Inferring the error variance in Metropolis-Hastings MCMC

One of the great joys of working with two talented post-docs in the research group – Mike Stout and Mudassar Iqbal – as well as a great collaboration with Theodore Kypraios, is that they are often one step ahead of me and I am playing catch-up. Recently, Theo has discussed with them how to estimate the error variance associated with the data used in Metropolis-Hastings MCMC simulations.

The starting point, usually, is that we have some data, let us say y_i for i=1, \cdots, n, and a model – usually, in our case, a dynamical system – which we are trying to fit to the data. For any given set of parameters \theta, our model will provide estimates for the data points that we will call \hat y_i. Now, assuming uniform Gaussian errors, our likelihood function L(\theta) looks like:

L(\theta) = \prod_{i=1}^n \frac{1}{\sqrt{2 \pi\sigma^2}}e^{-\frac{1}{2}(\frac{y_i - \hat y_i}{\sigma})^2}

where \sigma^2 is the error variance associated with the data. Now, when I first started using MCMC, I naively thought that we could use values for \sigma^2 provided by our experimental collaborators, and so we could use different values of \sigma^2 according to how confident our collaborators were in the measurements, equipment etc. What I found in practice was that these values rarely worked (in terms of convergence of the Markov chain) and we have had to make up error variances using trial and error.

So I was delighted when I heard that Theo had briefed both Mike and Mudassar about a method for estimating the error variance as part of the MCMC. Since I have not tried it before, I thought I would give it a go. I am posting the theory and some of my simulations, which are helpful results.

Theory

The theory behind estimating \sigma^2 is as follows. First, set

\tau = \frac{1}{\sigma^2}

We can then re-write the likelihood, now for the model parameters \theta and also the unknown value \tau, as

L(\bf{\theta}, \tau) = \frac{\tau^{(n/2)}}{\sqrt{2 \pi}^n}e^{-\frac{\tau}{2}\sum_{i=1}^n(y_i - \hat y_i)^2}

Now observe that this has the functional form of a Gamma distribution for \tau, as the p.d.f. for a Gamma distribution is given by:

f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}

So if we set a prior distribution for \tau as a Gamma distribution with parameters \alpha and \beta, then the conditional posterior distribution for \tau is given by:

p(\tau | \theta) \propto \tau^{(n/2)+ \alpha - 1}e^{-\tau(\frac{1}{2}\sum_{i=1}^n(y_i - \hat y_i)^2+\beta)}

We observe that this is itself a Gamma distribution, with parameters \alpha \prime = \alpha + n/2 and \beta \prime = \beta + \frac{1}{2} \sum_{i=1}^n (y_i - \hat y_i)^2. Thus the parameter \tau can be sampled with a Gibbs step as part of the MCMC simulation (usually using Metropolis-Hastings steps for the other parameters).

Simulations

The simulations I have run are with a toy model that I use a great deal for teaching. Consider a constitutively-expressed protein that is produced at constant rate k and degrades (or dilutes) at constant rate \gamma per protein. A differential equation for protein concentration P is given by:

\frac{dP}{dt} = k - \gamma P

This ODE has the closed form solution:

P = \frac{k}{\gamma} + (P_0 - \frac{k}{\gamma}) e^{-\gamma t}

where P_0 is the concentration of protein at t=0. For the purposes of MCMC estimation, mixing is improved by setting P_1 = \frac{k}{\gamma} so that the closed form solution is:

P = P_1 + (P_0 - P_1) e^{-\gamma t}

Some data I have used for teaching purposes comes from the paper Kim, J.M. et al. 2006. Thermal injury induces heat shock protein in the optic nerve head in vivo. Investigative ophthalmology and visual science 47: 4888-94. The data is quantitative Western blots of Hsp70 in the optic nerve of rats, as induced by laser damage. (Apologies for the unpleasantness of the experiment):

Time / hours Protein / au
3 1100
6 1400
12 1700
18 2100
24 2150

The aim is to use a Metropolis-Hastings MCMC, together with a Gibbs step for the \tau parameter, to fit the data. The issue that immediately arises is how to set the parameters \alpha and \beta. This may seem arbitrary, but it is already better than choosing a value for \sigma^2, as the Gamma distribution will exploring of that parameter. For my first simulation, I thought that \sigma = 100 would be sensible (this turned out to be a remarkably good choice, as we will see). So I set \alpha = 0.01 and \beta = 100 and lo and behold, the whole MCMC worked beautifully. (Incidentally, I used independent Gaussian proposals for the other three parameters, with standard deviations of 100 for the P_0 and P_1 and standard deviation of 0.01 for \gamma. These parameters were forced to be positive – Darren Wilkinson has an excellent post on doing that correctly. Use of log-normal proposals in this case leads to very poor mixing, with the chain taking some large excursions for the P_1 and \gamma parameters).

histograms

The median parameter values are P_0 = 786, P_1 = 2526, \gamma = 0.0686 and \tau = 0.000122. The latter corresponds to \sigma = 90.6. With these values, we can see a good fit to the data: below are plotted the data points (in red), the best fit (with median parameter values) in blue, and model predictions from a random sample of 50 parameter sets from the posterior distribution in black.

fit

Considerations

However, some questions obviously arise: how sensitive is this procedure to choices of \alpha and \beta? I will confess: I use Bayesian approaches fairly reluctantly, being more comfortable with classical frequentist statistics. What I like about Bayesian approaches are firstly the description of unknown parameters with a probability distribution, and secondly the availability of highly effective computer algorithms (i.e. MCMC). What makes me uncomfortable is the potential for introducing bias through the prior distributions. So I have carried out some investigations with different values of \alpha and \beta. In particular, I wanted to know: (i) what happens if I keep the mean (equal to \alpha / \beta) the same but vary the parameters? (ii) what happens if I vary the mean of the distribution? The table below summarizes positive results:

alpha beta P0 P1 gamma sigma
0.01 100 786 2526 0.0686 90.6
1 10000 747 2428 0.0795 98.0
0.0001 1 797 2533 0.0681 96.3
0.1 10 822 2603 0.0623 97.9
0.001 1000 760 2455 0.0758 94.8
1 1 792 2539 0.0676 64.9
0 0 805 2565 0.0653 98.3

As you can see (please ignore the last line for now), the results are robust to a very wide range of \alpha and \beta, even producing a good estimate for \sigma when that estimate is a long way from the mean of the prior distribution. But then we can make the following observation. Consider the sum of squares for a ‘best-fit’ model, for example using the parameters for the first row (this is 12748). So as long as \alpha \ll n/2 and \beta \ll 12748/2, the prior will introduce very little bias. But if you try to use values of \alpha and especially \beta very much larger than an estimated sum of squares from well-fitted model parameters, then things might go wrong. For example, when I set \alpha = 1 and \beta = 10^6 then my MCMC did not converge properly.

This leads to my final point, and the final row in the table. Would it be possible to remove prior bias altogether? If you look at the marginal posterior for \tau, we observe that if we set \alpha = \beta = 0, we obtain a Gamma distribution, whose mean is precisely the error variance, as, in this case,

\frac{\beta \prime}{\alpha \prime} = \frac{\sum_{i=1}^n(y_i - \hat y_i)^2}{n}

The algorithm should work perfectly well sampling from this Gamma distribution, and indeed it does, producing comparable results to when an informative prior is used.

Conclusions

In summary, I am happy to conclude that this method is good for estimating error variance. Clear advantages are:

  1. It is simple to implement and fairly fast to run – adding a Gibbs step is no big deal.
  2. It is clearly preferable to making up a fixed number for the error variance – which was what we were doing before.
  3. The prior parameters allow you to make use of information you might have from experimental collaborators on likely errors in the data.
  4. The level of bias from the priors is relatively low, and can be eliminated altogether.
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Matthias speaking at Computational Biology and Innovation PhD Symposium, Dublin

Today sees the start of the Computational Biology and Innovation PhD Symposium at University College, Dublin. Matthias Gerstgrasser will be giving a presentation in tomorrow’s (Wednesday’s) session.

Title and abstract are:

Parallelising Sequential Metropolis-Hastings: Implementing MCMC in multi-core and GPGPU environments.

Markov Chain Monte Carlo (MCMC) techniques have become popular in recent years to efficiently calculate complex posterior distributions in Bayesian statistics. In computational biology, these methods have a wide range of applications, and in particular lend themselves to parameter estimation in models of complex biological systems. The Metropolis-Hastings algorithm is one widely used routine in this context. (1)

Our research focuses on employing the computational power provided by multi-core CPUs and general-purpose graphics processing units (GPGPUs) to provide a speedup to the operation of this algorithm. Both multi-core and GPGPU architectures offer vast computing power compared to traditional single-core environments, but tapping into these resources presents additional complexities. Yet current computer systems rely increasingly on increasing core count rather than performance per core to provide improvements in computing power, a trend that is almost certain to continue in the future. While (2) provides a GPGPU algorithm applicable to Independent Metropolis-Hastings (IMH), a parallel implementation of general  MH instances has proven difficult due to the inherently sequential nature of this algorithm. In our own research, we are investigating possible speedups in automated model fitting and parameter estimation in large phenotype arrays of brewer’s yeast and other microorganisms. Our findings, however, would be equally applicable to other problems in systems biology.

We show how for some types of target distributions we can leverage independence in the structure of these distributions in order to partially parallelise the running of the MH algorithm. We furthermore discuss how this approach can be implemented efficiently on both multi-core CPUs as well as in GPGPU environments. In both cases we divide the workload of computing the acceptance probability in the MH algorithm’s main loop among several threads. Furthermore, we replicate the remaining instructions of the loop among these threads as well in order to minimise overhead incurred by thread creation, synchronisation and deletion. More importantly, in GPGPU environments this modification greatly decreases data transfers between GPU and main memory. Both our implementations show a significant speedup over a single-threaded classical MH algorithm for computationally expensive target distributions. We discuss limitations of these implementations and necessary conditions for them to provide a measurable speedup over single-threaded implementations. 

In conclusion we compare the performance of parallelising a single instance of the MH algorithm compared to running several instances in parallel on either a multi-core CPU or in a GPGPU environment. The latter approach is particularly applicable to the common situation of estimating e.g. parameters from a number of distinct, but similar, experiments. We show how GPGPU computing can be used in these situations to provide an even greater speedup compared to single-threaded implementations. 

1. Wilkinson, D J. Stochastic Modelling for Systems Biology, 2006.
2. Jacob, P, Robert CP, Murray HS. 2011; arXiv:1010.1595v3.