Linking predicted value from y2 into a downstream model y3

[Borizook]

Dear all,

I’m trying to build a joint Bayesian model with the following structure:

- y₁ ~ formula₁ + spatial_effect  
- y₂ ~ formula₂ + spatial_effect  # shares the same spatial latent field as y₁  
- y₃ ~ y₂ × values + β

Where:  
- `values` is a known covariate 
- y₂ is a **latent quantity** estimated from the second model.

My goal is to model this jointly, so that the uncertainty in y₂ is properly propagated into y₃.

So my questions are:

Is there a way to implement this structure in INLA, inlabru, or inlamemi (e.g., via zero-trick or other workaround)?  

Any suggestions or references would be greatly appreciated!

Thanks,  
Jun-Sik LIM

[Finn Lindgren]

From your text, it sounds like it’s not y2 itself that you want in the predictor for y3, but rather the predictor expression from the y2 model? If that’s not what you meant, you need to give more details on the intended mathematical definition of your model.

There are ways to do this with tricks, but also with inlabru.

For the y3 model, simply include the same predictor expression from y2 into your expression for y3. I can write more after you confirm more precisely what model you intend.

Finn

On 21 Aug 2025, at 06:43, Borizook <sgj...@gmail.com> wrote:

Dear all,

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[Borizook]

  Yes exactly. You are right.

I want to use linear predictor expression from y2 model in y3 model, and multiply it with value which is a known covariate.!

How could I implement this in inlabru?

Thanks

Best

Jun-Sik LIM

[Borizook]

Yes.

More precisely, I want to use the predicted linear predictor (η) from the y2​ model, projected into a new spatial area, as an input to the y3​ model — specifically, to multiply it by a known covariate (value).  

Thanks

Best

Jun-Sik LIM

On Thursday, August 21, 2025 at 1:57:58 PM UTC+8 Finn Lindgren wrote:

[Finn Lindgren]

I'll assume a model of the form
  eta1 = beta1 * covar1 + field
  eta2 = beta2 * covar2 + field
  eta3 = eta2 * covar3 + beta4 * covar4
where beta1, beta2, beta4 and the field are latent variables and a latent field, and covar1,covar2,covar3, and covar4 are known covariates.

(It's not clear what your "beta" in your y3 model was supposed to be; I assumed another covariate; the principle is the same if it's something else)

bru(~ 0 + covar1 + covar2 + covar4 + field(geometry, model = matern_model_object),
  c(
    Model1 = bru_obs(y1 ~ covar1 + field, data = data1),
    Model2 = bru_obs(y2 ~ covar2 + field, data = data2),
    Model3 = bru_obs(y3 ~ (covar2 + field) * covar3 + covar4, data = data3)
  )
)

This assumes data3 contains the values of the covar3 covariate, and it will estimate the beta1, beta2, beta4 coefficients, and the field.

Finn

To view this discussion, visit https://groups.google.com/d/msgid/r-inla-discussion-group/3315792d-c7a0-491c-acb2-fc06339c287fn%40googlegroups.com.

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Finn Lindgren
email: finn.l...@gmail.com