Probability: Classical and Bayesian 12/14/1998 12/14/98 Page 1 - PDF document

Probability: Classical and Bayesian 12/14/1998 12/14/98 Page 1 12/14/98 Page 2 P(h|e) P(h|e) PROBABILITY: Probability P(e|h) P(e|h) CLASSICAL AND BAYESIAN Classical and Bayesian P(e|~h) P(e|~h) Colloquium Statisticians are University of Northern Iowa December 14, 1998 • united on the axioms of statistics (mathematics) MILO SCHIELD • divided on the meaning of chance Augsburg College (philosophy) www.augsburg.edu/ppages/schield schield@augsburg.edu 12/14/98 Page 3 12/14/98 Page 4 P(h|e) P(h|e) United on United on P(e|h) P(e|h) Probability Axioms Bayes Theorems P(e|~h) P(e|~h) 1. P(a) ≥ 0 for all a in domain of P Bayes version: P(h|e) = P(e|h) P(h)/P(e) 2. P(t) = 1 if t is a tautology 3. P(a ∨ b) = P(a) + P(b) LaPlace version: if a, b and a ∨ b are all in domain of P P(h|e)= P(h)/[P(h)+P(~h) LR] and if a and b are mutually exclusive LR = Likelihood Ratio = P(e|~h)/P(e|h) 4. P(h|e) = P(h & e)/P(e) P(e) = P(e|h)P(h) + P(e|~h)P(~h) 12/14/98 Page 5 12/14/98 Page 6 P(h|e) P(h|e) Probability: Probability: P(e|h) P(e|h) Classical versus Bayesian Classical versus Bayesian P(e|~h) P(e|~h) Classical probability is objective: Bayesian probability is epistemic -- based on our context of knowledge • expresses fundamental laws regarding the assignment of objective physical • expresses numeric degrees of uncertainty probabilities to events in the outcome space of stochastic experiments • measures our strength of belief • independent of our feelings • can be applied to the truth of propositions • a property of the future: not of the past 1998-Schield-UNI-Slides.pdf 1

Probability: Classical and Bayesian 12/14/1998 12/14/98 Page 7 12/14/98 Page 8 P(h|e) P(h|e) Probability: Teaching Bayesian: Yes! P(e|h) P(e|h) Classical versus Bayesian Realistic approach P(e|~h) P(e|~h) “…differences of opinion are the norm in science and an approach [Bayesian] that • Classical (Purely objective) explicitly recognizes such differences is Hypothesis testing with p-values realistic.” [ Statistics: A Bayesian Perspective by Berry] Confidence that fixed parameter is in a range “The Bayesian approach is the only one • Bayesian strength of belief capable of representing faithfully the basic No hypothesis testing; no p-values principles of scientific reasoning.” Probability fixed parameter is in fixed range [ Scientific Reasoning by Howson and Urbach] 12/14/98 Page 9 12/14/98 Page 10 P(h|e) P(h|e) Teaching Bayesian: No! “Bayesian Interpretation of P(e|h) P(e|h) “at best, premature” Classical Hypothesis Tests” P(e|~h) P(e|~h) “ Surveys of the statistical methods actually in use • Combines classical hypothesis test with suggest that Bayesian techniques are little used. Bayesian strength of belief. Bayesians have not yet agreed on standard • If prior belief about truth of null is 50%, approaches to standard problems settings. P(alternate is false|reject null) = p-value Bayesian reasoning requires a grasp of conditional probability, a concept confusing to beginners. • Objectively determines prior strength of Finally, an emphasis on Bayesian inference might well belief necessary to achieve a 95% impede the trend toward experience with real data…” probability that the alternate is true. David Moore, 1997 Milo Schield, 1995 ASA JSM 12/14/98 Page 11 12/14/98 Page 12 P(h|e) P(h|e) “Bayesian Interpretation P(e|h) P(e|h) Conclusion of Classical Confidence” P(e|~h) P(e|~h) • Students take statistics to help them Interprets classical confidence as a make better decisions. Bayesian strength of belief. • Decision making is Bayesian -- based One should be indifferent in betting on on a strength of belief. • whether next ball is red (given 95% chance) • whether a particular 95% confidence • Elementary statistics should include a interval contains the population parameter Bayesian interpretation of classical Milo Schield, 1996 ASA JSM statistical inference. 1998-Schield-UNI-Slides.pdf 2

Probability: Classical and Bayesian 12/14/1998 12/14/98 Page 13 12/14/98 Page 14 P(h|e) P(h|e) “Statistical Literacy and “Statistical Literacy P(e|h) P(e|h) Evidential Statistics” and Simpson’s Paradox” P(e|~h) P(e|~h) • Focus on observational studies • Simpson’s Paradox: a reversal of an association due to a confounding • Focus on confounding factors factor. • Emphasize conditional probability • Objectively determines the minimum • Clearly identify role of chance: effect size for a reversal in the three • Highly unlikely if due to chance” variable case. • highly unlikely to be due to chance” Milo Schield, 1999 ASA JSM Milo Schield, 1998 ASA JSM 12/14/98 Page 15 12/14/98 Page 16 P(h|e) P(h|e) Elementary Statistics: Elementary Statistics: P(e|h) P(e|h) Technical versus Basic Technical versus Basic P(e|~h) P(e|~h) Elementary Statistics should be split: • Technical Statistics : • Technical statistics for majors that use Statistical inference: sampling distributions, hypothesis tests (psychology, sociology, confidence intervals and hypothesis tests education, etc.) • Basic statistics for majors that don’t • Basic Statistics : (humanities) and students that don’t (two- Reading tables, reading and interpreting year schools) graphs, and evaluating the results of observational studies. 12/14/98 Page 17 12/14/98 Page 18 P(h|e) P(h|e) Elementary Statistics: Elementary Statistics: P(e|h) P(e|h) Benefits of Changes (To be continued) P(e|~h) P(e|~h) • Goal is statistical literacy: critical Need more research on thinking about statistics • assessment of statistical literacy • student comprehension/retention • Opportunity to Improve: • selection of topics Statistical education • development of teaching materials Reputation of statistics • value added for other majors • Attract national attention • difficulty of training faculty Demonstrate leadership 1998-Schield-UNI-Slides.pdf 3

Probability: Classical and Bayesian 12/14/1998 12/14/98 Page 19 12/14/98 Page 20 P(h|e) P(h|e) Statistics Faculty Math program enrollments P(e|h) P(e|h) Bayesian: US and UK Two-year colleges P(e|~h) P(e|~h) • US & Canada: Enrollment in elementary statistics 0 - 10% Pure Bayesian** 10 - 30% Mixed Bayesian** • 11,000 in 1970 • UK, Australia, & New Zealand: • 20,000 in 1980 -- 6.0% growth/year 20 - 40% Pure Bayesian** • 47,000 in 1990 -- 8.5% growth/year 40 - 60% Mixed Bayesian** • 69,000 in 1995 -- 7.7% growth/year ** Estimated 12/14/98 Page 21 12/14/98 Page 22 P(h|e) P(h|e) Math program enrollments Math program enrollments: P(e|h) P(e|h) Four-year colleges Four-year colleges P(e|~h) P(e|~h) Enrollment in elementary statistics** Enrollment: 1995 versus 1990 • 117,000 in 1990 • 25% increase in elementary stats • 164,000 in 1995: 6.8% growth/year ** taught just in math programs • 10% decrease in math courses • 20% decrease in upper-level math 77% of all enrollment in elementary • 26% decrease in upper-level stats statistics is at the 4-year level 12/14/98 Page 23 P(h|e) Math program enrollments: P(e|h) Statistics P(e|~h) Why are more students taking stats? • Desire: Students have a greater interest in understanding mathematical concepts such as variable, function, slope and correlation. • Necessity: More students are required to take statistics for their major or graduation. 1998-Schield-UNI-Slides.pdf 4

12/14/98 Page 1 P(h|e) PROBABILITY: P(e|h) CLASSICAL AND BAYESIAN P(e|~h) Colloquium University of Northern Iowa December 14, 1998 MILO SCHIELD Augsburg College www.augsburg.edu/ppages/schield schield@augsburg.edu

12/14/98 Page 2 P(h|e) Probability P(e|h) Classical and Bayesian P(e|~h) Statisticians are • united on the axioms of statistics (mathematics) • divided on the meaning of chance (philosophy)

12/14/98 Page 3 P(h|e) United on P(e|h) Probability Axioms P(e|~h) 1. P(a) ≥ 0 for all a in domain of P 2. P(t) = 1 if t is a tautology 3. P(a ∨ b) = P(a) + P(b) if a, b and a ∨ b are all in domain of P and if a and b are mutually exclusive 4. P(h|e) = P(h & e)/P(e)

12/14/98 Page 4 P(h|e) United on P(e|h) Bayes Theorems P(e|~h) Bayes version: P(h|e) = P(e|h) P(h)/P(e) LaPlace version: P(h|e)= P(h)/[P(h)+P(~h) LR] LR = Likelihood Ratio = P(e|~h)/P(e|h) P(e) = P(e|h)P(h) + P(e|~h)P(~h)

12/14/98 Page 5 P(h|e) Probability: P(e|h) Classical versus Bayesian P(e|~h) Classical probability is objective: • expresses fundamental laws regarding the assignment of objective physical probabilities to events in the outcome space of stochastic experiments • independent of our feelings • a property of the future: not of the past

12/14/98 Page 6 P(h|e) Probability: P(e|h) Classical versus Bayesian P(e|~h) Bayesian probability is epistemic -- based on our context of knowledge • expresses numeric degrees of uncertainty • measures our strength of belief • can be applied to the truth of propositions

12/14/98 Page 7 P(h|e) Probability: P(e|h) Classical versus Bayesian P(e|~h) • Classical (Purely objective) Hypothesis testing with p-values Confidence that fixed parameter is in a range • Bayesian strength of belief No hypothesis testing; no p-values Probability fixed parameter is in fixed range