# Multi-modal distribution for the random effect (infinite mixture model with a Dirichlet process prior)- Multilevel model

**URL:** <https://discourse.mc-stan.org/t/multi-modal-distribution-for-the-random-effect-infinite-mixture-model-with-a-dirichlet-process-prior-multilevel-model/13518>\
**Category:** brms\
**Tags:** rstan\
**Created:** [March 4, 2020, 11:34am UTC](https://discourse.mc-stan.org/t/multi-modal-distribution-for-the-random-effect-infinite-mixture-model-with-a-dirichlet-process-prior-multilevel-model/13518 "2020-03-04T11:34:41Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![belay](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/belay/32/4288_2.png) [@belay](https://discourse.mc-stan.org/u/belay)\
**Post date:** [March 4, 2020, 11:34am UTC](https://discourse.mc-stan.org/t/multi-modal-distribution-for-the-random-effect-infinite-mixture-model-with-a-dirichlet-process-prior-multilevel-model/13518/1 "2020-03-04T11:34:41Z")

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Please also provide the following information in addition to your question:

- Operating System:
- brms Version:

I am working on a longitudinal data where I am fitting a multilevel/ Mixed-effect model expressed as follow  
y\_{it}|\alpha, \beta\_i^s, \sigma^2\_{\epsilon}, x\_{it} \sim N(y\_{it}; \alpha x\_{it}+ \beta\_i^s z\_{it}, \sigma^2\_{\epsilon})   
where \alpha - fixed effect and \beta\_i^s is the “random effect” for individual i. Often we assume \beta\_i^s to be distributed normally, i.e \beta\_i^s \sim N(0, \sigma^2\_{\beta^s}).  
Here, I want to relax the above assumption ([http://liu.diva-portal.org/smash/get/diva2:916319/FULLTEXT01.pdf](http://liu.diva-portal.org/smash/get/diva2:916319/FULLTEXT01.pdf)) by modelling \beta\_i^s as an infinite mixture of Gaussian. That is,  
 \beta\_i^s \sim \sum\_{k=1}^{\infty} \eta\_k N( \beta\_i^s;\beta\_k, Q\_k), \sum\_{k=1}^{\infty} \eta\_k =1 for some weights \eta\_k.

Is it possible to specify such kind of model in brms or rstan?

Thank you

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<div class="post-metadata">

**Author:** ![paul.buerkner](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/paul.buerkner/32/3303_2.png) [@paul.buerkner](https://discourse.mc-stan.org/u/paul.buerkner)\
**Post date:** [March 29, 2020, 10:54am UTC](https://discourse.mc-stan.org/t/multi-modal-distribution-for-the-random-effect-infinite-mixture-model-with-a-dirichlet-process-prior-multilevel-model/13518/2 "2020-03-29T10:54:33Z")

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In rstan, this is certainly possible. See the Stan manual for how to write mixture models.

In brms, this is not possible but you can perhaps build on brms’ stan code created via `make_stancode()`.