Bayesian statistics for biologists and ecologists
- 1. Bayesian for us Masahiro Ryo masahiroryo@gmail.com Matthias Rillig Lab. Freie Universität Berlin Berlin-Brandenburg Institute of Advanced Biodiversity Research 2016.07.27
- 2. 1. Bayes' theorem 2. What is same? 3. What is “practically” different? 4. p-value and our intuition 5. Assumptions 6. By the way… 7. prior distribution 8. Summary 2 Outline Practical part Technical part
- 3. 3 1. Bayes' theorem P(A|B) = P(B|A) P(A) P(B) P(A): the probability of observing event A P(A|B): a conditional probability, the probability of observing event A given that B is true
- 4. 4 1. Bayes' theorem P(A|B) = P(B|A) P(A) P(B) P(A): the probability of observing event A P(A|B): a conditional probability, the probability of observing event A given that B is true
- 5. 5 2. What is same? 1. To assume a hypothesis • M under the treatment is larger than M under control (comparison) • Y is increased by increasing X (correlation) 2. To collect samples 3. To test the hypothesis population samples Statistical test
- 6. 6 3. What is “practically” different? - test Frequentists Bayesians “if p < 0.05 or not” “how probable” 1. Set a null hypothesis H0 2. Estimate probability of type I error (p-value) 3. Evaluate how rarely H0 occurs 4. Reject H0 and support H1 1. Set a hypothesis H1 2. Evaluate how possibly H1 occurs Bayesian is more direct and intuitive to test a hypothesis e.g., my hypothesis was supported with 80% probability.
- 7. 7 3. What is “practically” different? Frequentists Bayesians “how probable”“if p < 0.05 or not” y = β0 + β1 x Parameter β1 Probability 0.89 H0: β1 = 0 is rejected H1: β1 ≠ 0 is supported (and β1= 0.89) The parameter β1 is rarely <0, so H1 is greatly supported. Most probably β1=0.89 and can span between 0.28 and 1.52 with 95% probability.
- 8. 8 4. p-value and our intuition Importantly, p-value does not indicate how reliable your hypothesis is but how safely reject the corresponding null hypothesis. This does not fit our intuition. In most cases, we want to know “how reliable my hypothesis is?”
- 9. Frequentists 9 5. Assumptions Bayesians population Your samples Assuming repetitive sampling from the population to find the single true value for a parameter governing the population population Your samples Assuming the probability distribution of a parameter value, and its reliability is increased by increasing the sample size.
- 10. 0.3 1.3 β=0.8 β What you estimated from your samples Frequentists 10 5. Assumptions Bayesians 95% Confidence interval (CI) assumes that if you conduct same sampling design 20 times and estimate a 95%CI for each, only once a 95%CI would not cross the true value β. β 95% Credible interval (CI) assumes that the parameter β governing the population would be within the range of the 95% CI with 95% probability. β 0.3 1.3 Any value you estimated from your samples can be the parameter (does not matter if it is true or not)
- 11. 11 5. Assumptions Using same data, fitting a same probability distribution, estimating the parameters based on likelihood…. Then, while the concepts are different, do we eventually get identical results from frequentism and Bayesianism?
- 12. 12 5. Assumptions Answer: Yes, if we consider only in our scientific field, they are “most likely“ same. Recommendation: If your interest is the effect size, use Bayesian If it is to support hypothesis, use Frequentist
- 13. 13 6. By the way… The answer becomes NO when “prior distribution“ is explicitly assumed P(β|D) = P(D|β) P(β) P(D) Likelihood posterior distribution fixed value (practically neglectable) prior distribution
- 14. 14 6. By the way… Note that this is much less important for us... P(β|D) ∝ P(D|β) P(β) posterior distribution prior distributionLikelihood
- 15. 15 What I have explained was the following case: P(β|D) ∝ P(D|β) P(β) posterior distribution prior distributionLikelihood β 0.3 1.3 This was the outcome (estimated parameter value & range) β 0.3 1.3 This was estimated β −∞ +∞ This was like nothing 6. By the way…
- 16. 16 7. prior distribution P(β|D) ∝ P(D|β) P(β) prior distribution β −∞ +∞ What‘s this?
- 17. 17 7. prior distribution Prior distribution • This is the other key point of Bayesian statistics • But interestingly we don‘t care this aspect for our analysis (and if you google Bayesian, most articles explain this firstly: Suck!) • We can include a subjective assumption of the probability distribution of parameters (i.e., prior distribution) before analysis (but don‘t have to and indeed we don‘t)
- 18. 18 7. prior distribution An example • Question: to quantify mean and variability of our height — Probability districution of height: Gaussian — Shall we assume the parameters beforehand? μ= XX? ±σ=XX? This is called “prior distribution“ μ 170 180 175 ±σ 3 8 5 — We measure individuals and calculate the likelihoods — Multiply the posterior distribution by the likelihoods to obtain the posterior distribution
- 19. 19 7. prior distribution An example • In case of average μ P(β|D) ∝ P(D|β) P(β) μ 170 180 175 prior dist. (your assumption) μ 160 178 171 Likelihood (measurement) Posterior dist. (output: updated distribution after measurement) μ 174 166 179
- 20. 20 7. prior distribution 170 180 175 P(β|D) ∝ P(D|β) P(β) 160 178 171174 166 179 In our study μ 160 178 171 μ 171 160 178 β −∞ +∞ To avoid subjectivity
- 21. 21 8. Summary Recommendation: If your interest is the effect size, use Bayesian If it is to support hypothesis, use Frequentist β 0.3 1.3 Any value you estimated from your samples can be the parameter, following a probability distribution P(β|D) ∝ P(D|β) P(β) Prior information can be important in other field but this is not important for us: so, let‘s ignore.