Category Archives: MCMC

MCMC in Python: A random effects logistic regression example

I have had this idea for a while, to go through the examples from the OpenBUGS webpage and port them to PyMC, so that I can be sure I’m not going much slower than I could be, and so that people can compare MCMC samplers “apples-to-apples”. But its easy to have ideas. Acting on them takes more time.

So I’m happy that I finally found a little time to sit with Kyle Foreman and get started. We ported one example over, the “seeds” random effects logistic regression. It is a nice little example, and it also gave me a chance to put something in the ipython notebook, which I continue to think is a great way to share code.

[py] [pdf]


Filed under MCMC, software engineering

PyMC+Pandas: Poisson Regression Example

When I was gushing about the python data package pandas, commenter Rafael S. Calsaverini asked about combining it with PyMC, the python MCMC package that I usually gush about. I had a few minutes free and gave it a try. And just for fun I gave it a try in the new ipython notebook. It works, but it could work even better. See attached:

[pdf] [ipynb]


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I learned last week about a Python Package for doing MCMC estimation, called PyMCMC. It sounds sort of like something I’m always writing about, doesn’t it?

From my quick look, it appears that pyMCMC has some advanced sampling methods (like Slice sampling) that are not yet implemented for PyMC. On the other hand, it seems like PyMC has a more flexible modeling language, which permits formulation of complex models without writing out likelihood functions explicitly.

Has anyone used PyMCMC? How did it go for you?

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Validating Statistical Models

I’ve been thinking a lot about validating statistical models. My disease models are complicated, there are many places to make a little mistake. And people care about the numbers, so they will care if I make mistakes. My concern is grounded in experience; when I was re-implementing my disease modeling system, I realized that I mis-parameterized a bit of the model, giving undue influence to observations with small sample size. Good thing I caught it before anything was published based on the resultsI published anything based on the results!

How do I avoid this trouble going forwards? A well-timed blog post from Statistical Modeling, Causal Inference, and Social Science highlights one way, described in a paper linked there. I like this and I partially replicated in PyMC. But I’m concerned about something, which the authors mention in their conclusion:

To help ensure that errors, when present, are apparent from the simulation results, we caution against using “nice” numbers for fixed inputs or “balanced” dimensions in these simulations. For example, consider a generic hyperprior scale parameter s. If software were incorrectly written to use s^2 instead of s, the software could still appear to work correctly if tested with the fixed value of s set to 1 (or very close to 1), but would not work correctly for other values of s.

How do I avoid nice numbers in practice? I have an idea, but I’m not sure I like it. Does anyone else have ideas?

Also, my replication only works part of the time for my simple example, I guess because one of my errors is not enough of an error:

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Causal Modeling in Python: Bayesian Networks in PyMC

While I was off being really busy, an interesting project to learn PyMC was discussed on their mailing list, beginning thusly:

I am trying to learn PyMC and I decided to start from the very simple discrete Sprinkler model.
I have found plenty of examples for continuous models, but I am not sure how should I proceed with conditional tables, especially when the condition is over more than a variable. How would you model a CPT with PyMC, and in particular the Sprinkler model?

I spend all my time on continuous models these days, and I miss my old topics from back in grad school. Not that this is one of them, exactly. But when I found myself with a long wait while running some validation code, I thought I’d give it a try. The model turned out to be simple, although using MCMC to fit it is probably not the best idea. Continue reading


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PyMC at SciPy 2011

I just returned from the SciPy 2011 conference in Austin. Definitely a different experience than a theory conference, and definitely different than the mega-conferences I’ve found myself at lately. I think I like it. My goal was to evangelize for PyMC a little bit, and I think that went successfully. I even got to meet PyMC founder Chris Fonnesbeck in person (about 30 seconds before we presented a 4 hour tutorial together).

For the tutorial, I put together a set of PyMC-by-Example slides and code to dig into that silly relationship between Human Development Index and Total Fertility Rate that foiled my best attempts at Bayesian model selection so long ago.

I’m not sure the slides stand on their own, but together with the code samples they should reproduce my portion of the talk pretty well. I even started writing it up for people who want to read it in paper form, but then I ran out of momentum. Patches welcome.

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Filed under education, global health, MCMC, statistics

My First Contribution to PyMC

I’m excited to report that my first contribution back to the PyMC codebase was accepted. 🙂

It is a slight reworking of the pymc.Matplot.plot function that make it include autocorrelation plots of the trace, as well as histograms and timeseries.  I also made the histogram look nicer (in my humble opinion).



In this example, I can tell that MCMC hasn’t converged from the trace of beta_2 without my changes, but it is dead obvious from the autocorrelation plot of beta_2 in the new version.

The process of making changes to the pymc sourcecode is something that has intimidated me for a while.  Here are the steps in my workflow, in case it helps you get started doing this, too.

# first fork a copy of pymc from on github
git clone

# then use virtualenv to install it and make sure the tests work
virtualenv env_pymc_dev
source env_pymc_dev/bin/activate

# then you can install pymc without being root
cd pymc
python install
# so you can make changes to it without breaking everything else

# to test that it is working
cd ..
>>> import pymc
>>> pymc.test()
# then make changes to pymc...
# to test changes, and make sure that all of the tests that use to pass still do repeat the process above
cd pymc
python install
cd ..
>>> import pymc
>>> pymc.test()

# once everything is perfect, push it to a public git repo and send a "pull request" to the pymc developers.

Is there an easier way?  Let me know in the comments.

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MCMC in Python: A simple random effects model and its extensions

A nice example of using PyMC for multilevel (aka “Random Effects”) modeling came through on the PyMC mailing list a couple of weeks ago, and I’ve put it into a git repo so that I can play around with it a little, and collect up the feedback that the list generates.

Here is the situation:

Hi All,

New to this group and to PyMC (and mostly new to Python). In any case, I’m writing to ask for help in specifying a multilevel model (mixed model) for what I call partially clustered designs. An example of a partially clustered design is a 2-condition randomized psychotherapy study where subjects are randomly assigned to conditions. In condition 1, subjects are treated in small groups of say 8 people a piece. In the condition 2, usually a control, subjects are only assessed on the outcome (they don’t receive an intervention). Thus you have clustering in condition 1 but not condition 2.

The model for a 2-condition study looks like (just a single time point to keep things simpler):
y_{ij} = b_0 + b_1 Tx + u_j Tx + e_{ij},
where y_ij is the outcome for the ith person in cluster j (in most multilevel modeling software and in PROC MCMC in SAS, subjects in the unclustered condition are all in clusters of just 1 person), b_0 is the overall intercept, b_1 is the treatment effect, Tx is a dummy coded variable coded as 1 for the clustered condition and 0 for the unclustered condition, u_j is the random effect for cluster j, and e_ij is the residual error. The variance among clusters is \sigma^2_u and the residual term is \sigma^2_e (ideally you would estimate a unique residual by condition).

Because u_j interacts with Tx, the random effect is only contributes to the clustered condition.

In my PyMC model, I expressed the model in matrix form – I find it easier to deal with especially for dealing with the cluster effects. Namely:
Y = XB + ZU + R,
where X is an n x 2 design matrix for the overall intercept and intervention effect, B is a 1 x 2 matrix of regression coefficients, Z is an n x c design matrix for the cluster effects (where c is equal to the number of clusters in the clustered condition), and U is a c x 1 matrix of cluster effects. The way I’ve written the model below, I have R as an n x n diagonal matrix with \sigma^2_e on the diagonal and 0’s on the off-diagonal.

All priors below are based on a model fit in PROC MCMC in SAS. I’m trying to replicate the analyses in PyMC so I don’t want to change the priors.

The data consist of 200 total subjects. 100 in the clustered condition and 100 in the unclustered. In the clustered condition there are 10 clusters of 10 people each. There is a single outcome variable.

I have 3 specific questions about the model:

  1. Given the description of the model, have I successfully specified the model? The results are quite similar to PROC MCMC, though the cluster variance (\sigma^2_u) differs more than I would expect due to Monte Carlo error. The differences make me wonder if I haven’t got it quite right in PyMC.
  2. Is there a better (or more efficient) way to set up the model? The model runs quickly but I am trying to learn Python and to get better at optimizing how to set up models (especially multilevel models).
  3. How can change my specification so that I can estimate unique residual variances for clustered and unclustered conditions? Right now I’ve estimated just a single residual variance. But I typically want separate estimates for the residual variances per intervention condition.

Here are my first responses:

1. This code worked for me without any modification. 🙂 Although when
I tried to run it twice in the same ipython session, it gave me
strange complaints. (for pymc version 2.1alpha, wall time 78s).
For the newest version in the git repo (pymc version 2.2grad,
commit ca77b7aa28c75f6d0e8172dd1f1c3f2cf7358135, wall time 75s) it
didn’t complain.

2. I find the data wrangling section of the model quite opaque. If
there is a difference between the PROC MCMC and PyMC results, this
is the first place I would look. I’ve been searching for the most
transparent ways to deal with data in Python, so I can share some
of my findings as applied to this block of code.

3. The model could probably be faster. This is the time for me to
recall the two cardinal rules of program optimization: 1) Don’t
Optimize, and 2) (for experts only) Don’t Optimize Yet.

That said, the biggest change to the time PyMC takes to run is in
the step methods. But adjusting this is a delicate operation.
More on this to follow.

4. Changing the specification is the true power of the PyMC approach,
and why this is worth the extra effort, since a random effects
model like yours is one line of STATA. So I’d like to write out
in detail how to change it. More on this to follow.

5. Indentation should be 4 spaces. Diverging from this inane detail
will make python people itchy.

Have a look in the revision history and the git repo README for more.

Good times. Here is my final note from the time I spent messing around:
Django and Rails have gotten a lot of mileage out of emphasizing
_convention_ in frequently performed tasks, and I think that PyMC
models could also benefit from this approach. I’m sure I can’t
develop our conventions myself, but I have changed all the file names
to move towards what I think we might want them to look like. Commit
linked here

My analyses often have these basic parts: data, model, fitting code,
graphics code. Maybe your do, too.

p.s. Scott and I have started making an automatic model translator, to generate models like this from the kind of concise specification that users of R and STATA are familiar with. More news on that in a future post.

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That Bayes Factor

A year or more ago, when I was trying to learn about model selection, I started writing a tutorial about doing it with PyMC. It ended up being more difficult than I expected though, and I left it for later. And later has become later and later. Yet people continue landing on this page and I’m sure that they are leaving disappointed because of its unfinishedness. Sad. But fortunately another tutorial exists, Estimating Bayes factors using PyMC by PyMC developer Chris Fonnesbeck. So try that for now, and I’ll work on mine more later (and later and later).

There is also an extensive discussion on the PyMC mailing list that you can turn to here. It produced this monte carlo calculation.

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Introducing the H-RAM

I haven’t had time to say much recently, due to some travel for work, but I did have a chance to prototype the Hit-and-Run/Adaptive Metropolis approach to MCMC that I mentioned in my last post.  Was that really three weeks ago?  How time flies.

Anyway, thanks to the tip Aram pointed out in the comments, Hit-and-Run can take steps in the correct direction without explicitly computing an approximation of the covariance matrix, just by taking a randomly weighted sum of random points.  It’s nearly magic, although Aram says that it actually makes plenty of sense. The code for this “H-RAM” looks pretty similar to my original Hit-and-Run step method, and it’s short enough that I’ll just show it to you:

class HRAM(Gibbs):
    def __init__(self, stochastic, proposal_sd=None, verbose=None):
        Metropolis.__init__(self, stochastic, proposal_sd=proposal_sd,
                            verbose=verbose, tally=False)
        self.proposal_tau = self.proposal_sd**-2.
        self.n = 0
        self.N = 11
        self.value = rnormal(self.stochastic.value, self.proposal_tau, size=tuple([self.N] + list(self.stochastic.value.shape)))

    def step(self):
        x0 = self.value[self.n]
        u = rnormal(zeros(self.N), 1.)
        dx = dot(u, self.value)

        self.stochastic.value = x0
        logp = [self.logp_plus_loglike]
        x_prime = [x0]

        for direction in [-1, 1]:
            for i in xrange(25):
                delta = direction*exp(.1*i)*dx
                    self.stochastic.value = x0 + delta
                    x_prime.append(x0 + delta)
                except ZeroProbability:
                    self.stochastic.value = x0

        i = rcategorical(exp(array(logp) - flib.logsum(logp)))
        self.value[self.n] = x_prime[i]
        self.stochastic.value = x_prime[i]

        if i == 0:
            self.rejected += 1
            self.accepted += 1

        self.n += 1
        if self.n == self.N:
            self.n = 0

Compare the results to the plain old Hit-and-Run step method:


Filed under MCMC