Writing Bayesian Hierarchical Models

Models for Socio-Environmental Data

Chris Che-Castaldo, Mary B. Collins, N. Thompson Hobbs

December 18, 2020

Che-Castaldo, Collins, Hobbs DBI-1052875, DBI-1639145, DEB 1145200 1 / 4

Examples of

I Multi-level models (aka random e↵ects)

I Sampling error in x’s and y’s

I Calibration error in y’s

All of these will appear in the exercises.

Che-Castaldo, Collins, Hobbs DBI-1052875, DBI-1639145, DEB 1145200 2 / 4

Things to watch for today

Partitioning uncertainty

I Process variance

I Sampling variance

I Calibration variance (aka observation variance)

I Group level variance

Che-Castaldo, Collins, Hobbs DBI-1052875, DBI-1639145, DEB 1145200 3 / 4

Steps in writing Bayesian models

1. Write your deterministic model. Be careful about support.

2. Draw Bayesian network (DAG) describing relationships

between observed and unobserved quantities.

3. Use the Bayesian network to write proportionality between

posterior and joint distributions using bracket notation [ | ].3.1 Posterior distribution: [unobserved quantities |data]3.2 Joint distribution

3.2.1 All nodes in Bayesian network at the heads of arrows (children)must be on the left hand side of a conditioning symbol.

3.2.2 All nodes in Bayesian network at the tails of arrows (parents)must be on the right hand side of a conditioning symbol |.

3.2.3 All nodes at the end of an arrow with no arrow coming intothem must be expressed unconditionally, i.e., they must havenumeric arguments.

4. Assign specific PDF or PMF to each of the brackets.

5. Choose numeric values for parameters of prior distributions.

Do this sensibly! Do not default to vague priors. (Do as I say,

not as I do.)

Che-Castaldo, Collins, Hobbs DBI-1052875, DBI-1639145, DEB 1145200 4 / 4

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