st5219: Bayesian hierarchical modelling lecture 2.1

 Priors: how to choose them, different types  The normal distribution in Bayesianism  Tutorial 1: over to you  Computing posteriors:  Monte Carlo  Importance Sampling  Markov chain Monte Carlo

FREQUENTISMBAYESIANISM  Something with a long run frequency distribution  E.g. coin tosses  Patients in a clinical trial  “Measurement” errors?  Everything  What you don’t know is random  Unobserved data, parameters, unknown states, hypotheses  Observed data still arise from probability model Knock on effects on how to estimate things and assess hypotheses

CHOOSING A PRIORDOING COMPUTATIONS  Very misunderstood  “How did you choose your priors?”  Please never answer “Oh, I just made them up”  For data analysis, you need strong rationale for choice of prior  (later)

 Following infection: body creates antibodies  These target pathogen and remain in the blood  Antibodies can provide data on historic disease exposure

Cook, Chen, Lim (2010) Emerg Inf Dis DOI: /EID

Singapore study longitudinal Chen et al (2010) J Am Med Assoc 303:

 Observation in ( x ij,2 x ij ) for individual i, observation j  Define “seroconversion” to be a “four-fold” rise in antibody levels, i.e.  y i = 1 if x i2 ≥ 4 x i1 and 0 otherwise  Out of 727 participants with follow up, we have 98 seroconversions Q: what proportion were infected?

 Seroconversion “test” not perfect: something about 80%  Infection rate should be higher than seroconversion rate Board work

 Need some priors  Last time: “U(0,1) good way to represent lack of knowledge of a probability”  Before we collected the JAMA data, we didn’t know what p would be, and a prior p~U(0,1) makes sense  But there are data out there on σ !

Zambon et al (2001) Arch Intern Med 161:

 m = 791  y = 629 This can give you a prior!!! σ~Be(630,163) Board work

NON-INFORMATIVEINFORMATIVE  p ~U(0,1)  σ ²~U(0,∞)  μ ~U(- ∞, ∞)  β ~N(0,1000²)  Should give you no information about that parameter except what is in the data  σ ~Be(630,163)  μ ~N(15.2,6.8²)  Lets you supplement natural information content of the data when not enough information on that aspect  Can give information on other parameters indirectly

Scenario 1. You are trying to reach an optimal decision in the presence of uncertainty: use whatever information you can, even if subjective, via informative priors Scenario 2. You are trying to estimate parameters for a scientific data analysis (you cannot or don’t want to use external data): use non-informative prior Scenario 3. You are trying to estimate parameters for a scientific data analysis (you have good external data): use non-informative priors for those bits you have no data for or in which you want your own data to speak for themselves; use informative priors elsewhere

Step 1: uniform prior for σ Step 2: fit model to Zambon data Step 3: posterior for that becomes prior for main analysis Board work

 The beta distribution is conjugate to a binomial model, in that if you start with a beta prior and use it in a binomial model for p and x, you end with a beta posterior of known form  I.e. if p~Be(a,b) and x~Bin(n,p), p|x~Be(a+x,b+n-x)  Other conjugate priors exist for simple models, e.g.... Board work

 It’s the incremental nature of accumulated knowledge  Eg Zambon study: StagePrior Data ( y, m )= Posterior 0Be(1,1)(0,0)Be(1,1) 1 (1,1)Be(2,1) 2 (1,2)Be(2,2) 3 (1,3)Be(2,3) 4 (2,4)Be(3,4)

 You can think of the parameters of the beta(a,b) as representing  a best guess of the proportion, a/(a+b)  a “sample size” that the prior is equivalent to (a+b)  This is an easy way to transform published results into beta priors: take the point estimate (MLE, say) and the sample size and transform to get a and b.  (So a uniform prior is like adding one positive and one negative value to your data set: is this fair???)

 Take a point estimate and CI and convert to 2 parameters to represent your prior.  Eg the infectious period is a popular parameter in infectious disease epidemiology: the average time from infection to recovery  For no good reason, often assumed to be exponential with mean λ, say  Fraser et al (2009) Science 324: suggest estimate of generation period of 1.91 with 95%CI (1.3,2.71) Board work

 I mentioned U(-∞, ∞) as a non-informative prior. What’s the density function for U(- ∞, ∞)? Board work

 A prior such as U(-∞, ∞) is called an improper prior as it does not have a proper density function.  Improper priors sometimes give proper posteriors: depending on the integral of the likelihood.  Not an improper prior is a proper one

 Just because a prior is flat in one representation does not mean it is flat in another  Eg for an exponential model (for survival analysis say) Board work