This document provides an introduction to Bayesian statistics. It discusses that Bayesian statistics takes a fundamentally different approach to probability than frequentist statistics by viewing parameters as random variables rather than fixed values. It also uses mathematical tools like Bayes' theorem, priors, posteriors, and Markov chain Monte Carlo simulations. The document explains Bayesian concepts and compares the Bayesian and frequentist perspectives. It argues that Bayesian methods are particularly useful for complex models with many interacting parameters.

1. An
Introduction
to Bayesian
Statistics
Paul Herendeen
April 2013

2. 0
2000
4000
6000
8000
1960 1970 1980 1990 2000 2010
WOS "Bayesian"
Citations by Year
The rise of Bayesian
statistics

3. So…What are Bayesian
Statistics?

4. So…What are Bayesian
Statistics?
1) A fundamentally different approach to
probability

5. So…What are Bayesian
Statistics?
1) A fundamentally different approach to
probability
2) An associated set of mathematical tools

6. Frequentists vs. Bayesian
Round 1
Parameters
fixed
Data
varies
Data
fixed
Parameters
Vary

7. Frequentists vs. Bayesian
Round 1

8. Frequentists vs. Bayesian
Round 1
Confidence
Interval
Credible
Interval

9. Conditional Probability
in 2 minutes

10. Conditional Probability
in 2 minutes
All possible outcomes

11. Conditional Probability
in 2 minutes

12. Conditional Probability
in 2 minutes

13. Conditional Probability
in 2 minutes

14. Conditional Probability
in 2 minutes

15. Conditional Probability
in 2 minutes

16. Bayes’ Theorem

17. Bayes’ Theorem
Prior

18. Bayes’ Theorem
Likelihood
Prior

19. Bayes’ Theorem
Likelihood
Prior
Evidence

20. Bayes’ Theorem
Likelihood
Prior
Posterior
Evidence

21. Frequentist vs. Bayesian
Round 2
“The Strength of the Prior”

22. Sparse
Data

23. Abundant
Data

24. Uniform
Prior

25. Where do Priors Come
From?

26. So…What are Bayesian
Statistics?
1) A fundamentally different approach to
probability
2) An associated set of mathematical tools

27. How do you actually
do this?

28. So how do you actually
do this?
1. Analytical methods

29. So how do you actually
do this?
1. Analytical methods
2. Grid approximation

30. So how do you actually
do this?
1. Analytical methods
2. Grid approximation
3. Markov Chain Monte Carlo

31. MCMC
• Algorithm for exploring parameter space

32. MCMC
• Algorithm for exploring parameter space
1.Pick a starting point

33. MCMC
• Algorithm for exploring parameter space
1.Pick a starting point
2.Propose a move

34. MCMC
• Algorithm for exploring parameter space
1.Pick a starting point
2.Propose a move
3.Accept or decline move based on
probability

35. MCMC
• Algorithm for exploring parameter space
1.Pick a starting point
2.Propose a move
3.Accept or decline move based on
probability
• Time spent at each point approximates
parameter distribution

36. MCMC
• Algorithm for exploring parameter space
1.Pick a starting point
2.Propose a move
3.Accept or decline move based on
probability
• Time spent at each point approximates
parameter distribution
• E.g. Metropolis-Hastings, Gibbs sampling

37. MCMC
2D example

38. MCMC
2D example

39. So what does all this
get us?

40. Bayesian methods
really shine in complex
(hierarchical) models…

41. For example,
Individual
Fecundity
Group
Effect
Population
Effect
Foraging
success
Environment

42. or…
Individual
Fecundity
Group
Effect
Population
Effect
Environment

43. Many benefits to this
approach
• Simultaneously estimate parameters
• …as well as parameter relationships
• “Borrow” strength across studies
• Model comparison

44. So, is it a Bayesian
Revolution?

45. Bayesian stats can do
most things
frequentist,

46. Bayesian stats can do
most things
frequentist, but…
• Many simple models don‟t gain much
• Better do something „boring‟ well than
something exciting poorly

47. Bayesian stats can do
most things
frequentist, but…
• Many simple models don‟t gain much
• Better do something „boring‟ well than
something exciting poorly
• Don‟t be this guy

48. DO use Bayesian
methods if
• You have a complex model with many
interacting parameters
• You have „messy‟ data
• You don‟t want to make assumptions
about distributions

49. In Conclusion
• Bayesian methods are powerful tools
for ecological research
• Like most things statistical, they are
no substitute for thinking
• They are here to stay, and you should
at least be familiar with them

50. Great, I want to learn
more!
JAGS
(Just Another Gibbs Sampler)