Matt Graham

Research software developer at UCL

[Matt Graham @mattgrahamuk]

@sam_power_825 @jeffNegrea Implementation of this on a 2D 'banana' distribution and comparison against the non-Metropolised discretisation at https://t.co/jzx5Ji9guK - I've only checked empirically so may well have something wrong somewhere! (5/5)

[Matt Graham @mattgrahamuk]

@sam_power_825 @jeffNegrea To ensure Φ is invertible and to deal with the possibility of the fixed point iteration not converging, we need to use a scheme similar to that in https://t.co/qLhGngnVh6 (4/5)

[Matt Graham @mattgrahamuk]

@martopix Agreed I always find reverse-i-search a pain use! It doesn't give quite the same functionality, but I found this tip of mapping keys to history-search-forward and history-search-backward to be very helpful https://t.co/i1vZyWZGU5

[Matt Graham @mattgrahamuk]

@sam_power_825 Specifically minimising KL[Q‖P] or KL[P‖Q] helps directly in terms of 'flattening' the (log-convex) marginal density p on β as the gradient of log p(β) at β=0 and β=1 depend directly on KL[Q‖P] and KL[P‖Q] respectively. (5/5)

[Matt Graham @mattgrahamuk]

@sam_power_825 A variational approach of fitting Q in some 'simple to explore' parametric family to P by minimising a divergence therefore is natural and also provides an estimate of (a bound of) the target normalising constant. (4/5)

[Matt Graham @mattgrahamuk]

For more details see the paper at https://t.co/1WxvWd54HO and a notebook with example code at https://t.co/w8JAoFszeD. The Python package Mici https://t.co/fXrL9RnChh can be used to straightforwardly apply manifold MCMC methods to your own problems.

7/7

[Matt Graham @mattgrahamuk]

A (belated!) tweet thread summary of 'Manifold lifting: scaling MCMC to the vanishing noise regime' recent work with @kxau_ and @alek_thiery. We propose an efficient MCMC method for posteriors concentrated on low-dimensional structures due to highly informative observations.

1/7

[Matt Graham @mattgrahamuk]

By applying a constrained HMC method to sample from this lifted distribution we obtain a method for which a fixed step size ε can be used as σ → 0 leading to improved sampling efficiency over competing methods for small σ.

6/7

[Matt Graham @mattgrahamuk]

@matto31415 @sam_power_825 In a dynamic integration time (NUTS-like) HMC scheme with multinomial sampling we terminate the trajectory tree expansion but can still 'accept' a move to one of the states in the valid sub-trees of the trajectory simulated so far.

[Matt Graham @mattgrahamuk]

@matto31415 @sam_power_825 In static integration time HMC scheme with Metropolis-correction if we reject a step due a detected non-reversibility we reject the whole proposal and so discard the trajectory simulated so far.