Contents
What is exchangeability in Bayesian?
Exchangeability is meant to capture symmetry in a problem, symmetry in a sense that does not require independence. Formally, a sequence is exchangeable if its joint probability distribution is a symmetric function of its n arguments.
What is positivity in causal inference?
In simple English, positivity means that if you want to compare two types of exposures, then you have to have people in your data who are able to & sometimes will receive any or all of the relevant exposure values! This is a fundamental requirement for causal inference, but it’s also just plain common sense.
What are the assumptions in Bayesian hierarchical modeling?
Chapters 7, 8, and 9 make an underlying assumption about the source of data: observations are assumed to be identically and independently distributed (i.i.d.) following a single distribution with one or more unknown parameters.
How are the prior distributions independent in a Bayesian model?
If one assumes that the prior distributions on the individual parameters for the schools are independent, one is essentially fitting five separate Bayesian models and one’s inferences about one particular school will be independent of the inferences on the remaining schools.
How to use Gibbs sampling in Bayesian hierarchical modeling?
If ones ignores the grouping variable, then the inference procedure described in Chapter 9 can be used. One constructs a prior for the parameters μ and σ and use Gibbs sampling to obtain a simulated sample from the posterior distribution of (μ, σ). Using this approach, one is effectively ignoring any differences between the five schools.
How to do Bayesian hierarchical modeling for SAT scores?
In our school example, one ignores the school variable and simply assumes that the SAT scores Y ′ is are distributed from a single Normal population with mean μ and standard deviation σ. Here, i = 1, ⋯, n where n is the total number of students from all five schools.