# Centered vs noncentered -

**URL:** <https://discourse.mc-stan.org/t/centered-vs-noncentered/1502>\
**Category:** General\
**Created:** [August 7, 2017, 11:21am UTC](https://discourse.mc-stan.org/t/centered-vs-noncentered/1502 "2017-08-07T11:21:24Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![felixmueller](https://avatars.discourse-cdn.com/v4/letter/f/258eb7/32.png) [@felixmueller](https://discourse.mc-stan.org/u/felixmueller)\
**Post date:** [August 7, 2017, 11:21am UTC](https://discourse.mc-stan.org/t/centered-vs-noncentered/1502/1 "2017-08-07T11:21:24Z")

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Heyo!

I am still learning and unsure about centered vs noncentered modelling. I have a simple linear regression model and am trying to understand the modelling with this before I move on to complex stuff.  
I had

```
model {
//// priors
beta_0 ~ cauchy(0, 5);
beta ~ cauchy(0, 5);
sigma ~ gamma(0.001, 0.001);
//// likelihood
price ~ normal(beta_0 + x * beta, sigma);
}

```

for now and was wondering if the correct way to non-center this would be:

```
transformed parameters{
vector[n_obs] price_mean;
price_mean = beta_0 + x * beta;
}

model{
//// priors
beta_0 ~ cauchy(0, 5);
beta ~ cauchy(0, 5);
sigma ~ gamma(0.001, 0.001);
//// likelihood
price ~ normal(price_mean, sigma);
}

```

I feel that I am not completely grasping something here. Would this already be “reparametrizing”? I feel that I only gave something a different name - then again, reparametrizing can be just that.

edit: Just for completeness:  
price and beta are vectors, beta\_0 is a real and x is a matrix. All have the right dimensions, both models run and produce the same results, I would say. As this is not a complex thing, I had expected that and was only looking to get the gist of it.

edit2:  
After thinking about it, I would say this is not about non-centering, as I did not split anything up in smaller parts - is that correct?

---

<div class="post-metadata">

**Author:** ![jonah](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/jonah/32/200_2.png) [@jonah](https://discourse.mc-stan.org/u/jonah)\
**Post date:** [August 7, 2017, 11:46pm UTC](https://discourse.mc-stan.org/t/centered-vs-noncentered/1502/2 "2017-08-07T23:46:36Z")

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> [@felixmueller](#):
>
> edit2:  
> After thinking about it, I would say this is not about non-centering, as I did not split anything up in smaller parts - is that correct?

Yeah what you’ve done here is _not_ a reparameterization. What you did is to store the result of `beta_0 + x * beta` in a variable called `price_mean`. So you’ve written the same model using slightly different Stan code but it’s not a reparameterization.

The non-centered parameterization that you refer to can be used when you have particular kinds of _hierarchical priors_. So in your case, imagine you had

```no-highlight
model {
...
beta ~ normal(mu_beta, sigma_beta);
}

```

where `mu_beta` and `sigma_beta` are parameters in the model. In that case you would be using a “centered” parameterization for `beta` because the prior is centered around the prior mean `mu_beta`. However, you could write a statistically equivalent model using the “non-centered” parameterization

```no-highlight
parameters {
  ... 
  real beta_raw;
}
transformed parameters {
  ...
  real beta = mu_beta + beta_raw * sigma_beta;
}
model {
  ...
  beta_raw ~ normal(0, 1); // implies beta ~ normal(mu_beta, sigma_beta)
}

```

in which case the prior for `beta_raw` is standard normal and then we compute `beta` by scaling `beta_raw` by `sigma_beta` and shifting it by `mu_beta`. That is, we take a `normal(0, 1)` and scale it by `sigma_beta` to get `normal(0, sigma_beta)` and then shift it by `mu_beta` to get `normal(mu_beta, sigma_beta)`.

I’m not saying this necessarily makes sense to do in your case, just trying to use your example.

I highly recommend @betanalpha’s case study at [Diagnosing Biased Inference with Divergences](http://mc-stan.org/users/documentation/case-studies/divergences_and_bias.html), which shows an example of centered and non-centered parameterizations and discusses when this type of reparameterization can be useful.

---

<div class="post-metadata">

**Author:** ![felixmueller](https://avatars.discourse-cdn.com/v4/letter/f/258eb7/32.png) [@felixmueller](https://discourse.mc-stan.org/u/felixmueller)\
**Post date:** [August 8, 2017, 9:21am UTC](https://discourse.mc-stan.org/t/centered-vs-noncentered/1502/3 "2017-08-08T09:21:00Z")

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Thank you! Yeah, I was unsure regarding this, but I agree that this is no reparametrization. And thank you for the example! I feel I am understanding it better and better the more I read and work with it - which is obviously not a surprise. ;)

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**Author:** ![Bob\_Carpenter](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/bob_carpenter/32/9230_2.png) [@Bob\_Carpenter](https://discourse.mc-stan.org/u/Bob_Carpenter)\
**Post date:** [August 14, 2017, 12:55pm UTC](https://discourse.mc-stan.org/t/centered-vs-noncentered/1502/4 "2017-08-14T12:55:02Z")

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This is all written up in the manual, along with a lot of other examples.

---

<div class="post-metadata">

**Author:** ![felixmueller](https://avatars.discourse-cdn.com/v4/letter/f/258eb7/32.png) [@felixmueller](https://discourse.mc-stan.org/u/felixmueller)\
**Post date:** [August 14, 2017, 1:32pm UTC](https://discourse.mc-stan.org/t/centered-vs-noncentered/1502/5 "2017-08-14T13:32:25Z")

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You’re right! I had read that, but was still unsure. (Now, a couple of days later, I believe I understand it a lot better)

I am getting more and more into Stan and also more and more in the ability to judge the scope of things I do not understand. Thanks for all the support here!