Bayesian Statistics
Bayesian Statistics by Duke University
Herbie Lee, Professor of Statistics at the University of California Santa Cruz.
way to define probabilities is under Bayesian perspective. Bayesian perspective is one of personal perspective.
Books and Notebook
Probabilistic Programming and Bayesian Methods for Hackers
Frequentism and Bayesianism: A Practical Introduction
Computational Statistics in Python0.1
Statistical Rethinking course and book
Courses
Bayesian-biostatistics-2015 STA663-2015 Bayes
#### minimizing loss functions
summary
‣ L0 minimized at mode
‣ L1 minimized at median
‣ L2 minimized at mean
‣ point estimate you report depends on your choice of loss function
posterior probabilities & decision
‣ suppose you have two competing hypotheses: H1 and H2
‣ then
| ‣ P(H1 is true | data) = posterior probability of H1 |
| ‣ P(H2 is true | data) = posterior probability of H2 |
‣ potential decision criterion: choose the hypothesis with the higher posterior probability
| reject H1 if P(H1 is true | data) < P(H2 is true | data) |
‣ alternative: consider a loss function
loss functions & decisions
‣ L(d) : loss that occurs when decision d is made
‣ Bayesian testing procedure then minimizes the posterior expected loss
‣ possible decisions (actions):
‣ d1 : choose H1 - decide that the patient doesn’t have HIV
‣ d2 : choose H2 - decide that the patient has HIV
summary
‣ Bayesian methodologies allow for the integration of losses into the decision making framework easily
‣ in Bayesian testing we minimize the posterior expected loss
Bayes factor
‣ quantifies the evidence of data arising from H1 vs. H2
‣ discrete case: ratio of the likelihoods of the observed data under the two hypotheses
‣ continuous case: ratio of the marginal likelihoods
#### interpreting the Bayes factor BF[H1 : H2]
Jeffreys (1961) Evidence against H2
1 to 3 Not worth a bare mention
3 to 20 Positive
30 to 150 Strong
150 Very strong
summary
‣ inference with paired or matched normal samples
‣default prior distributions under the two hypotheses
‣posterior distributions for proportions under H2
and H1
pooled prior distributions under H1 p(p) / pam
‣ Bayes factor for comparing H1 to H2 is 2.93
‣ slight evidence in favor of H1 but not “worth a bare mention”