cover these topics in a rather general way,
reflecting the increasing tendency of statisticians to work together with
applied scientists on complex problems.
Reading
There is a comprehensive handout for this course. The following are just
suggestions if youare interested in following up the topics in the lectures.
For additional reading, start with
- D.R. Cox, 2006, Principles of statistical inference, Oxford
University Press.
For background reading on basic probability and 'standard' statistics:
- David Freedman et al, 2007, Statistics, Norton, 4th edn (earlier
editions also good).
- J.A. Rice, 1999, Mathematical Statistics and Data Analysis, 2nd
edn, Duxbury Press.
- L.Wasserman, 2004, All of Statistics: A Concise Course in Statistical
Inference, New York: Springer.
- M. DeGroot and M. Schervish, 2002, Probability and Statistics, Addison Wesley, 3rd edn.
For more advanced material on applied statistics:
- A. Davison, 2003, Statistical Modelling, Cambridge, UK: Cambridge
University Press.
And for more advanced material on theoretical statistics:
- D.R. Cox and D.V. Hinkley, 1974, Theoretical Statistics, London:
Chapman and Hall.
- M. Schervish, 1995, Theory of Statistics, New York: Springer.
- C.P. Robert, 2007, The Bayesian Choice: From Decision-Theoretic
Foundations to Computational Implementation, New York: Springer.
- A. O'Hagan and J. Forster, 2004, Bayesian Inference, Kendall's
Advanced Theory of Statistics, vol2b, Hodder Arnold.
For the use of probability and statistics in society,
- G. Gigerenzer, 2003, Reckoning with Risk: Learning to Live with
Uncertainty, Penguin.
- S. Senn, 2003, Dicing with death: Chance, Risk and Health, Cambridge UK: Cambridge University Press.
The exam comprises three questions, of which your best two count. In my
experience it is much better to pick out your best two questions at
the start of the exam, and focus on these, than to try all three. If you
adopt the latter strategy, all of the time you put into your weakest question
is wasted.
Previous exam papers are available on Blackboard. You should be aware that
the course has changed this year and last year, and these questions cannot be taken as a
reliable quide to the questions that will be set this year.
Answers to previous exam papers will not be made available. The
exam is designed to assess whether you have attended the lectures, read and
thought about your lecture notes and the handouts, done the homework, and
read a bit more widely in the textbooks. Diligent students who have done
the above will gain no relative benefit from studying the answers to
previous exam questions. On the other hand, less diligent students may
suffer the illusion that they will do well in the exam, when probably they
will not.
Here is a lecture-by-lecture summary of the course, looking as far ahead as
seems prudent. This plan is subject to revision.
- Statistical models (2 lectures)
Handout, covering background probability
theory and statistical modelling approaches.
- Notation and concepts
- Statistical modelling, two approaches
- Bayesian inference (3 lectures)
Handout, covering the Bayesian approach to
inference.
- Bayes's theorem (don't mention the prior PDF)
- Sampling, marginalising, predicting
- Credible sets and and hypothesis tests
- Confidence sets (3 lectures)
Handout, covering confidence sets.
- Definition, pivotal functions
- Asymptotic approximations
- Bootstrapping
- Decision theory (4 lectures)
Handout, covering Decision Theory.
- Definition, the Bayes rule
- The value of information
- Admissibility
- Simple dichotomy
- Model criticism and hypothesis testing (4 lectures)
Handout, covering model criticism and
hypothesis testing.
- P-values, the P-value of a simple hypothesis
- Example with the Poisson distribution. R code
snippet. Warning! There is an unfortunately low P-value
in the middle of this script; I made a Type 1 error, and should have redone
the analysis with a different random seed. We'll explore this in the
homework.
- Hypothesis testing
- More on hypothesis testing. For the on-going debate
about P-values,
see Living
with statistics in observational research (S. Greenland and
C. Poole, Epidemiology, 2013, which also references two
previous papers in the journal), and Reconciling theory and
practice: What is to be done with P values? (David
A. Savitz, Epidemiology, 2013).
- Specifying priors (2 lectures)
Handout, covering specifying the prior PDF and
sensitivity analysis.
- The challenge of the prior; extension principles, and other rules.
- Hierarchical models, sensitivity testing.
There will be a homework every week, set on Mon and due back a week
later. Either hand in your homework after the Mon lecture, or leave it in the
box outside my office door (before 5pm). You are strongly encouraged
to do the homeworks and to hand in your efforts, to be commented on and
marked. Two of the homeworks will be assessed for credit points.
Homework 2 to be assessed for credit points.
Homework 5 to be assessed for credit points.
