Contents
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 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.
How to do Bayesian inference in school J?
One focuses on the observations in school j, {Y1j, Y2j, ⋯, Ynjj}, choose a prior distribution π(μj, σj) for the mean and the standard deviation parameters, follow the Bayesian inference procedure in Chapter 9 and obtain posterior inference on μj and σj.
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 calculate the posterior distribution using bayes’rule?
We now turn our attention back to our main goal, which is to find out, using Bayes’ rule, the posterior distribution of \\ ( heta\\) given our data: \\ (p ( heta|n,k)\\). In order to use Bayes’ rule to calculate this posterior distribution, we need to define a prior distribution over the parameter \\ ( heta\\).
Which is true of the hierarchical mixture model?
We begin in the upper level of the hierarchical model, which reflects the true expression level of genes in the transcriptome; this is a mixture distribution composed of inactive (blue) and active (red) genes ( Fig. 1 A ).
How is the prior belief in similarity of the means constructed?
The prior belief in similarity of the means is constructed in two stages. [Stage 1] The prior distribution for the j -th mean, μj is Normal, where the mean and standard deviation parameters are shared among all μj ’s: μj ∣ μ, τ ∼ Normal(μ, τ), j = 1,…, 5. [Stage 2] In the Stage 1 specification, the parameters μ and τ are unknown.