Sunday, February 25, 2018

Make model diagrams for human comprehension and ease of programming

There's a great new book by Farrell and Lewandowsky, Computational Modeling of Cognition and Behavior (at the publisher, at Amazon.com), that includes some chapters on Bayesian methods. Each chapter includes a little "in vivo" commentary by an outside contributor. My commentary accompanies their chapter regarding JAGS. The commentary is posted here in a succession of three blog posts; this is 3 of 3. (Part 1 is here, and part 2 is here.) Do check out their book!

Make model diagrams for human comprehension and ease of programming

While a JAGS model specification captures the full structure of the model, it can help human beings to have a diagrammatic representation of the model. A diagram can help the viewer achieve a comprehensive overview of the relations between parameters and their meanings with respect to each other and to the data. A good conceptual diagram of a model can also guide writing the JAGS model specification.

For example, Figure 8.13 (below) shows a representation of the normal model used in the previous section. Because of graphical conventions for probability distributions, the data must be shown at the bottom of the diagram. Starting with yi, the diagram shows that the data come from a normal distribution that has parameters μ and σ. Then the top of the diagram illustrates the prior distributions on the parameters.

Figure 8.13. Diagram of the normal model, in the style of the book, Doing Bayesian Data Analysis (Kruschke, 2015). Scan the diagram from the bottom up, that is, beginning with the data yi at the bottom. Notice that every arrow has a corresponding line of code in the JAGS model specification.
The type of diagram in Figure 8.13 has several helpful attributes. It spatially organizes related parameters in the same distribution. For example, we can see that parameters μ and σ are both participating in the same distribution, and the icon also suggests the μ is for the central tendency and σ is for the scale (standard deviation). Moreover, the diagram completely captures all the structure of the model, showing the form of the prior distribution along with the likelihood function. Indeed, every arrow in the diagram has a corresponding line of code in the JAGS model specification, as shown in the previous post's Listing 8.11 and repeated here for convenience:
model {
  for ( i in 1:N ) { y[i] ~ dnorm( mu , 1/sigma^2 ) }
  mu ~ dunif( -100 , 100 )
  sigma ~ dunif( 0 , 100 )
}
   (Listing 8.11. Describe data with a normal distribution in JAGS.)
Often when I’m creating a new model, I first sketch out a diagram in the style of Figure 8.13, and after I’m sure I have a coherent structure, then I type the model into JAGS, scanning the diagram from the bottom up.

There is another convention that is sometimes used to illustrate Bayesian models. This convention has historical roots in general treatments of statistical models that specify probabilistic dependencies between parameters such that no dependencies cycle back on themselves. Such structures are called directed acyclic graphs (DAG’s). In particular, the DAG diagrammatic convention was used by the software DoodleBUGS, which was a component of WinBUGS (Spiegelhalter, Thomas, Best, & Lunn, 2003), the precursor to JAGS.

Figure 8.14. Diagram of the normal model, in the style of conventional graphical models. Shaded node indicates observed (not estimated) values. Plate indicates repetition. Notice that the arrows have no relation to lines of code in the JAGS model specification.
Figure 8.14 shows a DAG diagram for the normal model. The arrows between variables indicate that the data, yi, are dependent on parameter μ and on parameter σ. But the diagram does not indicate whether or not the two parameters participate in the same distribution or come from different distributions. The diagram does not show the prior distributions at all. Importantly, the diagram provides no clue how to express the model in JAGS because there is no relation between the arrows in the diagram and the lines of code in JAGS. Often when DAGs are used for illustration, the diagram will be accompanied by a list of all the equations that specify the model. While the equations provide complete information, the reader must scan back and forth between equations and diagram to make sense of the diagram.

For more disucssion, see p. 197 of Kruschke (2015). It’s repeatedly emphasized for many different models in that book that every arrow in a model diagram (usually) has a corresponding line of code in JAGS. See another comparison of diagrams at this blog post. See tools for creating diagrams at this blog post

(Ironically, the next chapter of Farrell and Lewandowsky's book violates this advice. You can judge for yourself whether you think the DAGs have any useful correspondence to the JAGS code. Despite the use of DAGs, their book is a great resource!)

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