Allows a principled approach to the exploitation of all available data …

with an emphasis on continually updating one’s models as data accumulate

as seen in the consideration of what is learned from a positive mammogram

Bayesian Reasoning

- Casscells, Schoenberger & Grayboys, 1978

- Eddy, 1982

Gigerenzer & Hoffrage, 1995, 1999

Butterworth, 2001

Hoffrage, Lindsey, Hertwig & Gigerenzer, 2001

When PREVALENCE, SENSITIVITY, and FALSE POSITIVE RATES are given, most physicians misinterpret the information in a way that could be potentially disastrous for the patient.

The conditional probability slides the revised probability in its direction but doesn’t replace the prior probability

A NATURAL FREQUENCIES presentation is one in which the information about the prior probability is embedded in the conditional probabilities (the proportion of people using Bayesian reasoning rises to around half).

Test sensitivity issue (or: “if two conditional probabilities are equal, the revised probability equals the prior probability”)

Where do the priors come from?

-----> Bayes’ theorem

p(X|A)*p(A)

p(A|X) = ______________________

P(X|A)*p(A) + p(X|~A)*p(~A)

Given some phenomenon A that we want to investigate, and an observation X that is evidence about A, we can update the original probability of A, given the new evidence X.

It relates the conditional density of a parameter (posterior probability) with its unconditional density (prior, since depends on information present before the experiment).

The likelihood is the probability of the data given the parameter and represents the data now available.

Frequentist measures depend on design; require that design be followed.

Bayesian view: update continually as data accumulate (only requirement is honesty). Sample size: need not choose in advance. Weigh costs/benefits; decide whether to start experiment. After experiment starts, decide whether to continue—stop at any time, for any reason.

5. Decision making

5. Decision making

Frequentist: historically avoided.

Bayesian: tailored to decision analysis; losses and gains considered explicitly.

Statistics: A Bayesian Perspective D. Berry, 1996, Duxbury Press.

Statistics: A Bayesian Perspective D. Berry, 1996, Duxbury Press.

excellent introductory textbook, if you really want to understand what it’s all about.

http://ftp.isds.duke.edu/WorkingPapers/97-21.ps

“Using a Bayesian Approach in Medical Device Development”, also by Berry

http://www.pnl.gov/bayesian/Berry/

a powerpoint presentation by Berry

http://yudkowsky.net/bayes/bayes.html

Extremely clear presentation of the mammography example; highly polemical and fun too!

http://www.stat.ucla.edu/history/essay.pdf

Bayes’ original essay

Jaynes, E. T., 1956, `Probability Theory in Science and Engineering,' (No. 4 in `Colloquium Lectures in Pure and Applied Science,' Socony-Mobil Oil Co. USA. http://bayes.wustl.edu/etj/articles/mobil.pdf

A physicist’s take on Bayesian approaches. Proposes an interesting metric of probability using decibels (yes, the unit used for sound levels!).

http://www.sportsci.org/resource/stats/

a skeptical account of Bayesian approaches. The rest of the site is very informative and sensible about basic statistical issues.