Amortized Bayesian inference trains neural networks to solve stochastic inference problems using model simulations, thereby making it possible to rapidly perform Bayesian inference for any newly observed data. However, current simulation-based amortized inference methods are simulation-hungry and inflexible: They require the specification of a fixed parametric prior, simulator, and inference tasks ahead of time. Here, we present a new amortized inference method -- the Simformer -- which overcomes these limitations. By training a probabilistic diffusion model with transformer architectures, the Simformer outperforms current state-of-the-art amortized inference approaches on benchmark tasks and is substantially more flexible: It can be applied to models with function-valued parameters, it can handle inference scenarios with missing or unstructured data, and it can sample arbitrary conditionals of the joint distribution of parameters and data, including both posterior and likelihood. We showcase the performance and flexibility of the Simformer on simulators from ecology, epidemiology, and neuroscience, and demonstrate that it opens up new possibilities and application domains for amortized Bayesian inference on simulation-based models.
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Many methods for estimating integrated volatility and related functionals of semimartingales in the presence of jumps require specification of tuning parameters for their use in practice. In much of the available theory, tuning parameters are assumed to be deterministic and their values are specified only up to asymptotic constraints. However, in empirical work and in simulation studies, they are typically chosen to be random and data-dependent, with explicit choices often relying entirely on heuristics. In this paper, we consider novel data-driven tuning procedures for the truncated realized variations of a semimartingale with jumps based on a type of random fixed-point iteration. Being effectively automated, our approach alleviates the need for delicate decision-making regarding tuning parameters in practice and can be implemented using information regarding sampling frequency alone. We show our methods can lead to asymptotically efficient estimation of integrated volatility and exhibit superior finite-sample performance compared to popular alternatives in the literature.
Building on the recent development of the model-free generalized fiducial (MFGF) paradigm (Williams, 2023) for predictive inference with finite-sample frequentist validity guarantees, in this paper, we develop an MFGF-based approach to decision theory. Beyond the utility of the new tools we contribute to the field of decision theory, our work establishes a formal connection between decision theories from the perspectives of fiducial inference, conformal prediction, and imprecise probability theory. In our paper, we establish pointwise and uniform consistency of an {\em MFGF upper risk function} as an approximation to the true risk function via the derivation of nonasymptotic concentration bounds, and our work serves as the foundation for future investigations of the properties of the MFGF upper risk from the perspective of new decision-theoretic, finite-sample validity criterion, as in Martin (2021).
One of the main theoretical challenges in learning dynamical systems from data is providing upper bounds on the generalization error, that is, the difference between the expected prediction error and the empirical prediction error measured on some finite sample. In machine learning, a popular class of such bounds are the so-called Probably Approximately Correct (PAC) bounds. In this paper, we derive a PAC bound for stable continuous-time linear parameter-varying (LPV) systems. Our bound depends on the H2 norm of the chosen class of the LPV systems, but does not depend on the time interval for which the signals are considered.
The Lamport diagram is a pervasive and intuitive tool for informal reasoning about "happens-before" relationships in a concurrent system. However, traditional axiomatic formalizations of Lamport diagrams can be painful to work with in a mechanized setting like Agda. We propose an alternative, inductive formalization -- the causal separation diagram (CSD) -- that takes inspiration from string diagrams and concurrent separation logic, but enjoys a graphical syntax similar to Lamport diagrams. Critically, CSDs are based on the idea that causal relationships between events are witnessed by the paths that information follows between them. To that end, we model happens-before as a dependent type of paths between events. The inductive formulation of CSDs enables their interpretation into a variety of semantic domains. We demonstrate the interpretability of CSDs with a case study on properties of logical clocks, widely-used mechanisms for reifying causal relationships as data. We carry out this study by implementing a series of interpreters for CSDs, culminating in a generic proof of Lamport's clock condition that is parametric in a choice of clock. We instantiate this proof on Lamport's scalar clock, on Mattern's vector clock, and on the matrix clocks of Raynal et al. and of Wuu and Bernstein, yielding verified implementations of each. The CSD formalism and our case study are mechanized in the Agda proof assistant.
Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov subspace methods, which are among the most popular iterative solvers for large sparse linear systems. We develop a deflation strategy for symmetric saddle point matrices by taking advantage of their underlying block structure. The vectors used for deflation come from an elliptic singular value decomposition relying on the generalized Golub-Kahan bidiagonalization process. The block targeted by deflation is the off-diagonal one since it features a problematic singular value distribution for certain applications. One example is the Stokes flow in elongated channels, where the off-diagonal block has several small, isolated singular values, depending on the length of the channel. Applying deflation to specific parts of the saddle point system is important when using solvers such as CRAIG, which operates on individual blocks rather than the whole system. The theory is developed by extending the existing framework for deflating square matrices before applying a Krylov subspace method like MINRES. Numerical experiments confirm the merits of our strategy and lead to interesting questions about using approximate vectors for deflation.
In many modern regression applications, the response consists of multiple categorical random variables whose probability mass is a function of a common set of predictors. In this article, we propose a new method for modeling such a probability mass function in settings where the number of response variables, the number of categories per response, and the dimension of the predictor are large. Our method relies on a functional probability tensor decomposition: a decomposition of a tensor-valued function such that its range is a restricted set of low-rank probability tensors. This decomposition is motivated by the connection between the conditional independence of responses, or lack thereof, and their probability tensor rank. We show that the model implied by such a low-rank functional probability tensor decomposition can be interpreted in terms of a mixture of regressions and can thus be fit using maximum likelihood. We derive an efficient and scalable penalized expectation maximization algorithm to fit this model and examine its statistical properties. We demonstrate the encouraging performance of our method through both simulation studies and an application to modeling the functional classes of genes.
Generalization is the ability of machine learning models to make accurate predictions on new data by learning from training data. However, understanding generalization of quantum machine learning models has been a major challenge. Here, we introduce the data quantum Fisher information metric (DQFIM). It describes the capacity of variational quantum algorithms depending on variational ansatz, training data and their symmetries. We apply the DQFIM to quantify circuit parameters and training data needed to successfully train and generalize. Using the dynamical Lie algebra, we explain how to generalize using a low number of training states. Counter-intuitively, breaking symmetries of the training data can help to improve generalization. Finally, we find that out-of-distribution generalization, where training and testing data are drawn from different data distributions, can be better than using the same distribution. Our work provides a useful framework to explore the power of quantum machine learning models.
When modeling a vector of risk variables, extreme scenarios are often of special interest. The peaks-over-thresholds method hinges on the notion that, asymptotically, the excesses over a vector of high thresholds follow a multivariate generalized Pareto distribution. However, existing literature has primarily concentrated on the setting when all risk variables are always large simultaneously. In reality, this assumption is often not met, especially in high dimensions. In response to this limitation, we study scenarios where distinct groups of risk variables may exhibit joint extremes while others do not. These discernible groups are derived from the angular measure inherent in the corresponding max-stable distribution, whence the term extreme direction. We explore such extreme directions within the framework of multivariate generalized Pareto distributions, with a focus on their probability density functions in relation to an appropriate dominating measure. Furthermore, we provide a stochastic construction that allows any prespecified set of risk groups to constitute the distribution's extreme directions. This construction takes the form of a smoothed max-linear model and accommodates the full spectrum of conceivable max-stable dependence structures. Additionally, we introduce a generic simulation algorithm tailored for multivariate generalized Pareto distributions, offering specific implementations for extensions of the logistic and H\"usler-Reiss families capable of carrying arbitrary extreme directions.
The Bayesian evidence, crucial ingredient for model selection, is arguably the most important quantity in Bayesian data analysis: at the same time, however, it is also one of the most difficult to compute. In this paper we present a hierarchical method that leverages on a multivariate normalised approximant for the posterior probability density to infer the evidence for a model in a hierarchical fashion using a set of posterior samples drawn using an arbitrary sampling scheme.
WISER: multimodal variational inference for full-waveform inversion without dimensionality reduction
We present a semi-amortized variational inference framework designed for computationally feasible uncertainty quantification in 2D full-waveform inversion to explore the multimodal posterior distribution without dimensionality reduction. The framework is called WISER, short for full-Waveform variational Inference via Subsurface Extensions with Refinements. WISER leverages the power of generative artificial intelligence to perform approximate amortized inference that is low-cost albeit showing an amortization gap. This gap is closed through non-amortized refinements that make frugal use of acoustic wave physics. Case studies illustrate that WISER is capable of full-resolution, computationally feasible, and reliable uncertainty estimates of velocity models and imaged reflectivities.
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