# Non centered parametrization of lognormal Distribution
**URL:** https://discourse.mc-stan.org/t/non-centered-parametrization-of-lognormal-distribution/20117
**Category:** Modeling
**Tags:** specification
**Created:** [January 8, 2021, 11:03am UTC](https://discourse.mc-stan.org/t/non-centered-parametrization-of-lognormal-distribution/20117 "2021-01-08T11:03:34Z")
**Posts on this page:** 5
**Page:** 1
### Author:  [@ants007](https://discourse.mc-stan.org/u/ants007)
#### Post date: [January 8, 2021, 11:03am UTC](https://discourse.mc-stan.org/t/non-centered-parametrization-of-lognormal-distribution/20117/1 "2021-01-08T11:03:34Z")
Hello, All,
Happy New Year to everyone.
I have a question about how to write the Non-centered parametrization of lognormal distribution in STAN.
Suppose I have a variable, u ~ lognormal(mu, sigma\_u), where mu and sigma\_u both \> 0 -----\> (1)
I also understand that this can also be written as exp(u) ~ normal (mu\_normal, sigma\_u\_normal) —\> (2)
So my question is how can write Eq. (1) and (2) as non-centered parameterization in STAN?
For Eq(1), I am not sure, but can I write is -\> mu + u\_raw\*sigma\_u? with both mu and u\_raw having lognormal distribution priors? I am not sure about it.
And for Eq(2). the non-centered version may be written as exp(mu\_normal) + exp(u\_raw)\*sigma\_u\_normal ? with normal priors for mu\_normal and u\_raw?
Any guidance will be highly appreciated.
cheers
Antony
---
### Author:  [@mike-lawrence](https://discourse.mc-stan.org/u/mike-lawrence)
#### Post date: [January 8, 2021, 1:59pm UTC](https://discourse.mc-stan.org/t/non-centered-parametrization-of-lognormal-distribution/20117/2 "2021-01-08T13:59:57Z")
In Stan (note it’s a name, not an acronym, so the last letters don’t need to be capitalized, and often folks don’t bother even with the first letter), you can achieve selection of centered/non-centered parameterization of a lognormal prior via:
```stan
data {
// K: number of identifiable-units-of-observation (IUOO)
int K ;
// N: number of observations total (must be at least one per IUOO)
int N;
// which_K: index associating each observation with its IUOO
int which_K[N] ;
// Y: vector of observations
vector[N] Y ;
// centered: binary toggle for intercept centered/non-centered parameterization
int centered ;
}
parameters {
// mu: mean (across-IUOOs) of X
real mu ;
// sigma: sd (across-IUOOs) of X
real sigma ;
// X: mean (across observations) for each IUOO
vector<
offset = (centered ? 0 : mu)
, multiplier = (centered ? 1 : sigma)
>[K] X ;
}
model {
//hyper-priors
mu ~ std_normal() ; //must be changed to reflect domain expertise
sigma ~ std_normal() ; //must be changed to reflect domain expertise
//hierarchical *log-normal* prior for X:
X ~ lognormal( mu, sigma ) ;
//likelihood:
Y ~ normal( X[which_K], 1 ) ;
}
```
That is, you don’t really have to do anything special relative to the [normal case](https://discourse.mc-stan.org/t/fyi-how-to-easily-toggle-between-centered-non-centered/19904/) other than express X as `lognormal` rather than `normal`
---
### Author:  [@ants007](https://discourse.mc-stan.org/u/ants007)
#### Post date: [January 10, 2021, 11:09pm UTC](https://discourse.mc-stan.org/t/non-centered-parametrization-of-lognormal-distribution/20117/3 "2021-01-10T23:09:22Z")
Thanks Mike for both the correction with name and suggestion on non-centered parameterization.
Cheers
Antony
---
### Author:  [@yaniva](https://discourse.mc-stan.org/u/yaniva)
#### Post date: [March 13, 2021, 8:51pm UTC](https://discourse.mc-stan.org/t/non-centered-parametrization-of-lognormal-distribution/20117/4 "2021-03-13T20:51:38Z")
Thanks for this thread and reply - most useful.
Could you actually explain the behavior of using ``?
In the case of a lognormally distributed variable, the offset and multiplier would be applied on the log scale, I assume?
It’s a bit confusing, since you can’t see the `exp` operation.
Also - loosing `` on your lognormally distributed parameter in order to use offset and multiplier won’t case the sampler to suggest negative values for the parameter?
---
### Author:  [@mike-lawrence](https://discourse.mc-stan.org/u/mike-lawrence)
#### Post date: [March 13, 2021, 10:53pm UTC](https://discourse.mc-stan.org/t/non-centered-parametrization-of-lognormal-distribution/20117/5 "2021-03-13T22:53:07Z")
> [@yaniva](#):
>
> In the case of a lognormally distributed variable, the offset and multiplier would be applied on the log scale, I assume?
Correct.
> [@yaniva](#):
>
> It’s a bit confusing, since you can’t see the `exp` operation.
Ah, you should think of the `y ~ lognormal()` expression as being expanded behind the scenes to `log(y) ~ normal()` and not any kind of expression involving `exp` like `y ~ exp(normal())`.
This is important (and this is a point that it took me forever to realize/understand) because the `y ~ distribution(...)` is actually not expressing a structural relationship in the sense that you can exponentiate the right hand side, but instead a shorthand for `target += distribution_lupdf( y | ...)`, which in turn is expressing that we want to compute the log-probability-density (log of the height of the curve) of at each value in y given the distribution and it’s parameters. Then whatever value that yields (if y is a vector, then the individual log-pdfs are computed then summed) is added to a special variable called `target` that reflects, once all the prior log-pdfs and likelihood log-pdfs are added to it, the posterior probability of the model for a given configuration of parameter values and the data (almost; to get that you have to use the `_lpdf` functions rather than the `_lupdf`, but these will differ by a constant for a given dataset, so since we’re usually only interested in the relative posterior probability across different parameter values for a given dataset, that constant can be left off for faster compute and no harm). Stan’s HMC-based sampler will then attempt to explore different configurations of parameter values, obtaining for each the sum-log-posterior-probability value, and do fancy beyond-me stuff to explore parameter values in a way that the history of visited values itself eventually serves as samples from the full posterior probability distribution.
Most users (myself included!) see the `~` syntax and assume it’s expressing a sampling event of some sort, and for simpler models this isn’t such a big deal, but with more complicated stuff involving things like transformations, mixtures, etc, it can be important to have a good grasp on the fact that it’s a shorthand for `target +=` .
I really should redo my [“Bayes by hand” lecture](https://www.youtube.com/watch?v=kbJqD2PP4eU&list=PLu77iLvsj_GNmWqDdX-26kZ56dCZqkzBO&index=9) to illustrate these points (ex. using Stan-like `target += ` syntax in R to do the computations for the single-parameter case after showing it by hand).