This document discusses teaching Bayesian reasoning through short training sessions. It finds that representing probability information as natural frequencies aligns better with human cognitive algorithms and enables simpler Bayesian calculations. Studies show training people to use frequency representations significantly improves Bayesian inference both immediately and over time compared to traditional probability training or no training. Frequency grids and trees are effective teaching methods that produce high learning gains and long-term retention of Bayesian reasoning skills.

1. Behavioral Economics – Decision Support
Teaching Bayesian Reasoning
Birte Gröger

2. Agenda
Teaching Bayesian Method in Less Than Two Hours
1. Bayesian Method/Inference
2. Information Formats
3. Teaching Methods
4. Training Effectiveness
5. Studies and Experiments
6. Results and Conclusion

3. Bayesian Method/Inference
Bayes Rule in Theory
• Named after Thomas Bayes, published
1763
• Describing conditional probabilities (A|B)
given another event (B)
• Update beliefs in light of new evidence
• Transfer prior probability P(A) into
posterior probability

4. Bayesian Method/Inference
The Problems
• Studies show: Bayesian inference is alien to
human inference
– Neglect or overweighing of base rates
(conservatism)
– Cognitive illusions = systematic deviations
• Studies attempting to teach Bayesian
reasoning with no success

5. Information Formats
Probability vs. Natural Frequencies
• Cognitive algorithms work on information 
information needs representation format
• Mathematical probability and percentage =
recent developments
• Input format for human minds:
natural frequencies

6. Information Formats
Crucial Theoretical Results
1. Bayesian computations = simpler, when
information represented in natural
frequencies
2. Natural frequencies = corresponding to the
information format encountered throughout
most of our evolutionary development

7. Information Formats
Example Comparison – Mammography Problem
The probability that a woman
Ten of every 1,000 women who
who undergoes a
undergo a mammography have
mammography will have breast
breast cancer.
cancer is 1%.
Eight of every 10 women with
If a woman undergoing a
breast cancer who undergo a
mammography has breast
mammography will test
cancer, the probability that she
positive.
will test positive is 80%.
Ninety-nine of every 990
If a woman undergoing a
women without breast cancer
mammography does not have
who undergo a mammography
cancer, the probability that she
will test positive.
will test positive is 10%.

8. Teaching Methods
Overview
• Teaching: showing people how to construct
frequency representations
• Mechanism: tutorial, practices, feedback
Rule Training Frequency Grid Frequency Tree

9. Teaching Methods
Rule Training
• Explanation how to extract
numerical information by computer
system
• Translation of base-rate information
in components of Bayes’ formula
• Insert probabilities
• Calculation of result

10. Teaching Methods
Rule Training

11. Teaching Methods
Frequency Grid
• Representation cases by squares
• Indicate squares according to base
rates
– Shaded percentage of population
– Circled pluses (+) for hit rate on shaded
squares
– Circled pluses for false alarm rate on
non-shaded squares
• Calculate ratio: pluses in shaded
squares divided by all circled pluses

12. Teaching Methods
Frequency Grid

13. Teaching Methods
Frequency Tree
• Constructing reference class and
breaking-down into four subclasses
• System: explanation how to obtain
frequencies
• Inserting into corresponding nodes
• Calculation by dividing number of
true positive by sum of all positives

14. Teaching Methods
Frequency Tree

15. Training Effectiveness
Evaluation
• Explanation of program and instructions
• Answer format/solution as a formula
• Systematically varied order of problems
• Scoring criteria
strict liberal
• Match exact value • Match value +/- 5%
• Obscure fact that • Increased
participants created possibility including
sound but inexact non-Bayesian
response algorithms

16. Training Effectiveness
Measures
• Comparing solution rates
At baseline Immediately About a week 1 to 3 months
(w/o training after training after training after training
– Test 1) (Test 2) (Test 3) (Test 4)
• Traditional: steep decay curve
• Expectation now: decay not as quick with
frequency training

17. Studies and Experiments
Structure
Study 1a Study 1b Study 2
• 62 University of • 56 Free University of • 72 University of
Chicago students Berlin students Munich students
• 4 groups in 3 training • Prevent high attrition • Issue of used graphical
methods and one w/o rates with later aids in frequency
training as control payments and bonus conditions
• All 4 tests with 10 based on results • Longer period of time
problems each • 2 groups with the between Test 3 and 4
• Old and new problems different frequency • Use also graphical aid
• High attrition rates trainings for rule training 
(increasing # of • Reduced number of probability tree
participants) problems
• No attrition

18. Studies and Experiments
Results – Study 1a
• Substantial improvement in
Bayesian reasoning
• High level of transfers:
average performance in
new problems almost as
god as in old problems
• Increase in median number
of inferences in the
frequency grid condition

19. Studies and Experiments
Results – Studies 1b and 2
Study 1b Study 2

20. Conclusion
Teaching Bayesian Reasoning is possible
• Prove that Bayesian computations are simpler
using natural frequencies
• Environmental change  illusions
• Idea: teach people to represent information
according to cognitive algorithms
• Translation in representation format = major
tool for helping to attain insight
• High immediate effects, better transfer to
other problems and long-term stability