Losses and decision making

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En provenance du cours de Duke University

Bayesian Statistics

517 notes

Duke University

Bayesian Statistics

517 notes

Cours 4 sur 5 dans Specialization Statistics with R

This course describes Bayesian statistics, in which one's inferences about parameters or hypotheses are updated as evidence accumulates. You will learn to use Bayes’ rule to transform prior probabilities into posterior probabilities, and be introduced to the underlying theory and perspective of the Bayesian paradigm. The course will apply Bayesian methods to several practical problems, to show end-to-end Bayesian analyses that move from framing the question to building models to eliciting prior probabilities to implementing in R (free statistical software) the final posterior distribution. Additionally, the course will introduce credible regions, Bayesian comparisons of means and proportions, Bayesian regression and inference using multiple models, and discussion of Bayesian prediction. We assume learners in this course have background knowledge equivalent to what is covered in the earlier three courses in this specialization: "Introduction to Probability and Data," "Inferential Statistics," and "Linear Regression and Modeling."

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Decision Making

In this module, we will discuss Bayesian decision making, hypothesis testing, and Bayesian testing. By the end of this week, you will be able to make optimal decisions based on Bayesian statistics and compare multiple hypotheses using Bayes Factors.

Losses and decision making3:17

Working with loss functions6:34

Minimizing expected loss for hypothesis testing5:24

Posterior probabilities of hypotheses and Bayes factors6:18

Rencontrer les enseignants

Mine Çetinkaya-Rundel
Associate Professor of the Practice
Department of Statistical Science
David Banks
Professor of the Practice
Statistical Science
Colin Rundel
Assistant Professor of the Practice
Statistical Science
Merlise A Clyde
Professor
Department of Statistical Science

In the previous modules, we discussed posterior distribution and

credible intervals.

This lesson discussed Bayesian point estimates, losses and decision making.

To a Bayesian, the posterior distribution is the basis of any inference,

since it integrates both her prior opinions and knowledge and

the new information provided by the data.

It also contains everything she believes about the distribution of

the unknown parameter of interest.

But the posterior distribution on its own is not always sufficient.

Sometimes one wants to express one's inference is a credible interval

to indicate a range of likely values for the parameter.

That would be helpful if you wanted to say that you are 95% certain the probability

of an RU-486 pregnancy lies between some number L and some number U.

And on other occasions,

one needs to make a single number guess about the value of the parameter.

For example, you might want to declare the average payoff for

an insurance claim or tell a patient how much longer she has to live.

In such cases, the Bayesian perspective leads directly to decision theory.

And in decision theory, one seeks to minimize one's expected loss.

Loss can be tricky.

If you're declaring the average payoff for an insurance claim, and

if you are linear in how you value money, that is,

twice as much money is exactly twice as good, then one can prove that the optimal

one-number estimate is the median of the posterior distribution.

But in different situations, other measures of loss might apply.

If you are advising a patient on her life expectancy,

it is easy to imagine that large errors are far more problematic than small ones.

And perhaps the loss increases as the square of how far off your single number

estimate is from the truth.

For example, if she's told that her average life expectancy is two years, and

it is actually ten, then her estate planning will be catastrophically bad, and

she will die in poverty.

In the case when the loss is proportional to the quadratic error, one can show

that the optimal one-number estimate is the mean of the posterior distribution.

Finally, in some cases, the penalty is 0 if If you're exactly correct,

but constant if you're at all wrong.

This is the case with the old saying that close only counts with horseshoes and

hand grenades.

And it would apply if you want a prize for

correctly guessing the number of jelly beans in a jar.

Here, of course, instead of minimizing expected losses,

we want to maximize the expected gain.

If a Bayesian is in such a situation, then her best one-number estimate

is the mode of her posterior distribution, which is the most likely value.

There's a large literature on decision theory, and

it is directly link to risk analysis, which arises in many fields.

Although it is possible for frequentists to employ

a certain kind of decision theory, It is much more natural for Bayesians.

In this lesson,

we've seen how Bayesians make point estimates of unknown parameters,

we introduced the idea that one should make the choices that minimize the loss,

and found that the best estimate depends on the kind of loss function one is using.

In the next video,

we will discuss in more depth how these best estimates are determined.

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