# Posterior density of nonlinear random effects

Consider the nonlinear mixed effects model for the $j$th observation $y_{ij}$, $j=1,\dots,n_i$, $i=1,\dots,N$, of individual $i$ at time $t_{ij}$ : $$y_{ij} = f(\alpha_i, t_{ij}) + g(\alpha_i, t_{ij}) \varepsilon_{ij}.$$ Here $\alpha_i$ is the random effect, $\alpha_i \sim \mathcal{N}(\mu,\Sigma)$ and $\varepsilon_{ij} \sim \mathcal{N}(0,\sigma^2)$. The parameters of this model are $\theta = (\mu, \Sigma, \sigma^2)$.

My question: how can one estimate (realy, I mean estimate) the conditional distribution $p(\alpha_i| y_i; \theta)$ with $y_i = (y_{i1}, \dots, y_{in_i})$?

Thanks

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So $\theta=\Sigma$ ? Or are there other parmaters ? And you consider a Bayesian approach ? –  Stéphane Laurent Oct 5 '12 at 23:14
Sorry about that two mistakes now corrected in my post: $\theta = (\mu, \sigma, \Sigma)$. –  donguzman Oct 5 '12 at 23:58
I would actually consider any approach, Bayesian could indeed be one of them. –  donguzman Oct 6 '12 at 0:01
For the Bayesian framework I think its easier to do if you use conditional distributions rather than equations. You've done this for the random effects, but not for the linking model. Also you've used $\sigma$ twice - probably better to write $g(\alpha_i,t_{ij})$ instead or use a different parameter for the error variance. –  probabilityislogic Oct 6 '12 at 1:08
Is $\alpha$ a vector or a scalar? –  probabilityislogic Oct 6 '12 at 1:12
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## 1 Answer

This conditional distribution does not have a closed-form expression because your function $f$ is non-linear in $\alpha_i$. Estimating the whole conditional distribution itself may seem difficult, but you may estimate $$\mathbb{E} (\alpha_ i|y_i ; \theta)\qquad (*)$$ for example. As far as I know, this may be done by considering one of these solutions:

1. Approximate your model: Take a first order taylor expansion of your model function $f$ and use linear mixed model tools to estimate the best linear unbiased predictor (see for instance pinheiro and bates, 2000, chap 7.)
2. Consider numerical MCMC integration in a Bayesian frame of work, and using the Gaussian $p(\alpha_i; \hat\theta)$ as an informative prior, with $\hat\theta$ an estimate obtained on your sample. Here, you do not have to approximate your model but you will have to deal with potential problems related to MCMC procedures (choice of an algorithm, convergence, mixing...)

These are the 2 traditional approaches to estimate (*) and are implemented in most recent the recent libraries for nonlinear mixed models. However, there may be other ways, especially if you really want to estimate the whole distribution.

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I would clarify point 1 by saying that rather than approximating the model, you are approximating the posterior distribution of the parameter. You will get a different "approximate model" for different parameters. –  probabilityislogic Oct 6 '12 at 1:52
Thank you for your replies. I want to avoid method 1 because of the first order approximation. I also want to get closer to the full distribution, I need more than just the expectation. –  donguzman Oct 6 '12 at 11:14
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