We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic ...Fokker–Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. In particular, we are able to recover kinetic behaviour in terms of explicit lower bounds for the contraction rate. For example, for a rescaled double-well potential with local minima at distance a, we obtain a lower bound for the contraction rate of order Ω(a
−1) provided the friction coefficient is of order Ξ(a
−1).
In recent years important progress has been achieved towards proving the validity of the replica predictions for the (asymptotic) mutual information (or “free energy”) in Bayesian inference problems. ...The proof techniques that have emerged appear to be quite general, despite they have been worked out on a case-by-case basis. Unfortunately, a common point between all these schemes is their relatively high level of technicality. We present a new proof scheme that is quite straightforward with respect to the previous ones. We call it the
adaptive interpolation method
because it can be seen as an extension of the interpolation method developped by Guerra and Toninelli in the context of spin glasses, with an interpolation path that is adaptive. In order to illustrate our method we show how to prove the replica formula for three non-trivial inference problems. The first one is symmetric rank-one matrix estimation (or factorisation), which is the simplest problem considered here and the one for which the method is presented in full details. Then we generalize to symmetric tensor estimation and random linear estimation. We believe that the present method has a much wider range of applicability and also sheds new insights on the reasons for the validity of replica formulas in Bayesian inference.
Choosing a probability distribution to represent daily precipitation depths is important for precipitation frequency analysis, stochastic precipitation modeling and in climate trend assessments. ...Early studies identified the two-parameter gamma (G2) distribution as a suitable distribution for wet-day precipitation based on the traditional goodness-of-fit tests. Here, probability plot correlation coefficients and L-moment diagrams are used to examine distributional alternatives for the wet-day series of daily precipitation for hundreds of stations at the point and catchment scales in the United States. Importantly, both Pearson Type-III (P3) and kappa (KAP) distributions perform very well, particularly for point rainfall. Our analysis indicates that the KAP distribution best describes the distribution of wet-day precipitation at the point scale, whereas the performance of G2 and P3 distributions are comparable for wet-day precipitation at the catchment scale, with P3 generally providing the improved goodness of fit over G2. Since the G2 distribution is currently the most widely used probability density function, our findings could be considerably important, especially within the context of climate change investigations.
Slepian and Sudakov–Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in ...probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the Lévy concentration function for the Gaussian maximum. The bound is dimension-free and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.
We prove the existence and uniqueness of a local in time solution to the periodic
Φ
3
4
model of stochastic quantisation using the method of paracontrolled distributions introduced recently by M. ...Gubinelli, P. Imkeller and N. Perkowski in Forum Math., Pi 3 (2015) e6.
Elicitation is a key task for subjectivist Bayesians. Although skeptics hold that elicitation cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and ...subject-matter expert colleagues. This article reviews the state of the art, reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively and how they respond to questions about that information. In a discussion of the elicitation process, the first issue to address is what it means for an elicitation to be successful; that is, what criteria should be used. Our answer is that a successful elicitation faithfully represents the opinion of the person being elicited. It is not necessarily "true" in some objectivistic sense, and cannot be judged in that way. We see that elicitation as simply part of the process of statistical modeling. Indeed, in a hierarchical model at which point the likelihood ends and the prior begins is ambiguous. Thus the same kinds of judgment that inform statistical modeling in general also inform elicitation of prior distributions. The psychological literature suggests that people are prone to certain heuristics and biases in how they respond to situations involving uncertainty. As a result, some of the ways of asking questions about uncertain quantities are preferable to others, and appear to be more reliable. However, data are lacking on exactly how well the various methods work, because it is unclear, other than by asking using an elicitation method, just what the person believes. Consequently, one is reduced to indirect means of assessing elicitation methods. The tool chest of methods is growing. Historically, the first methods involved choosing hyperparameters using conjugate prior families, at a time when these were the only families for which posterior distributions could be computed. Modern computational methods, such as Markov chain Monte Carlo, have freed elicitation from this constraint. As a result, now both parametric and nonparametric methods are available for low-dimensional problems. High-dimensional problems are probably best thought of as lacking another hierarchical level, which has the effect of reducing the as-yet-unelicited parameter space. Special considerations apply to the elicitation of group opinions. Informal methods, such as Delphi, encourage the participants to discuss the issue in the hope of reaching consensus. Formal methods, such as weighted averages or logarithmic opinion pools, each have mathematical characteristics that are uncomfortable. Finally, there is the question of what a group opinion even means, because it is not necessarily the opinion of any participant.
Low front-end cost and rapid accrual make Web-based surveys and enrolment in studies attractive, but participants are often self-selected with little reference to a well-defined study base. Of ...course, high quality studies must be internally valid (validity of inferences for the sample at hand), but Web-based enrolment reactivates discussion of external validity (generalization of within-study inferences to a target population or context) in epidemiology and clinical trials. Survey research relies on a representative sample produced by a sampling frame, prespecified sampling process and weighting that maps results to an intended population. In contrast, recent analytical epidemiology has shifted the focus away from survey-type representativity to internal validity in the sample. Against this background, it is a good time for statisticians to take stock of our role and position regarding surveys, observational research in epidemiology and clinical studies. The central issue is whether conditional effects in the sample (the study population) may be transported to desired target populations. Success depends on compatibility of causal structures in study and target populations, and will require subject matter considerations in each concrete case. Statisticians, epidemiologists and survey researchers should work together to increase understanding of these challenges and to develop improved tools to handle them.
We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for V : R d → R V : \mathbb {R}^d \to \mathbb {R} a potential function to minimize, we ...consider the stochastic differential equation d Y t = − σ σ ⊤ ∇ V ( Y t ) dY_t = - \sigma \sigma ^\top \nabla V(Y_t) d t + a ( t ) σ ( Y t ) d W t + a ( t ) 2 Υ ( Y t ) d t dt + a(t)\sigma (Y_t)dW_t + a(t)^2\Upsilon (Y_t)dt , where ( W t ) (W_t) is a Brownian motion, where σ : R d → M d ( R ) \sigma : \mathbb {R}^d \to \mathcal {M}_d(\mathbb {R}) is an adaptive (multiplicative) noise, where a : R + → R + a : \mathbb {R}^+ \to \mathbb {R}^+ is a function decreasing to 0 0 and where Υ \Upsilon is a correction term. This setting can be applied to optimization problems arising in Machine Learning; allowing σ \sigma to depend on the position brings faster convergence in comparison with the classical Langevin equation d Y t = − ∇ V ( Y t ) d t + σ d W t dY_t = -\nabla V(Y_t)dt + \sigma dW_t . The case where σ \sigma is a constant matrix has been extensively studied; however little attention has been paid to the general case. We prove the convergence for the L 1 L^1 -Wasserstein distance of Y t Y_t and of the associated Euler scheme Y ¯ t \bar {Y}_t to some measure ν ⋆ \nu ^\star which is supported by argmin ( V ) \operatorname {argmin}(V) and give rates of convergence to the instantaneous Gibbs measure ν a ( t ) \nu _{a(t)} of density ∝ exp ( − 2 V ( x ) / a ( t ) 2 ) \propto \exp (-2V(x)/a(t)^2) . To do so, we first consider the case where a a is a piecewise constant function. We find again the classical schedule a ( t ) = A log − 1 / 2 ( t ) a(t) = A\log ^{-1/2}(t) . We then prove the convergence for the general case by giving bounds for the Wasserstein distance to the stepwise constant case using ergodicity properties.
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