# Multivariate reparameterisation (Stan example) - Identifiability issues?

**URL:** <https://discourse.mc-stan.org/t/multivariate-reparameterisation-stan-example-identifiability-issues/24841>\
**Category:** Modeling\
**Tags:** specification\
**Created:** [October 14, 2021, 12:20pm UTC](https://discourse.mc-stan.org/t/multivariate-reparameterisation-stan-example-identifiability-issues/24841 "2021-10-14T12:20:50Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![cao](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/cao/32/13567_2.png) [@cao](https://discourse.mc-stan.org/u/cao)\
**Post date:** [October 14, 2021, 12:20pm UTC](https://discourse.mc-stan.org/t/multivariate-reparameterisation-stan-example-identifiability-issues/24841/1 "2021-10-14T12:20:50Z")

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Hi all,

In Chapter 23 - Efficiency Tuning under Stan users-guide, there is an example for multivariate reparameterisation (with codes provided below).

The covariance matrix Sigma is a data input, and from that the cholesky factor L was obtained. Suppose if Sigma was not a data input and L was an unknown parameter, wouldn’t multiplying the two unknown parameters, L with the alpha, make the model non-identifiable? If the model is still identifiable, can someone explain why it may still be identifiable? If it is non-identifiable, would introducing a new parameter so that \theta \equiv L \alpha, and omitting L and alpha as a parameter solve the identifiability issue?

* * *

```stan
data {
  int<lower=2> K;
  vector[K] mu;
  cov_matrix[K] Sigma;
  ...
}
transformed data {
  matrix[K, K] L;
  L = cholesky_decompose(Sigma);
}
parameters {
  vector[K] alpha;
  ...
}
transformed parameters {
  vector[K] beta;
  beta = mu + L * alpha;
}
model {
  alpha ~ std_normal();
  // implies: beta ~ multi_normal(mu, Sigma)
}

```

* * *

Many thanks.

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<div class="post-metadata">

**Author:** ![mike-lawrence](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/mike-lawrence/32/59_2.png) [@mike-lawrence](https://discourse.mc-stan.org/u/mike-lawrence)\
**Post date:** [October 15, 2021, 1:14pm UTC](https://discourse.mc-stan.org/t/multivariate-reparameterisation-stan-example-identifiability-issues/24841/2 "2021-10-15T13:14:39Z")

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`L` and `alpha` as parameters is shown in [SUG Section 1.13](https://mc-stan.org/docs/2_27/stan-users-guide/multivariate-hierarchical-priors-section.html).

If you have two parameters that are of type `real`, then yes, having them multiplied in the structures leading to the likelihood would induce an identifiability issue. But because `L` and `alpha` have structure in their types, the multiplication does not induce an identifiability issue.

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<div class="post-metadata">

**Author:** ![cao](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/cao/32/13567_2.png) [@cao](https://discourse.mc-stan.org/u/cao)\
**Post date:** [October 17, 2021, 7:18am UTC](https://discourse.mc-stan.org/t/multivariate-reparameterisation-stan-example-identifiability-issues/24841/3 "2021-10-17T07:18:16Z")

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sounds clear, thanks Mike!