Matti Vihola has posted a new paper on arXiv about adaptive (random walk) Metropolis-Hastings algorithms. The update in the (lower diagonal) scale matrix is
![S_nS_n^\text{T}=S_{n-1}\left(\mathbf{I}_d-\eta_n[\alpha_n-\alpha^\star]\dfrac{U_nU_n^\text{T}}{||U_n||^2}\right)S^\text{T}_{n-1}](https://s0.wp.com/latex.php?latex=S_nS_n%5E%5Ctext%7BT%7D%3DS_%7Bn-1%7D%5Cleft%28%5Cmathbf%7BI%7D_d-%5Ceta_n%5B%5Calpha_n-%5Calpha%5E%5Cstar%5D%5Cdfrac%7BU_nU_n%5E%5Ctext%7BT%7D%7D%7B%7C%7CU_n%7C%7C%5E2%7D%5Cright%29S%5E%5Ctext%7BT%7D_%7Bn-1%7D&bg=000000&fg=B0B0B0&s=0&c=20201002 "S_nS_n^\text{T}=S_{n-1}\left(\mathbf{I}_d-\eta_n[\alpha_n-\alpha^\star]\dfrac{U_nU_n^\text{T}}{||U_n||^2}\right)S^\text{T}_{n-1}")
where
is the current acceptance probability and 
the target acceptance rate;
is the current random noise for the proposal, 
;
is a step size sequence decaying to zero.
The spirit of the adaptation is therefore a Robbins-Monro type adaptation of the covariance matrix in the random walk, with a target acceptance rate. It follows the lines Christophe Andrieu and I had drafted in our [most famous!] unpublished paper, Controlled MCMC for optimal sampling. The current paper shows that the fixed point for 
is proportional to the scale of the target if the latter is elliptically symmetric (but does not establish a sufficient condition for convergence). It concludes with a Law of Large Numbers for the empirical average of the 
under rather strong assumptions (on f, the target, and the matrices ). The simulations run on formalised examples show a clear improvement over the existing adaptive algorithms (see above) and the method is implemented within Matti Vihola’s Grapham software. I presume Matti will present this latest work during his invited talk at Adap’skiii.
Ps-Took me at least 15 minutes to spot the error in the above LaTeX formula, ending up with S^\text{T}_{n−1}: Copy-pasting from the pdf file had produced an unconventional minus sign in n−1 that was impossible to spot!
This entry was posted on November 23, 2010 at 9:09 am and is filed under R, Statistics with tags acceptance rate, Adapski, adaptive MCMC methods, controlled MCMC, Grapham, LaTeX, Metropolis-Hastings. You can follow any responses to this entry through the RSS 2.0 feed.
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April 7, 2011 at 7:12 am
[…] arXiv:0903.4061 On the stability and ergodicity of adaptive scaling Metropolis algorithms, by Matti Vihola […]