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Hierarchical models of variance

Marcus Brubaker writes:

I am currently working on a problem in computational biology using Bayesian inference and I’ve come to a question for which I hope you have an answer. In this problem there are a large number of noisy 2D images of a molecule, from which we wish to infer the 3D structure. Much of the modeling is straightforward but I have hit a roadblock. Specifically, the noise parameters for these images.

Each image is assumed to have IID zero-mean gaussian noise added to each pixel. The variance of this noise would be the same for each image except that this noise also needs to account for modeling errors in the 3D representation of the molecule which may vary between images. This suggests that a hierarchical model of the variance might be appropriate. However, I have been unable to find any good discussion of hierarchical models of variance. Clearly I can define something arbitrary but I would like something which makes a modicum of sense. Ideally, it would also have a directly sample-able conditional distribution of the hyperparameters given the variances.

My reply: It sounds like what you want is something like the model of Besag, York, and Mollie (1991), which has two additive variance component: a spatially correlated error plus white noise. The key contribution of this paper was that it can make sense to fit such an additive model, rather than trying to model the autocorrelation structure of the data directly. The Banerjee, Carlin, and Gelfand book on spatial statistics is probably the best place to start for this sort of model. In any case, you’d probably have to adapt this to your particular problem, but that’s fine: you could fit the model, then create random simulations from the fitted model and see how they differ from the data, in order to get insight into where to go next.


  1. Martin says:

    The book Random Heterogeneous Materials by Torquato has a chapter on "Computer Simulations, Image Analysis, and Reconstructions".

  2. Richard says:

    I'm confused as to how there's modelling error in the 3D representation. Do you mean each 2D image has Gaussian error from the imaging device plus additional error from the molecule? Or do you mean the 3D shape of the molecule changes slightly from image to image?

    Inferring a 3D image from multiple 2D images is done all the time in medical imaging (Radon transform), but your situation sounds a little more complicated. Althought patient motion is often a big issue in medical imaging, so you could look into reconstruction schemes for that problem. Cardiac CT has a lot of motion issues, for example.

  3. Jor says:

    sounds like protien NMR

  4. Marcus says:


    Thanks for the answer! Spatially correlated white noise sounds about right for this problem. I will take a closer look at the references.


    The modeling error in 3D is due to approximations in the model. I.E., we are modeling the electron density of a protein using radial basis functions. This is an approximation which isn't going to be exact for a number of reasons. Plus, there are a few other complications regarding how the images relate to protein. For instance, each image is actually from a different instance of the same protein. Each instance will have almost exactly the same 3D structure but can have differences.


    It's actually protein Cryo-EM. That said, people have done a fairly successful Bayesian treatment of protein NMR.