Hurdle Prior or Piecewise Function?

Modeling
,
1

Hello,

I’m trying to use brms to fit a nonlinear model of the form

y \sim Lognormal(\mu , \sigma) \\ \mu = a - (b_1x)/(1 + b_2x^2)

The part of the model that I’m unsure how to specify is the idea that the parameter b_1 depends on x, such that when x is below a certain threshold value, b_1 = 0, (and thus \mu = a), and above the threshold b_1 is some positive value. Additionally, there is some grouping in my data, so both the threshold location and the value of b_1 above the threshold vary among groups (as a random effect).

It seems to me that the distribution of b_1 across all groups should follow something like a hurdle-lognormal distribution. Is there a way to specify this as a prior for a parameter or a random effect?

Alternatively, maybe it could be a piecewise function where below the threshold x = t_{[i]} the it’s just \mu = a and above it’s the full function. Can anyone provide insight on the best way to specify this model, and how it can be coded in brms?

2

If b1 is always zero below a certain threshold, then I would think a piecewise function is the way to go:

if (x < t) {
 mu = a;
} else {
 mu = a - (b1 * x) / (1 + b2 * x^2);
}

If it has some probability of being zero or not below a certain threshold, then I think you need a hurdle model.

However, as you probably noted from my use of Stan’s native code, I do not have any knowledge of how to specify a piecewise model in brms.