Archive for stationarity
Venice time series workshop
Posted in pictures, Statistics, Travel, University life with tags asymptotics, Ca' Foscari University, distribution-free inference, econometrics, ERC, instrumental variables, Italia, stationarity, time series, unit roots, Università Ca' Foscari Venezia, Venice, workshop on May 25, 2026 by xi'an
the problem with non-stationary processes [xkcd]
Posted in Statistics with tags Agent Orange, Donald Trump, political polls, predictions, Stand Up To Trump, stationarity, US elections 2024, US politics, USA, xkcd on November 7, 2024 by xi'an
bias reduction in self-normalised importance sampling
Posted in Books, Statistics with tags bias, Gibbs sampling, importance sampling, Markov chains, sampling resampling, self-normalised importance sampling, stationarity on December 22, 2023 by xi'anGabriel Cardoso and coauthors (among whom Éric Moulines and Achille Thin, with whom I collaborated on the inFINE/NEO algorithm), have arXived a nice entry on a cheap way to reduce bias in the famously biased self-normalised importance sampling estimator. Which is a standard solution when the target density is not normalised. They reconsider a 2004 technical paper by Tjemeland—I remember reading at the time—, which constructs a sampling resampling algorithm by creating a Markov chain and choosing between the current value and a pool of M proposed values (from the importance function), according to the importance weight, which, thanks to Tjemeland’s reformulation with two copies of the current state, constitutes a Gibbs sampler with the correct target. As in Tjemeland (2004), they propose to recycle all proposed values into the integral estimate, which then turned being unbiased under stationarity, rather unexpectedly. The paper then proceeds to analyse convergence towards this expectation, linearly in the size of the pool and exponentially in the number of Markov iterations.
exact yet private MCMC
Posted in Statistics with tags Arrowleaf Cellars, differential privacy, ergodicity, ICML 2023, Lake Okanagan, MCMC, Metropolis-Hastings algorithm, Okanagan vineyards, Poisson subsampling, reversibility, spectral gap, stationarity on August 9, 2023 by xi'an
“at each iteration, DP-fast MH first samples a minibatch size and checks if it uses a minibatch of data or full-batch data. Then it checks whether to require additional Gaussian noise. If so, it will instantiate the Gaussian mechanism which adds Gaussian noise to the energy difference function. Finally, it chooses accept or reject θ′ based on the noisy acceptance probability.”
Private, Fast, and Accurate Metropolis-Hastings for Large-Scale Bayesian Inference is an(other) ICML²³ paper, written by Wanrong Zhang and Ruqi Zhang. Who are running MCMC under DP constraints. For one thing, they compute the MH acceptance probability with a minibatch, which is Poisson sampled (in order to guarantee privacy). It appears as a highly calibrated algorithm (see, e.g., Algorithm 1). Under the assumption (1) that the difference between individual log densities for two values of the parameter is upper bounded (in the data), differential privacy is established as failing to detect for certain a datapoint from the MCMC output. Interestingly, the usual randomisation leading to pricacy is operated on the energy level, rather than on observations or summary statistics, although this may prove superfluous when there is enough randomness provided by the MH step itself: “inherent privacy guarantees in the MH algorithm”
“when either the privacy hyperparameter ϵ or δ becomes small, the convergence rate becomes small, characterizing how much the privacy constraint slows down the convergence speed of the Markov chain”
The major results of the paper are privacy guarantees (at each iteration) and preservation of the proper target distribution, in contrast with earlier versions. In particular, adding the Gaussian noise to the energy does not impact reversibility. (Even though I am not 100% sure I buy the entire argument about reversibility (in Appendix C) as it sounds too easy!) The authors even achieve a bound on the relative spectral gaps.
informed proposals for local MCMC in discrete spaces
Posted in Books, Kids, Statistics, University life with tags acceptance probability, Barker's algorithm, discrete distribution, informed proposals, JASA, local MCMC, Markov kernel, Peskun ordering, Rao-Blackwellisation, stationarity, vanilla Rao-Blackwellisation on April 17, 2020 by xi'an
Last year Giacomo Zanella published a paper entitled informed proposals for local MCMC in discrete spaces in JASA. Which I had missed somehow and only discovered through another paper, and which we recently discussed at Paris-Dauphine with graduate students, marooned by COVID-19 . Probability targets in discrete spaces are intrinsically hard[er] to simulate in my opinion if only because there is no natural distance, hence no natural neighbourhood. A random walk proposal like the reference kernel in the paper is not directly calibrated. Without demarginalisation there is neither a clear version of calculus for implementing MALA or HMC. What indeed is HMC on a discrete space? If this requires “embedding the binary space in a continuous space”, it does not sound very enticing if the construct is context dependent.
“This would allow for more moves to be accepted and longer moves to be performed, thus improving the algorithm’s efficiency.”
A interesting aspect of the paper is that for near atomic transition kernels K, informally for small σ’s, the proposal switch to Q finds target x normalising constant as new stationary and close to the actual target. Which incidentally reminded me of our vanilla Rao-Blackwellisation with Randal Douc. This however begets the worry that it may prove unwieldy in continuous cases, as except for Gaussian kernels, the proposal switch to Q may prove intractable and requires further MCMC steps, in a form of infinite regress. Plus a musing that, were the original kernel K to be replaced with the new Q, another informed proposal transform could be applied to Q. Further infinite regress…
“[The optimality of the Metropolis-Hastings choice of acceptance probability] does not translate to the context of balancing functions.”
The paper indeed exhibits a setting that is rehabilitating Barker’ (1965) version of the acceptance probability, but I never was very much convinced there was a significant difference in using one or the other. During our virtual (?) discussion, we also wondered at the adaptive abilities of the approach, e.g., selecting among a finite family of g’s (according to which criterion) or parameterising g towards an optimal choice of its parameter. And at the capacity for Rao-Blackwellisation since the proposal have to consider the entire set of neighbours prior to moving to a likely one.
