Archive for November, 2008

In this case, sample.

November 25, 2008

In this previous post I described the debate over whether to sample or collapse missing observation variables. I have now run the experiments, the results of which I will present here as promised. These experiments are based on my infinite sparse Factor Analysis model, which you can read about here, run on the gene expression dataset you can find as part of Mike West’s BFRM package here.

mv1

Figure 1. Twenty runs of a 1000 MCMC iterations of my infinite sparse Factor Analysis model with different approaches to handling the missing values.

Figure 1 shows that the uncollapsed sampler, with no added noise (red) performs best: it achieves the lowest predictive error in the shortest time. Adding noise to the missing values (like you should for a genuine Gibbs sampling scheme) for this version (green) both decreases the performance in absolute terms, and has a surprisingly detrimental effect on the run time as well (this could just be a result of the time it takes to sample noise at each iteration). The collapsed sampler performs better in absolute terms than the collapsed sampler with noise and has better run time.

Figure 2. Boxplot of prediction error after 1000 MCMC steps for each missing value approach.

Figure 2. Boxplot of prediction error after 1000 MCMC steps for each missing value approach.

Figure 2 confirms this conclusion: sampling the missing values and not adding any noise gives the best performance.

On a related note, I’ve been looking at the effect of removing the assumption of isotropic noise. This seems to be quite a reasonable thing to do, and doesn’t make the calculations much more involved at all.

Figure 3. 20 MCMC runs of 1000 iterations with isotropic vs. non-isotropic noise model.

Figure 3. 20 MCMC runs of 1000 iterations with isotropic vs. non-isotropic noise model.

Figure 4. Boxplot of predictive error with diagonal or isotropic noise model.

Figure 4. Boxplot of predictive error with diagonal or isotropic noise model.

Figures 3-4 confirm the intuitive that including a full diagonal noise model does improve the predictive performance of the model.

A Bayesian view of the Stein estimator

November 17, 2008

I’ve been attending Statistical Theory, a course in Part III Maths at Cambridge, taught by Richard Samworth. In the latest lecture, Richard showed the Stein estimator for the mean of multivariate Gaussian has lower expected loss (measured as the Euclidean distance between the true and estimated values of the mean) than the MLE for any value of \theta. From a frequentist perspective this appears completely unintuitive, but from a Bayesian perspective it appears much more reasonable.

Assume we observe vector X drawn from a multivariate normal of dimension p, with mean \theta and identity covariance matrix. The MLE of \theta is then just X, but the Stein estimator

\theta_s=\left( 1-\frac{p-2}{||X||^2} \right) X

The fact that this estimator performs better than the ML is termed shrinkage, because the estimator is shrunk towards 0.

How would a Bayesian approach this problem? First let’s put a Gaussian prior on \theta, so

\theta \sim N_p(\theta;0,\lambda^{-1}I)

where \lambda is a precision (inverse variance). In a fully Bayesian setting we would then put a Gamma prior on \lambda, but unfortunately we would then have to resort to sampling to infer the posterior over \theta. Assuming \lambda is known, then the posterior of \theta is

P(\theta|X,\lambda) \propto P(X|\theta) P(\theta|\lambda) = N_p(\theta;(1+\lambda)^{-1}X,(1+\lambda)^{-1})

Thus the expected value of \theta is
E(\theta|X)=(1+\lambda)^{-1}X

Now let’s find the MLE of \lambda. This is not ideal, but is tractable at least. To do this we’ll first integrate out \theta:

P(X|\lambda)=\int P(X|\theta) P(\theta|\lambda) d\theta
=N_p(X;0,\lambda_x^{-1}I)
where
\lambda_x=\frac{\lambda}{1+\lambda}.

An unbiased estimate of \lambda_x is given by
\lambda_x^{ML}=\frac{|X|^2}{p-1}.

Substituting for \lambda and rearranging gives
\lambda^{ML} = \left( \frac{|X|^2}{p-1}-1 \right)^{-1}

Substituting into the expression for E(\theta|X) above and rearranging gives
E(\theta|X) = \left( 1-\frac{p-1}{|X|^2} \right) X,
which is very close to the Stein estimate. I suspect that some choice of prior on \lambda would result in a MAP estimate which would give the p-2 term.

The conclusion is that an estimator which has unintuitively desirable properties in a frequentist framework, is intuitively a sensible estimator using under a Bayesian framework.

To sample or not to sample? Missing values in Gibbs sampling

November 12, 2008

I ran into an interesting debate last week. Consider the factor analysis model:

y=Gx+\epsilon

where y is an observed vector, G is the mixing matrix, x is a vector of latent variables and the last term is isotropic Gaussian noise. Assume we observe a sequence of n y’s drawn iid from the model. Then writing Y=[y_1 \dots y_n], X=[x_1 \dots x_n] and E=[\epsilon_1 \dots \epsilon_n] we have

Y=GX+E

A very naive way to infer the posterior P(G,X|Y) would be to perform Gibbs sampling: successively sample P(G|X,Y), then P(X|G,Y). Assume conjugate Gaussian priors on G and X, this becomes particularly easy. Note that the likelihood function, assuming Gaussian noise with precision \lambda_e, is

\left(\frac{\lambda_e}{2\pi}\right)^\frac{ND}{2}\exp{\left\{-\frac{\lambda_e}{2}[-tr((Y-GX)^T(Y-GX))]\right\}}

Now assume however that some of the elements of Y are missing, at random. How should we cope with this? Two, superficially quite different approaches are possible.

In the first, the simplest way, would be to consider the unobserved elements of Y as latent variables. Then in our Gibbs sampling scheme, have initialized G and X, we simply sample P(Y^u|G,X), i.e. set the unobserved elements of Y to the corresponding elements of GX+E (sampling E as noise). Then our sampling steps for G and X are the same as before. This does not change the model structure in any way, and is a completely valid Gibbs sampling scheme.

The second approach is to exclude terms involving the missing values from the likelihood function. We can achieve this algebraically by element-wise multiplication of the (Y-GX) term by a binary matrix H. Element (i,j) of H is equal to one iff element (i,j) of Y was observed. Now the likelihood function becomes:

\left(\frac{\lambda_e}{2\pi}\right)^\frac{N_o}{2}\exp{\left\{-\frac{\lambda_e}{2}[-tr(((Y-GX)\circ H)^T((Y-GX)\circ H)]\right\}}

where N_o is the number of observed elements of Y. This approach makes sampling G and X a little more tricky because H affects the covariance structure of the conditional distributions. To deal with the algebra (specifically to get rid of the troublesome Hadamard product) I use some ideas from Tom Minka’s matrix algebra notes.

Having had two pretty smart people arguing the case for both sides, we decided to give this some more thought. Both schemes are valid Gibbs samplers for the same model. The first is considerably easier to implement, but intuitively should not perform as well. Once we have a sample for the unobserved elements of Y, when we sample G and X the algorithm has no knowledge about which elements of Y are observed, so we must be introducing more noise into the algorithm than we would leaving these terms out.

Jurgen made the very good point that leaving the terms out of the likelihood function is equivalent to integrating out the unobserved variables. This leads to the intuition that the question of which approach to take is actually equivalent to the much broader question of how much to collapse a Gibbs sampler. This question has garnered a great deal of interest of late, and seems to be very much model dependent. If you have two highly correlated variables (such as G and X above) then integrating out one would seem very beneficial, since exploring the joint parameter space will be very slow otherwise, whereas Jurgen and Finale have found cases (specifically the linear Gaussian IBP model) where the uncollapsed Gibbs sampler is much faster.

I’ll report back again when I have some hard numbers to support these ideas!

Hi!

November 12, 2008

On this blog I’ll be putting up ideas I have about machine learning in general, maybe with a slight twist towards bioinformatics.