Hi everyone, I’m currently using NUTS (not Stan but my own implementation) to sample from a distribution which describes the probability of the parameters \theta_n for a set of sinusoidal basis functions, given some time series data d (where the number of basis functions is fixed). This corresponds…

[image] betanalpha: this is an experimental design problem more than a computational one The experimental design definitely does make this a tricky problem. However, people have been successfully applying beamforming methods to these sort of problems for a long time and I’d like to think a co…

[image] PhilClemson: since Stan uses NUTS I suspect @betanalpha will take issue with this passage, but he might be able to also provide some insight into these pathologies.

Does your HMC implementation do something like Stan’s metric adaptation, allowing the integrator to explore anisotropically along the ridges? What I’m suggesting here is that HMC on this surface might require a transient period of metric adaption to the geometry of the ridge on which the sampler fi…

[image] mike-lawrence: Are you maybe encountering the multimodality of periodic likelihoods that I describe here ? This might be part of the issue. Small peaks occur where both parameters are equal to the actual value of one of the sinusoids. There are also some small sidelobes to the main pe…

@PhilClemson : I think we need to assess the version of NUTS that is in Stan (i.e. use mass matrix and step-size adaption). Maybe the thing to do is to describe this pdf as a Stan file and see what happens with Stan. @jsocolar : are you confident that Stan’s variant of NUTS would navigate this kind o…

@s.maskell nope! But that seems like a good candidate for what’s going wrong?

@jsocolar : while Phil’s example problem is 2D and involves going up a ridge then along to a peak, in general, we’re interested in a higher-dimensional version of the problem that involves the sample having to navigate between a sequence of saddle points before it gets to the peak. As I see it, we th…

@PhilClemson the sinusoids are of substantially different frequencies, right? Can you post the model?

Sure, the actual problem I’m considering is for data measured by a spatial sensor array. The design matrix for an array of L sensors measuring M sinusoids is: G_{\theta,\omega}=\left[\begin{array}{c c c c} e^{j(\omega t+\alpha_{1}(\theta_{1},\omega))} & e^{j(\omega t+\alpha_{1}(\theta_{2},\omega)…

So the thetas defining your axes in the plots reflect the phase, right? Operationalized with a parameter in the model bound to 0-2*pi? Are you doing anything special to relate the two phases such that the theta2 seems to always be greater than theta1?

NUTS and ridges

Algorithms

mike-lawrence April 27, 2022, 8:56pm 2

Are you maybe encountering the multimodality of periodic likelihoods that I describe here?

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NUTS and ridges

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