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differential privacy for Bayesian inference

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on July 30, 2024 by xi'an

As I was reading it in preparation for my JSM²⁴ lecture, I found anew that, in this landmark paper of Dimitrikakis et al. (2017), some limitations of the concept of differential privacy were most apparent:

– a requirement to bend both the model and the prior to fit differential privacy, like switching to Lipschitz constraints or using new (e.g., truncated) priors, which runs contrary to Bayesian principles, although the former can be seen as a form of randomization akin to ABC when the randomization itself is accounted for in the derivation of the “exact” posterior distribution (as in the paper of Berah, Favaro, and Rao (2023) on running MCMC for Bayesian non-parametric estimation on privatized (noisy) data I discussed a few days ago);

– a subtle switch of the randomness from the (privatization) procedure itself (as in Dwork (2006)) to the uncertainty about the parameter, not that it clashes per se with Bayesian principles (even though there is an unclear randomness statement in Theorem 9, when the prior itself seems to become random (?)). Which actually means that producing one realisation from the posterior is the (privatization) procedure, as I realised when discussing with Shenggang Hu in Warwick;

– a linear degradation of the privacy parameter ε when moving from one realisation of the posterior to a simulated sample, assuming iid realisations (I wonder whether or not releasing a dependent sample could involve the ESS instead of the number of MCMC iterations). The upper bound means that no privacy whatsoever is guaranteed for an infinite posterior sample, hence for delivering de facto the posterior (despite the paper producing an (ε,0) bound on the Kullback-Leibler measure of the difference between posteriors);

– an absence of prior knowledge or modelling on the data itself, unless the distance from the actual data to an hypothetical alternative, ρ(x,y), can be interpreted as minus a score function conditional on the actual data, eg the opposite of the log predictive

  • the occurrence of an “exponential prior” that is exp p/m ell (theta) with ell a Lipschitz constant for the associated likelihood

  • the assumption that the data user is not an adversary of the data keeper, in that their utility function is about the parameter (and publicly available)

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approximate computation for exact statistical inference from differentially private data

Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , on September 10, 2023 by xi'an

“the employment of ABC for differentially private data serendipitously eradicates the “approximate” nature of the resulting posterior  samples, which otherwise would be the case if the data were noise-free.”

In parallel or conjunction with the 23w5601 workshop, I was reading some privacy literature and came across this Exact inference with approximate computation for differentially private data via perturbation by Ruobin Gong  that appeared in the Journal of Privacy and Confidentiality last year (2022). When differential privacy is implemented by perturbation, i.e. by replacing the private data with a randomised, usually Gaussian, version, the exact posterior distribution is a convolution which, if unavailable, can be approximated by a standard ABC step. Which, most interestingly does not impact the accuracy of the (public) posterior, i.e. it does not modify this posterior when the probability of acceptance in ABC is the density of the perturbation noise at the public data given the pseudo-data. Which follows from the 1984 introduction of the ABC idea. On the opposite, EM does not enjoy an exact version, as the E step must be (unbiasedly) approximated by a Monte Carlo representation that relies on the same ABC algorithm. Which usually makes the M step harder, although a Monte Carlo version of the gradient is also available.

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O’Bayes 19/1 [snapshots]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , , on June 30, 2019 by xi'an

Although the tutorials of O’Bayes 2019 of yesterday were poorly attended, albeit them being great entries into objective Bayesian model choice, recent advances in MCMC methodology, and the multiple layers of BART, for which I have to blame myself for sticking the beginning of O’Bayes too closely to the end of BNP as only the most dedicated could achieve the commuting from Oxford to Coventry to reach Warwick in time, the first day of talks were well attended, despite weekend commitments, conference fatigue, and perfect summer weather! Here are some snapshots from my bench (and apologies for not covering better the more theoretical talks I had trouble to follow, due to an early and intense morning swimming lesson! Like Steve Walker’s utility based derivation of priors that generalise maximum entropy priors. But being entirely independent from the model does not sound to me like such a desirable feature… And Natalia Bochkina’s Bernstein-von Mises theorem for a location scale semi-parametric model, including a clever construct of a mixture of two Dirichlet priors to achieve proper convergence.)

Jim Berger started the day with a talk on imprecise probabilities, involving the society for imprecise probability, which I discovered while reading Keynes’ book, with a neat resolution of the Jeffreys-Lindley paradox, when re-expressing the null as an imprecise null, with the posterior of the null no longer converging to one, with a limit depending on the prior modelling, if involving a prior on the bias as well, with Chris discussing the talk and mentioning a recent work with Edwin Fong on reinterpreting marginal likelihood as exhaustive X validation, summing over all possible subsets of the data [using log marginal predictive].Håvard Rue did a follow-up talk from his Valencià O’Bayes 2015 talk on PC-priors. With a pretty hilarious introduction on his difficulties with constructing priors and counseling students about their Bayesian modelling. With a list of principles and desiderata to define a reference prior. However, I somewhat disagree with his argument that the Kullback-Leibler distance from the simpler (base) model cannot be scaled, as it is essentially a log-likelihood. And it feels like multivariate parameters need some sort of separability to define distance(s) to the base model since the distance somewhat summarises the whole departure from the simpler model. (Håvard also joined my achievement of putting an ostrich in a slide!) In his discussion, Robin Ryder made a very pragmatic recap on the difficulties with constructing priors. And pointing out a natural link with ABC (which brings us back to Don Rubin’s motivation for introducing the algorithm as a formal thought experiment).

Sara Wade gave the final talk on the day about her work on Bayesian cluster analysis. Which discussion in Bayesian Analysis I alas missed. Cluster estimation, as mentioned frequently on this blog, is a rather frustrating challenge despite the simple formulation of the problem. (And I will not mention Larry’s tequila analogy!) The current approach is based on loss functions directly addressing the clustering aspect, integrating out the parameters. Which produces the interesting notion of neighbourhoods of partitions and hence credible balls in the space of partitions. It still remains unclear to me that cluster estimation is at all achievable, since the partition space explodes with the sample size and hence makes the most probable cluster more and more unlikely in that space. Somewhat paradoxically, the paper concludes that estimating the cluster produces a more reliable estimator on the number of clusters than looking at the marginal distribution on this number. In her discussion, Clara Grazian also pointed the ambivalent use of clustering, where the intended meaning somehow diverges from the meaning induced by the mixture model.

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selecting summary statistics [a tale of two distances]

Posted in Books, Statistics with tags , , , , , , , , , , , , , , on May 23, 2019 by xi'an

As Jonathan Harrison came to give a seminar in Warwick [which I could not attend], it made me aware of his paper with Ruth Baker on the selection of summaries in ABC. The setting is an ABC-SMC algorithm and it relates with Fearnhead and Prangle (2012), Barnes et al. (2012), our own random forest approach, the neural network version of Papamakarios and Murray (2016), and others. The notion here is to seek the optimal weights of different summary statistics in the tolerance distance, towards a maximization of a distance (Hellinger) between prior and ABC posterior (Wasserstein also comes to mind!). A sort of dual of the least informative prior. Estimated by a k-nearest neighbour version [based on samples from the prior and from the ABC posterior] I had never seen before. I first did not get how this k-nearest neighbour distance could be optimised in the weights since the posterior sample was already generated and (SMC) weighted, but the ABC sample can be modified by changing the [tolerance] distance weights and the resulting Hellinger distance optimised this way. (There are two distances involved, in case the above description is too murky!)

“We successfully obtain an informative unbiased posterior.”

The paper spends a significant while in demonstrating that the k-nearest neighbour estimator converges and much less on the optimisation procedure itself, which seems like a real challenge to me when facing a large number of particles and a high enough dimension (in the number of statistics). (In the examples, the size of the summary is 1 (where does the weight matter?), 32, 96, 64, with 5 10⁴, 5 10⁴, 5 10³ and…10 particles, respectively.) The authors address the issue, though, albeit briefly, by mentioning that, for the same overall computation time, the adaptive weight ABC is indeed further from the prior than a regular ABC with uniform weights [rather than weighted by the precisions]. They also argue that down-weighting some components is akin to selecting a subset of summaries, but I beg to disagree with this statement as the weights are never exactly zero, as far as I can see, hence failing to fight the curse of dimensionality. Some LASSO version could implement this feature.

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ABC²DE

Posted in Books, Statistics with tags , , , , , , , , , , , , , on June 25, 2018 by xi'an

A recent arXival on a new version of ABC based on kernel estimators (but one could argue that all ABC versions are based on kernel estimators, one way or another.) In this ABC-CDE version, Izbicki,  Lee and Pospisilz [from CMU, hence the picture!] argue that past attempts failed to exploit the full advantages of kernel methods, including the 2016 ABCDE method (from Edinburgh) briefly covered on this blog. (As an aside, CDE stands for conditional density estimation.) They also criticise these attempts at selecting summary statistics and hence failing in sufficiency, which seems a non-issue to me, as already discussed numerous times on the ‘Og. One point of particular interest in the long list of drawbacks found in the paper is the inability to compare several estimates of the posterior density, since this is not directly ingrained in the Bayesian construct. Unless one moves to higher ground by calling for Bayesian non-parametrics within the ABC algorithm, a perspective which I am not aware has been pursued so far…

The selling points of ABC-CDE are that (a) the true focus is on estimating a conditional density at the observable x⁰ rather than everywhere. Hence, rejecting simulations from the reference table if the pseudo-observations are too far from x⁰ (which implies using a relevant distance and/or choosing adequate summary statistics). And then creating a conditional density estimator from this subsample (which makes me wonder at a double use of the data).

The specific density estimation approach adopted for this is called FlexCode and relates to an earlier if recent paper from Izbicki and Lee I did not read. As in many other density estimation approaches, they use an orthonormal basis (including wavelets) in low dimension to estimate the marginal of the posterior for one or a few components of the parameter θ. And noticing that the posterior marginal is a weighted average of the terms in the basis, where the weights are the posterior expectations of the functions themselves. All fine! The next step is to compare [posterior] estimators through an integrated squared error loss that does not integrate the prior or posterior and does not tell much about the quality of the approximation for Bayesian inference in my opinion. It is furthermore approximated by  a doubly integrated [over parameter and pseudo-observation] squared error loss, using the ABC(ε) sample from the prior predictive. And the approximation error only depends on the regularity of the error, that is the difference between posterior and approximated posterior. Which strikes me as odd, since the Monte Carlo error should take over but does not appear at all. I am thus unclear as to whether or not the convergence results are that relevant. (A difficulty with this paper is the strong dependence on the earlier one as it keeps referencing one version or another of FlexCode. Without reading the original one, I spotted a mention made of the use of random forests for selecting summary statistics of interest, without detailing the difference with our own ABC random forest papers (for both model selection and estimation). For instance, the remark that “nuisance statistics do not affect the performance of FlexCode-RF much” reproduces what we observed with ABC-RF.

The long experiment section always relates to the most standard rejection ABC algorithm, without accounting for the many alternatives produced in the literature (like Li and Fearnhead, 2018. that uses Beaumont et al’s 2002 scheme, along with importance sampling improvements, or ours). In the case of real cosmological data, used twice, I am uncertain of the comparison as I presume the truth is unknown. Furthermore, from having worked on similar data a dozen years ago, it is unclear why ABC is necessary in such context (although I remember us running a test about ABC in the Paris astrophysics institute once).

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