# Which variables show up in sampling chain and which don't?

**URL:** <https://discourse.pymc.io/t/which-variables-show-up-in-sampling-chain-and-which-dont/1569>\
**Category:** Questions\
**Created:** [July 24, 2018, 4:21pm UTC](https://discourse.pymc.io/t/which-variables-show-up-in-sampling-chain-and-which-dont/1569 "2018-07-24T16:21:42Z")\
**Posts on this page:** 6\
**Page:** 1

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**Author:** ![phildias](https://avatars.discourse-cdn.com/v4/letter/p/a6a055/32.png) [@phildias](https://discourse.pymc.io/u/phildias)\
**Post date:** [July 24, 2018, 4:21pm UTC](https://discourse.pymc.io/t/which-variables-show-up-in-sampling-chain-and-which-dont/1569/1 "2018-07-24T16:21:42Z")

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Hi there,

I have a section of my code where I estimate a covariance matrix using the LKJCholeskyCov. However, this is making me really confused as to what randomly distributed variables in my model get a proper “name” and get explicitly sampled in my posterior chain.

Let me show what I mean. Here is an example of a section of my code:

```
# Priors for error covariance matrix
sd_dist = pm.HalfNormal.dist(sd=1, shape=num_dep_vars)
packed_chol = pm.LKJCholeskyCov('chol_cov', eta=2, n=num_dep_vars, sd_dist=sd_dist)
chol = pm.expand_packed_triangular(num_dep_vars, packed_chol, lower=True)

# Generate covariance matrix from Cholesky factor
cov_mtx = tt.dot(chol, chol.T)
cov_terms = pm.Deterministic('cov_terms',cov_mtx[cov_term_indices])

# Generate the standard deviations from the cholesky factor
sd = pm.Deterministic('sd',tt.sqrt(tt.diag(cov_mtx)))

# Tensor manipulation used to extract the correlation terms
cor_mtx = tt.diag(sd**-1).dot(cov_mtx).dot(tt.diag(sd**-1))
cor_terms = pm.Deterministic('cor_terms',cor_mtx[cor_term_indices])

```

So I first create a half-normally distributed set of standard deviations, build up a vector of cholesky factors, build up a covariance matrix. Then, i start “going backwards”, extracting the variances, standard deviations and finally, the actual correlations.

But why do I have to do this to get to the standard deviations? Aren’t I already calculating them at the top of the code, when I say `sd_dist = pm.HalfNormal.dist(sd=1, shape=num_dep_vars)`? Why isn’t sd\_dist being “watched” as a variable to be sampled from in the posterior, as is the `packed_chol` variable? Is it because `sd_dist` isn’t named? Also, why can I **not** give `sd_dist` a name?

I’m just getting a bit confused here with what gets a proper name and what gets sampled from in my posterior chains.

Thanks!

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**Author:** ![junpenglao](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/junpenglao/32/8_2.png) [@junpenglao](https://discourse.pymc.io/u/junpenglao)\
**Post date:** [July 24, 2018, 4:42pm UTC](https://discourse.pymc.io/t/which-variables-show-up-in-sampling-chain-and-which-dont/1569/2 "2018-07-24T16:42:12Z")

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`LKJCholeskyCov` is defined based on the LKJ Correlation matrix, which does not contain information of the variance (diagonal being 1). Here we give it a distribution so that the diagonal elements could be evaluated on.  
I _think_ you can do below to keep track the sd, but not sure - please verify yourself.

```python
sd_dist = pm.HalfNormal('sd', sd=1, shape=num_dep_vars)
packed_chol = pm.LKJCholeskyCov('chol_cov', eta=2, n=num_dep_vars, sd_dist=sd.distribution)

```

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<div class="post-metadata">

**Author:** ![phildias](https://avatars.discourse-cdn.com/v4/letter/p/a6a055/32.png) [@phildias](https://discourse.pymc.io/u/phildias)\
**Post date:** [July 24, 2018, 5:59pm UTC](https://discourse.pymc.io/t/which-variables-show-up-in-sampling-chain-and-which-dont/1569/3 "2018-07-24T17:59:05Z")

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> [@junpenglao](#):
>
> sd\_dist = pm.HalfNormal(‘sd’, sd=1, shape=num\_dep\_vars)  
> packed\_chol = pm.LKJCholeskyCov(‘chol\_cov’, eta=2, n=num\_dep\_vars, sd\_dist=sd.distribution)

In that second line, when you have `sd_dist=sd.distribution`, I think you meant `sd_dist=sd_dist.distribution` =)

That worked! But I’ve noticed something odd. Here are the traces for `sd_dist`, `sd` and `chol_cov`:

 ![image](https://canada1.discourse-cdn.com/flex036/uploads/pymc3/original/1X/90376ca6bfc61636468f43a1bdd46ac44968af5e.png)

My question is: why does `sd_dist` look like a distribution while `sd` and `chol_cov` look like actual parameters? I’ve defined all three of these things so similarly in my model. Why are they behaving so differently?

Thanks!

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**Author:** ![junpenglao](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/junpenglao/32/8_2.png) [@junpenglao](https://discourse.pymc.io/u/junpenglao)\
**Post date:** [July 24, 2018, 6:48pm UTC](https://discourse.pymc.io/t/which-variables-show-up-in-sampling-chain-and-which-dont/1569/4 "2018-07-24T18:48:07Z")

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Hmm I was wrong,`sd_dist` is just a free node on the graph and the sample you see is from the prior. LKJCholeskyCov is evaluating the logp using the log-likelihood but the RV `sd_dist` is not actually connect to any other variables in the model.

In that case, there probably no easier way to log the `sd_dist` into the trace than what you are already doing with the deterministic.

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<div class="post-metadata">

**Author:** ![phildias](https://avatars.discourse-cdn.com/v4/letter/p/a6a055/32.png) [@phildias](https://discourse.pymc.io/u/phildias)\
**Post date:** [July 24, 2018, 7:15pm UTC](https://discourse.pymc.io/t/which-variables-show-up-in-sampling-chain-and-which-dont/1569/5 "2018-07-24T19:15:30Z")

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Thanks for the clarification, @junpenglao! But my question still remains: if I defined `sd_dist` just like the other variables, why is PYMC drawing from it’s prior instead of its posterior?

I think i’m missing something super basic and silly here.

Thanks again!

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<div class="post-metadata">

**Author:** ![junpenglao](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/junpenglao/32/8_2.png) [@junpenglao](https://discourse.pymc.io/u/junpenglao)\
**Post date:** [July 24, 2018, 7:22pm UTC](https://discourse.pymc.io/t/which-variables-show-up-in-sampling-chain-and-which-dont/1569/6 "2018-07-24T19:22:52Z")

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Because the `sd_dist` is never actually used in the model, you can think of it as a potential function, which is evaluated and added to the model logp, but will not leave samples in the MCMC chain (in another word, it is not a free parameter that other parameters depends on)