# 732A66 Decision theory

### Course information

#### Making decisions under uncertainty

This course is about the decision-theoretic approach to statistical inference. Expressing it simply the course is about making decisions under uncertainty. One may argue that almost all decision-making is under some kind of uncertainty, otherwise there would not be a matter of decision. For instance, if you are about to take a longer walk outside it might be that you will choose between bringing an umbrella with you or not. This is a decision problem when there are reasons for you to hesitate. If you are certain that there will be no rain during your walk you would probably not even consider taking an umbrella. Analogously you would bring the umbrella (or use rain-proof clothing) for certain if it is raining when you are about to walk.A possibly clearer example is when there is an instruction
for how to carry out something, how to proceed etc. Then you are
supposed to check a number of points and the instruction will tell
you what to do (what decision to make). Assume there is a roadwork
going on for a distance of 200 metres, and assume this implies that
there is only one lane to drive on. Hence, the traffic could only
pass in one direction at a time. In most cases you will find
temporary traffic lights at each end of the roadwork distance. As a
driver you then understand that you must wait when the read light is
on and you (must) drive when the green light is on. The decision
about in which direction the traffic should go is automatically
controlled, and as a driver you seldom consider it as a decision
problem. You simply follow the rules (the instruction).

The situation is of course
different when there are no traffic lights. This may happen if the
whole roadwork distance and a bit longer is visible from both ends of
it. If you before arriving to "your" end-point see a vehicle oncoming
on the roadwork distance you will make the (almost automatic)
decision to stop and wait. If you cannot see any vehicle oncoming
(on the whole distance visible for you) you might as easily make the
decision to drive on. The problem occurs when you see a vehicle that
still haven't reached the other end of the roadwork distance. Then
you must make a quick analysis about whether your or the other
vehicle will be the first to arrive to the respective end-point of the
roadwork distance, and upon that analysis make a decision on whether
you should drive on or wait.

One could say that decisions are made under uncertainty when the
decision-maker either has to analyse
incoming data to obtain grounds for the decision, or in absence of
data when the decision-maker has some preference order for different
decisions (actions) based on how probable each action is to be the
best action. In the latter case the probabilities guiding the
decision process are by natural reasons subjective to the
decision-maker.

When it stands clear that a decision is to be made under uncertainty, i.e. when there is no formal instruction to follow and/or no certain consequence of each possible action to take, it also becomes obvious that there must be probabilistic reasoning with or without statistical inference involved. The latter can be said to be involved as soon as there is incoming data that can be be analysed.

#### Inferential objectives

Statistical inference can be made with different objectives. In a first course in statistics the objectives are generally descriptive or explanatory. In descriptive inference results are reported without probabilistic reasoning (i.e. no confidence intervals, credible intervals, p-values etc. are reported). The purpose is to give good presentations of data that will assist end-users of the data, but leaving the drawing of conclusions to them. In explanatory inference the purpose is to find and evaluate a model for data that can be interpreted in terms of the application for which the data has been collected. Much of the introductory courses in statistics comprise methods for point estimation, interval estimation and classical hypothesis testing. Such inferential steps are made both when the data has been collected to learn about a particular quantity of interest (e.g. the average income in a specific group of people, the average time of recovery from a disease, the total volume of pulpwood in a forest area), and when the mutual relations between several variables are of primary interest (e.g. multiple linear regression, factor analysis, time series).

When the objective is neither descriptive, nor explanatory it can be predictive or decisive. Predictive inference is used when the data collected should provide knowledge about and assist in assessing values of objects not yet observed. Predictive inference is very common in forecasting, e.g. of economic time series or near-future weather, but the purpose may also be interpolation of an estimated pattern to obtain a complete picture of some phenomenon, e.g. the prevalance of a specific weed in a grassland. What is of particular interest when comparing explanatory and predictive inference is that the models estimated in each of these two stages can be very different despite the collected data being exactly the same. In introductory courses in statistics the predictive stage mostly appears when studying linear regression models. While the model is usually the same both for the explanatory inference and the predictive inference, it is possible to demonstrate why it could be wise to work with different models. In courses on time series analysis there is often a clear distinction between which models should be used for explanatory inference and which should be used for forecasting (i.e. predictive inference).

#### Decisive inference

Decisive inference is another way of going beyond the descriptive and explanatory stages. The model used in the explanatory stage is usually retained in the decision stage, but while the explanatory inference can point towards a particular set of parameter values - e.g. the outcome of a classical hypothesis test can be to reject or not reject a hypothesis specifying these values - the subsequent decisive inference can give the opposite result. The data is thus analysed with help of the explanatory model and the outcome is given in terms of probability statements, but while the explanatory stage stops at concluding which estimated model is best supported by the data the decisive stage involves consequences of using that estimated model and consequences of using other estimates. For instance, assume the explanatory inference has come to the conclusion that the amount of pulpwood in a forest area is at least 50 000 cubic metres. The uncertainty comes in terms of a p-value for classical hypothesis testing or in terms of posterior probabilities for a Bayesian analysis. If that figure (50 000) is used in a budget for cutting the area there will be receipts and costs that depend on it. If the final amount of pulpwood is shown to be lower than 50 000 cubic metres while the budget was for at least 50 000, there may be severe extra costs. If on the other hand precautions are made leading to a budget for less than 50 000 cubic metres while the amount showed to be more than 50 000, there is a loss in not having processed the surplus. Hence the probabilistic outcome of the explanatory inference must be processed further taking consequences into account.

Consequences need not however be given in terms of losses from the application. Loss functions are part of statistical inference itself and can as such define the inferential procedure, hence formulating the statistical inference as a decision problem. This is essentially what is meant by decisive inference.

#### This course

In this course we will take up the general framework for decisive
inference. Since decision theory is very closely related to Bayesian
inference we will take our standpoint from the subjective
interpretation of probability, but for sake of completeness also
discuss non-Bayesian decision theory. The decisive inferential steps
will mainly be about point estimation and hypothesis
testing/evaluation.

* Course content*

The course content comprises:

- Probability interpretation, Probabilistic reasoning and likelihood theory
- Elements of Baysian inference
- Elements of decision theory (actions, consequences, utility and loss)
- The value of information
- Decisive inference and sampling from a decision-making perspective
- Applications in forensic science

*Teaching*

The teaching comprises meetings that are mixtures of lectures, seminars and tutorials. The lecture parts are devoted to presentations of theories, concepts, and methods. At the seminars more complex decision problems will be discussed in smaller groups and class. The tutorials may include using computer software for problem solving. Homework and independent study are necessary complements to the course.

*Examination*

The course will be examined by

- Assignments encompassing both theoretical and computer-based exercises
- One final oral examination

*Course literature*

- Peterson M. (2017). An Introduction to Decision Theory. 2nd ed. Cambridge University Press. ISBN 9781316606209 (Paper back).
- Winkler R.L. (2003). An Introduction to Bayesian Inference and
Decision. 2nd ed. Probabilistic Publishing. ISBN 0964793849

Electronic version available for purchase or lending: https://archive.org/details/introductiontoba00robe/page/n8/mode/1up - Taroni F., Bozza S., Garbolino P., Biedermann A., Aitken C. (2010). Data Analysis in Forensic Science: A Bayesian Decision Perspective. Chichester: Wiley-Blackwell.
- Gittelson S. (2013). Evolving from Inferences to Decisions in the Interpretation of Scientific Evidence. Thčse de Doctorat, Série criminalistique LVI, Université de Lausanne. ISBN 2-940098-60-3. Available at http://www.unil.ch/webdav/site/esc/shared/These_Gittelson.pdf.
Berger J.O. (1980). Statistical decision theory and Bayesian analysis. 2nd ed.New York: Springer.__Advanced text:__

The text book below from previous years can also be used, but it is old, and difficult/expensive to order. There is a possibility to by an electronic version or lend it for 14 days access via the link provided.

*Supplementary literature*

For more reading on this topic the following texts are useful (but do not constitute any indispensable complement to the course literature as defined above):

*Course responsible and tutor*

Anders Nordgaard. E-mail: Anders.Nordgaard@liu.se. Phone: +46 10 562 8013.

*Course start*

Monday 5 September 2022 at 15.15 in room T15 (Building C on Campus Valla)

Page responsible: Anders Nordgaard

Last updated: 2022-11-13