Likelihood-free methods are an essential tool for performing inference for implicit models which can be simulated from, but for which the corresponding likelihood is intractable. However, common likelihood-free methods do not scale well to a large number of model parameters. A promising approach to high-dimensional likelihood-free inference involves estimating low-dimensional marginal posteriors by conditioning only on summary statistics believed to be informative for the low-dimensional component, and then combining the low-dimensional approximations in some way. In this paper, we demonstrate that such low-dimensional approximations can be surprisingly poor in practice for seemingly intuitive summary statistic choices. We describe an idealized low-dimensional summary statistic that is, in principle, suitable for marginal estimation. However, a direct approximation of the idealized choice is difficult in practice. We thus suggest an alternative approach to marginal estimation which is easier to implement and automate. Given an initial choice of low-dimensional summary statistic that might only be informative about a marginal posterior location, the new method improves performance by first crudely localising the posterior approximation using all the summary statistics to ensure global identifiability, followed by a second step that hones in on an accurate low-dimensional approximation using the low-dimensional summary statistic. We show that the posterior this approach targets can be represented as a logarithmic pool of posterior distributions based on the low-dimensional and full summary statistics, respectively. The good performance of our method is illustrated in several examples.

An open question in the study of emergent behaviour in multi-agent Bayesian systems is the relationship, if any, between individual and collective inference. In this paper we explore the correspondence between generative models that exist at two distinct scales, using spin glass models as a sandbox system to investigate this question. We show that the collective dynamics of a specific type of active inference agent is equivalent to sampling from the stationary distribution of a spin glass system. A collective of specifically-designed active inference agents can thus be described as implementing a form of sampling-based inference (namely, from a Boltzmann machine) at the higher level. However, this equivalence is very fragile, breaking upon simple modifications to the generative models of the individual agents or the nature of their interactions. We discuss the implications of this correspondence and its fragility for the study of multiscale systems composed of Bayesian agents.

Human perception is based on unconscious inference, where sensory input integrates with prior information. This phenomenon, known as context dependency, helps in facing the uncertainty of the external world with predictions built upon previous experience. On the other hand, human perceptual processes are inherently shaped by social interactions. However, how the mechanisms of context dependency are affected is to date unknown. If using previous experience - priors - is beneficial in individual settings, it could represent a problem in social scenarios where other agents might not have the same priors, causing a perceptual misalignment on the shared environment. The present study addresses this question. We studied context dependency in an interactive setting with a humanoid robot iCub that acted as a stimuli demonstrator. Participants reproduced the lengths shown by the robot in two conditions: one with iCub behaving socially and another with iCub acting as a mechanical arm. The different behavior of the robot significantly affected the use of prior in perception. Moreover, the social robot positively impacted perceptual performances by enhancing accuracy and reducing participants overall perceptual errors. Finally, the observed phenomenon has been modelled following a Bayesian approach to deepen and explore a new concept of shared perception.

One aim of representation learning is to recover the original latent code that generated the data, a task which requires additional information or inductive biases. A recently proposed approach termed Independent Mechanism Analysis (IMA) postulates that each latent source should influence the observed mixtures independently, complementing standard nonlinear independent component analysis, and taking inspiration from the principle of independent causal mechanisms. While it was shown in theory and experiments that IMA helps recovering the true latents, the method's performance was so far only characterized when the modeling assumptions are exactly satisfied. Here, we test the method's robustness to violations of the underlying assumptions. We find that the benefits of IMA-based regularization for recovering the true sources extend to mixing functions with various degrees of violation of the IMA principle, while standard regularizers do not provide the same merits. Moreover, we show that unregularized maximum likelihood recovers mixing functions which systematically deviate from the IMA principle, and provide an argument elucidating the benefits of IMA-based regularization.

Weighted Majority Voting (WMV) is a well-known optimal decision rule for collective decision making, given the probability of sources to provide accurate information (trustworthiness). However, in reality, the trustworthiness is not a known quantity to the decision maker - they have to rely on an estimate called trust. A (machine learning) algorithm that computes trust is called unbiased when it has the property that it does not systematically overestimate or underestimate the trustworthiness. To formally analyse the uncertainty to the decision process, we introduce and analyse two important properties of such unbiased trust values: stability of correctness and stability of optimality. Stability of correctness means that the decision accuracy that the decision maker believes they achieved is equal to the actual accuracy. We prove stability of correctness holds. Stability of optimality means that the decisions made based on trust, are equally good as they would have been if they were based on trustworthiness. Stability of optimality does not hold. We analyse the difference between the two, and bounds thereon. We also present an overview of how sensitive decision correctness is to changes in trust and trustworthiness.

Simulation-Based Inference (SBI) is a promising Bayesian inference framework that alleviates the need for analytic likelihoods to estimate posterior distributions. Recent advances using neural density estimators in SBI algorithms have demonstrated the ability to achieve high-fidelity posteriors, at the expense of a large number of simulations ; which makes their application potentially very time-consuming when using complex physical simulations. In this work we focus on boosting the sample-efficiency of posterior density estimation using the gradients of the simulator. We present a new method to perform Neural Posterior Estimation (NPE) with a differentiable simulator. We demonstrate how gradient information helps constrain the shape of the posterior and improves sample-efficiency.

We present a new probabilistic method for detecting, deblending, and cataloging astronomical sources called the Bayesian Light Source Separator (BLISS). BLISS is based on deep generative models, which embed neural networks within a Bayesian model. For posterior inference, BLISS uses a new form of variational inference known as Forward Amortized Variational Inference. The BLISS inference routine is fast, requiring a single forward pass of the encoder networks on a GPU once the encoder networks are trained. BLISS can perform fully Bayesian inference on megapixel images in seconds, and produces highly accurate catalogs. BLISS is highly extensible, and has the potential to directly answer downstream scientific questions in addition to producing probabilistic catalogs.

Despite their advantages, normalizing flows generally suffer from several shortcomings including their tendency to generate unrealistic data (e.g., images) and their failing to detect out-of-distribution data. One reason for these deficiencies lies in the training strategy which traditionally exploits a maximum likelihood principle only. This paper proposes a new training paradigm based on a hybrid objective function combining the maximum likelihood principle (MLE) and a sliced-Wasserstein distance. Results obtained on synthetic toy examples and real image data sets show better generative abilities in terms of both likelihood and visual aspects of the generated samples. Reciprocally, the proposed approach leads to a lower likelihood of out-of-distribution data, demonstrating a greater data fidelity of the resulting flows.

This primer is an attempt to provide a detailed summary of the different facets of lifelong learning. We start with Chapter 2 which provides a high-level overview of lifelong learning systems. In this chapter, we discuss prominent scenarios in lifelong learning (Section 2.4), provide 8 Introduction a high-level organization of different lifelong learning approaches (Section 2.5), enumerate the desiderata for an ideal lifelong learning system (Section 2.6), discuss how lifelong learning is related to other learning paradigms (Section 2.7), describe common metrics used to evaluate lifelong learning systems (Section 2.8). This chapter is more useful for readers who are new to lifelong learning and want to get introduced to the field without focusing on specific approaches or benchmarks. The remaining chapters focus on specific aspects (either learning algorithms or benchmarks) and are more useful for readers who are looking for specific approaches or benchmarks. Chapter 3 focuses on regularization-based approaches that do not assume access to any data from previous tasks. Chapter 4 discusses memory-based approaches that typically use a replay buffer or an episodic memory to save subset of data across different tasks. Chapter 5 focuses on different architecture families (and their instantiations) that have been proposed for training lifelong learning systems. Following these different classes of learning algorithms, we discuss the commonly used evaluation benchmarks and metrics for lifelong learning (Chapter 6) and wrap up with a discussion of future challenges and important research directions in Chapter 7.

We develop a multiscale approach to estimate high-dimensional probability distributions from a dataset of physical fields or configurations observed in experiments or simulations. In this way we can estimate energy functions (or Hamiltonians) and efficiently generate new samples of many-body systems in various domains, from statistical physics to cosmology. Our method -- the Wavelet Conditional Renormalization Group (WC-RG) -- proceeds scale by scale, estimating models for the conditional probabilities of "fast degrees of freedom" conditioned by coarse-grained fields. These probability distributions are modeled by energy functions associated with scale interactions, and are represented in an orthogonal wavelet basis. WC-RG decomposes the microscopic energy function as a sum of interaction energies at all scales and can efficiently generate new samples by going from coarse to fine scales. Near phase transitions, it avoids the "critical slowing down" of direct estimation and sampling algorithms. This is explained theoretically by combining results from RG and wavelet theories, and verified numerically for the Gaussian and $\varphi^4$ field theories. We show that multiscale WC-RG energy-based models are more general than local potential models and can capture the physics of complex many-body interacting systems at all length scales. This is demonstrated for weak-gravitational-lensing fields reflecting dark matter distributions in cosmology, which include long-range interactions with long-tail probability distributions. WC-RG has a large number of potential applications in non-equilibrium systems, where the underlying distribution is not known {\it a priori}. Finally, we discuss the connection between WC-RG and deep network architectures.