start article on variable precision

[venpopov]

I've began the variable precision article. I'm opening a draft pull request so that changes are tracked over time. My plan is:

1. Brief intro to variable precision. How it was done before (special distributions), and note that it's much easier with bmm. Can be applied to any model

2. Introduce different levels of variable precision

   • same variability for all participants (complete pooling for variability parameter)
   • each participant gets their own variability, no pooling at all
   • hierarchical pooling in which the variability itself is drawn from a distribution over participants

3. Dig down into hierarchical pooling and how it can be implemented within the formula

4. Illustrate an example fit with hierarchical pooling

5. Miscellaneous

   • explain that in the original model variability followed a gamma distribution, whereas with this approach it follows a log-normal distribution. Explain that this depends on the link function

[GidonFrischkorn]

Generally, I like the structure.

I wondered if we should frame this more general, as in a general way to implement trial-to-trial variation in model parameters. This could also be useful in SDT models that assume variable memory strength over trials, and evidence accumulation models. Granted these are not yet implemented in bmm, so we could also think about generalizing the article once these models have been added to the package.

[GidonFrischkorn]

One thing, we would then need to also explain is that the distribution of trial-to-trial variability on the native scale depends on the link function, as it will always be estimated as a normal-distribution on the parameter space. We shortly discussed this for the variable precision model and how assuming a gaussian on the parameter space with a log-link function results in a log-normal that is reasonably similar to the gamma originally assumed by van den Berg.

[venpopov]

Generally, I like the structure.

I wondered if we should frame this more general, as in a general way to implement trial-to-trial variation in model parameters. This could also be useful in SDT models that assume variable memory strength over trials, and evidence accumulation models. Granted these are not yet implemented in bmm, so we could also think about generalizing the article once these models have been added to the package.

I was wondering about the same. One idea I had is to split this into two shorter articles. One is about the general trick of including trial-by-trial variability where the variability itself has a random effect over subjects. Then the variable precision article can just link to that for the final step and show how to do it without reexplaining the logic and details

[venpopov]

One thing, we would then need to also explain is that the distribution of trial-to-trial variability on the native scale depends on the link function, as it will always be estimated as a normal-distribution on the parameter space. We shortly discussed this for the variable precision model and how assuming a gaussian on the parameter space with a log-link function results in a log-normal that is reasonably similar to the gamma originally assumed by van den Berg.

Great point, I forgot about that. Will add it to the structure