# What exactly is meant with "data parameters" in Bayesian hierarchical models?

Please excuse me if the question in unclear, but I am seeing much nomenclature for the first time and am somewhat confused.

I am reading "Statistics for Spatio-temporal data" by Noel Cressie and Christopher K. Wikle, and I'm having a hard time understanding what exactly is intended as "data parameters" in this book.

In the text, if $$A$$ is a random quantity, then $$[A]$$ is its probability distribution. On page 21, one reads:

Consider three generic quantities of interest, $$Z$$, $$Y$$, and $$\theta$$, in the HM [Hierarchical Model]; for expository purposes we often consider these simply to be random variables. Think of $$Z$$ as the data, $$Y$$ as a (hidden) process that we wish to predict, and $$\theta$$ as unknown parameters. [...] Based on $$Z$$, we wish to make inference on $$Y$$ and $$\theta$$.

This seems logic. The next section, $$2.1.1$$, starts with

The basic representation of a BHM [Bayesian Hierarchical Model] is obtained by splitting up the model into three levels (Berliner, 1996):

Data model: $$[Z\mid Y,\theta]$$

Process model: $$[Y\mid \theta]$$

Parameter model: $$[\theta]$$

The part that is confusing to me comes next.

[...] sometimes we write $$[Z\mid Y, \theta_D]$$ and $$[Y \mid \theta_P]$$ to emphasize the data-model parameters $$\theta_D$$ and the process-model parameters $$\theta_P$$. Then $$\theta = \{\theta_D, \theta_P\}$$ and the parameter model is $$[\theta_D, \theta_P]$$.

This is confusing because I would have expected $$Y$$ to represent exactly the "data parameters", i.e. that random variable that governs the generation of $$Z$$.

What would be an example of BHM (possibly in a spatial setting) where the "data-model parameters" are different from the process itself? Could these for example be nuisance parameters? But then do we really "wish to make inference" on them?

• Perhaps you have a random process Y that is observed with additional normal noise (to get Z). In that case, wouldn't the variance of that noise distribution be an example of a "data-model parameter"? Dec 19, 2019 at 16:30
• @Jason eh, I could see that. I would have said that that additional normal noise is simply part of the process $Y$. That is, one simply states $Y = (Y_1, Y_2)$, where $Y_2$ is normal noise and $Y_1$ is the "main" process. Then one simply has process parameters. Dec 20, 2019 at 12:55

Imagine you have 20 time series of abundance of some species of fish coming from 20 different sites and you are interested in uncovering their shared common trends.

In this example, Z would denote the collection of 20 time series - the observed data. Y would denote the shared common trends - these trends are not directly observable (they are considered hidden), but they can be estimated from the observed data. Each observed time series of abundance is a 'concoction' of these long-term trends plus some noise - for some time series of abundance, one trend is dominant, for other time series of abundance, two or more trends can be at play simultaneously, etc.

You can formulate a dynamic factor analysis model for estimating the underlying common trends and determining how they combine to produce each time series of abundance (see https://nwfsc-timeseries.github.io/atsa-labs/sec-dfa.html). This model will include:

1. An "observation model" (think of it as a "data model") which will describe the connection between the observed time series of abundance and the unobservable underlying common trends;

2. A "process model" which will detail how each common trend evolves over time and whether or not these trends are correlated with each other over time.

The two models are qualitatively different - in particular, in the "observation model", the response variables are observed; in the "process model", the response variables are unobserved (and unobservable).

Most importantly, the two models include different sets of unknown parameters which must be estimated from the observed time series abundance data. The parameters for the "observation model" are the so-called "data model parameters", while the parameters for the "process model" are the so-called "process model parameters".

This example illustrates that the process is something that happens behind the scenes and cannot be directly observed. However, whatever happens behind the scenes conspires to affect what data we observe.