Tag Archives: integrated systems modeling

MCMC in Python: Statistical model stuck on a stochastic system dynamics model in PyMC

My recent tutorial on how to stick a statistical model on a systems dynamics model in PyMC generated a good amount of reader interest, as well as an interesting comment from Anand Patil, who writes:

Something that might interest you is that, in continuous-time stochastic differential equation models, handling the unobserved sample path between observations is really tricky. Roberts and Stramer’s On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm explains why. This difficulty can apply to discrete-time models with loads of missing data as well. Alexandros Beskos has produced several really cool solutions.

This body of research is quite far from the vocabulary I am familiar with, so I’m not sure how serious a problem this could be for me. It did get me interested in sticking my statistical model to a systems model with stochastic dynamics, though, something which took only a few additional lines… thanks PyMC!

## Stochastic SI model

from pymc import *
from numpy import *

#observed data
T = 10
susceptible_data = array([999,997,996,994,993,992,990,989,986,984])
infected_data = array([1,2,5,6,7,18,19,21,23,25])

# stochastic priors
beta = Uniform('beta', 0., 1., value=.05)
gamma = Uniform('gamma', 0., 1., value=.001)
tau = Normal('tau', mu=.01, tau=100., value=.01)

# stochastic compartmental model
S = {}
I = {}

## uninformative initial conditions
S[0] = Uninformative('S_0', value=999.)
I[0] = Uninformative('I_0', value=1.)

## stochastic difference equations for later times
for i in range(1,T):
    def E_S_i(S=S[i-1], I=I[i-1], beta=beta):
        return S - beta * S * I / (S + I)
    S[i] = Normal('S_%d'%i, mu=E_S_i, tau=tau, value=E_S_i.value)

    def E_I_i(S=S[i-1], I=I[i-1], beta=beta, gamma=gamma):
        return I + beta * S * I / (S + I) - gamma * I
    I[i] = Normal('I_%d'%i, mu=E_I_i, tau=tau, value=E_I_i.value)

# data likelihood
A = Poisson('A', mu=[S[i] for i in range(T)], value=susceptible_data, observed=True)
B = Poisson('B', mu=[I[i] for i in range(T)], value=infected_data, observed=True)

This ends up taking a total of 6 lines more than the deterministic version, and all the substantial changes are from lines 24-34. So, question one is for Anand, do I have to worry about unobserved sample paths here? If I’ve understood Roberts and Stramer’s introduction, I should be ok. Question two returns to a blog topic from one year ago, that I’ve continued to try to educate myself about: how do I decide if and when this more complicated model should be used?

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Filed under global health, MCMC, statistics

MCMC in Python: How to stick a statistical model on a system dynamics model in PyMC

A recent question on the PyMC mailing list inspired me.  How can you estimate transition parameters in a compartmental model?  I did a lit search for just this when I started up my generic disease modeling project two years ago.  Much information, I did not find.  I turned up one paper which said basically that using a Bayesian approach was a great idea and someone should try it (and I can’t even find that now!).

Part of the problem was language.  I’ve since learned that micro-simulators call it “calibration” when you estimate parameter values, and there is a whole community of researchers working on “black-box modeling plug-and-play inference” that do something similar as well.  These magic phrases are incantations to the search engines that help find some relevant prior work.

But I started blazing my own path before I learned any of the right words; with PyMC, it is relatively simple.  Consider the classic SIR model from mathematical epidemology.  It’s a great place to start, and it’s what Jason Andrews started with on the PyMC list.  I’ll show you how to formulate it for Bayesian parameter estimation in PyMC, and how to make sure your MCMC has run for long enough. Continue reading


Filed under global health, MCMC, statistics