Publications

Selected publications below. For a complete list see my Google Scholar or ORCID profile.

2026

  1. Position: Agentic AI orchestration should be Bayes-consistent

    Papamarkou, Theodore and Alquier, Pierre and Bauer, Matthias and Buntine, Wray and Davison, Andrew and Dziugaite, Gintare Karolina and Filippone, Maurizio and Foong, Andrew Y. K. and Fortuin, Vincent and Fouskakis, Dimitris and Frellsen, Jes and Hüllermeier, Eyke and Karaletsos, Theofanis and Khan, Mohammad Emtiyaz and Kotelevskii, Nikita and Lahlou, Salem and Li, Yingzhen and Liu, Fang and Lyle, Clare and Möllenhoff, Thomas and Palla, Konstantina and Panov, Maxim and Sale, Yusuf and Schweighofer, Kajetan and Shelmanov, Artem and Swaroop, Siddharth and Trapp, Martin and Waegeman, Willem and Wilson, Andrew Gordon and Zaytsev, Alexey

    43rd International Conference on Machine Learning ICML

    LLMs excel at predictive tasks and complex reasoning tasks, but many high-value deployments rely on decisions under uncertainty, for example, which tool to call, which expert to consult, or how many resources to invest. While the usefulness and feasibility of Bayesian approaches remain unclear for LLM inference, this position paper argues that the control layer of an agentic AI system (that orchestrates LLMs and tools) is a clear case where Bayesian principles should shine. Bayesian decision theory provides a framework for agentic systems that can help to maintain beliefs over task-relevant latent quantities, to update these beliefs from observed agentic and human-AI interactions, and to choose actions. Making LLMs themselves explicitly Bayesian belief-updating engines remains computationally intensive and conceptually nontrivial as a general modeling target. In contrast, this paper argues that coherent decision-making requires Bayesian principles at the \theoorchestration level of the agentic system, not necessarily the LLM agent parameters. This paper articulates practical properties for Bayesian control that fit modern agentic AI systems and human-AI collaboration, and provides concrete examples and design patterns to illustrate how calibrated beliefs and utility-aware policies can improve agentic AI orchestration.

    Bayesian deep learning agentic AI
  2. Post-hoc probabilistic vision-language models

    Baumann, Anton and Li, Rui and Klasson, Marcus and Mentu, Santeri and Karthik, Shyamgopal and Akata, Zeynep and Solin, Arno and Trapp, Martin

    14th International Conference on Learning Representations ICLR

    Vision-language models (VLMs), such as CLIP and SigLIP, have found remarkable success in classification, retrieval, and generative tasks. For this, VLMs deterministically map images and text descriptions to a joint latent space in which their similarity is assessed using the cosine similarity. However, a deterministic mapping of inputs fails to capture uncertainties over concepts arising from domain shifts when used in downstream tasks. In this work, we propose post-hoc uncertainty estimation in VLMs that does not require additional training. Our method leverages a Bayesian posterior approximation over the last layers in VLMs and analytically quantifies uncertainties over cosine similarities. We demonstrate its effectiveness for uncertainty quantification and support set selection in active learning. Compared to baselines, we obtain improved and well-calibrated predictive uncertainties, interpretable uncertainty estimates, and sample-efficient active learning. Our results show promise for safety-critical applications of large-scale models.

    Bayesian deep learning approximate inference uncertainty quantification

2025

  1. LogSumExp: Efficient Approximate Logarithm Acceleration for Embedded Tractable Probabilistic Reasoning

    Yao, Lingyun and Zhao, Shirui and Trapp, Martin and Leslin, Jelin and Verhelst, Marian and Andraud, Martin

    IEEE Transactions on Circuits and Systems I IEEE TCAS I

    Probabilistic models (PMs) have become an alternative to complement or replace deep learning in applications where transparency and trustworthiness are crucial. As PMs compute explicit high-resolution probabilities, ensuring numerical stability legitimates the need for logarithmic (log) computing. As exact log computation on hardware is typically costly, existing hardware accelerators stick to high-resolution linear computation with, e.g., floating point (FP). From the perspective of efficient execution on edge devices, using such generic linear hardware for log operations is prone to underflow and ill-suited for operations such as log addition. Hence, the log-domain computing of PMs requires new hardware solutions, combining numerical stability and energy-efficient execution. Inspired by the Log-Sum-Exp (LSE) function used in existing PM software tools transferring data between log and linear domains to compute log additions, this work proposes an LSE Processing Element (LSE-PE). LSE-PE allows for efficient log computation, through an innovative double approximation for log addition, while ensuring numerical stability with an error compensation method using a compact error correction Look-Up Table (CLUT). Hardware synthesis results using a 16nm technology show that the proposed 24-bit LSE-PE hardware consumes 46% area and 32% power of 32-bit floating point, using only 16 LUT entries with 10 bits in each entry. Moreover, our experiments on various PM benchmarks show that LSE-PE prevents underflow even for large models, which exist in all other 32-bit number systems, with less than 0.2% accuracy loss. We also demonstrate an outlier detection task for uncertainty estimation of image classification models using the LSE-PE, for a fraction of the main model’s computing cost (0.06 to 20% of representative DNN architectures for MNIST).

    probabilistic circuits hardware acceleration
  2. Approximate Bayesian Inference via Bitstring Representations

    Sladek, Aleksanteri and Trapp, Martin and Solin, Arno

    41st Conference on Uncertainty in Artificial Intelligence UAI

    The machine learning community has recently put effort into quantized or low-precision arithmetics to scale large models. This paper proposes performing probabilistic inference in the quantized, discrete parameter space created by these representations, effectively enabling us to learn a continuous distribution using discrete parameters. We consider both 2D densities and quantized neural networks, where we introduce a tractable learning approach using probabilistic circuits. This method offers a scalable solution to manage complex distributions and provides clear insights into model behavior. We validate our approach with various models, demonstrating inference efficiency without sacrificing accuracy. This work advances scalable, interpretable machine learning by utilizing discrete approximations for probabilistic computations.

    probabilistic circuits quantization bitblasting approximate Bayesian inference
  3. Streamlining Prediction in Bayesian Deep Learning

    Li, Rui and Klasson, Marcus and Solin, Arno and Trapp, Martin

    13th International Conference on Learning Representations ICLR

    The rising interest in Bayesian deep learning (BDL) has led to a plethora of methods for estimating the posterior distribution. However, efficient computation of inferences, such as predictions, has been largely overlooked with Monte Carlo integration remaining the standard. In this work we examine streamlining prediction in BDL through a single forward pass without sampling. For this we use local linearisation on activation functions and local Gaussian approximations at linear layers. Thus allowing us to analytically compute an approximation to the posterior predictive distribution. We showcase our approach for both MLP and transformers, such as ViT and GPT-2, and assess its performance on regression and classification tasks.

    Bayesian deep learning approximate inference uncertainty quantification
  4. Flatness Improves Backbone Generalisation in Few-Shot Classification

    Li, Rui and Trapp, Martin and Klasson, Marcus and Solin, Arno

    IEEE/CVF Winter Conference on Applications of Computer Vision WACV Oral

    Deployment of deep neural networks in real-world settings typically requires adaptation to new tasks with few examples. Few-shot classification (FSC) provides a solution to this problem by leveraging pre-trained backbones for fast adaptation to new classes. However, approaches for multi-domain FSC typically result in complex pipelines aimed at information fusion and task-specific adaptation without consideration of the importance of backbone training. In this work, we introduce an effective strategy for backbone training and selection in multi-domain FSC by utilizing flatness-aware training and fine-tuning. Our work is theoretically grounded and empirically performs on par or better than state-of-the-art methods despite being simpler. Further, our results indicate that backbone training is crucial for good generalisation in FSC across different adaptation methods.

    few-shot classification flatness deep neural networks
  5. Turing.jl: a general-purpose probabilistic programming language

    Fjelde, Tor Erlend and Xu, Kai and Widmann, David and Tarek, Mohamed and Pfiffer, Cameron and Trapp, Martin and Axen, Seth D. and Sun, Xianda and Hauru, Markus and Yong, Penelope and Tebbutt, Will and Ghahramani, Zoubin and Ge, Hong

    ACM Transactions on Probabilistic Machine Learning TOPML

    Probabilistic programming languages (PPLs) are becoming increasingly important in many scientific disciplines, such as economics, epidemiology, and biology, to extract meaning from sources of data while accounting for one’s uncertainty. The key idea of probabilistic programming is to decouple inference and model specification, thus allowing the practitioner to approach their task at hand using Bayesian inference, without requiring extensive knowledge in programming or computational statistics. At the same time, the complexity of problem settings in which PPLs are employed is steadily increasing, both in terms of project size and model complexity, calling for more flexible and efficient systems.In this work, we describe Turing.jl, a general-purpose PPL, which is designed to be flexible, efficient, and easy to use. Turing.jl is built on top of the Julia programming language, which is known for its high performance and ease-of-use. We describe the design of Turing.jl, contextualizing it within different types of users and use cases, its key features, and how it can be used to solve a wide range of problems. We also provide a brief overview of the ecosystem around Turing.jl, including the different libraries and tools that can be used in conjunction with it. Finally, we provide a few examples of how Turing.jl can be used in practice.

2024

  1. On Hardware-efficient Inference in Probabilistic Circuits

    Yao, Lingyun and Trapp, Martin and Leslin, Jelin and Singh, Gaurav and Zhang, Peng and Periasamy, Karthekeyan and Andraud, Martin

    40th Conference on Uncertainty in Artificial Intelligence UAI

    Probabilistic circuits (PCs) offer a promising avenue to perform embedded reasoning under uncertainty. They support efficient and exact computation of various probabilistic inference tasks by design. Hence, hardware-efficient computation of PCs is highly interesting for edge computing applications. As computations in PCs are based on arithmetic with probability values, they are typically performed in the log domain to avoid underflow. Unfortunately, performing the log operation on hardware is costly. Hence, prior work has focused on computations in the linear domain, resulting in high resolution and energy requirements. This work proposes the first dedicated approximate computing framework for PCs that allows for low-resolution logarithm computations. We leverage Addition As Int, resulting in linear PC computation with simple hardware elements. Further, we provide a theoretical approximation error analysis and present an error compensation mechanism. Empirically, our method obtains up to 357\texttimes and 649\texttimes energy reduction on custom hardware for evidence and MAP queries respectively with little or no computational error.

    probabilistic circuits hardware-efficient inference approximate computing
  2. Subtractive Mixture Models via Squaring: Representation and Learning

    Loconte, Lorenzo and Sladek, Aleksanteri M. and Mengel, Stefan and Trapp, Martin and Solin, Arno and Gillis, Nicolas and Vergari, Antonio

    12th International Conference on Learning Representations ICLR Spotlight (5%)

    Mixture models are traditionally represented and learned by adding several distributions as components. Allowing mixtures to subtract probability mass or density can drastically reduce the number of components needed to model complex distributions. However, learning such subtractive mixtures while ensuring they still encode a non-negative function is challenging. We investigate how to learn and perform inference on deep subtractive mixtures by squaring them. We do this in the framework of probabilistic circuits, which enable us to represent tensorized mixtures and generalize several other subtractive models. We theoretically prove that the class of squared circuits allowing subtractions can be exponentially more expressive than traditional additive mixtures; and, we empirically show this increased expressiveness on a series of real-world distribution estimation tasks.

    probabilistic circuits subtractive mixtures representation learning
  3. Fixing Overconfidence in Dynamic Neural Networks

    Meronen, Lassi and Trapp, Martin and Pilzer, Andrea and Yang, Le and Solin, Arno

    IEEE/CVF Winter Conference on Applications of Computer Vision WACV

    Dynamic neural networks are a recent technique that promises a remedy for the increasing size of modern deep learning models by dynamically adapting their computational cost to the difficulty of the inputs. In this way, the model can adjust to a limited computational budget. However, the poor quality of uncertainty estimates in deep learning models makes it difficult to distinguish between hard and easy samples. To address this challenge, we present a computationally efficient approach for post-hoc uncertainty quantification in dynamic neural networks. We show that adequately quantifying and accounting for both aleatoric and epistemic uncertainty through a probabilistic treatment of the last layers improves the predictive performance and aids decision-making when determining the computational budget. In the experiments, we show improvements on CIFAR100, ImageNet, and Caltech-256 in terms of accuracy, capturing uncertainty, and calibration error.

    Bayesian deep learning uncertainty quantification dynamic neural networks

2023

  1. Characteristic Circuits

    Yu, Zhongjie and Trapp, Martin and Kersting, Kristian

    36th Conference on Neural Information Processing Systems NeurIPS Oral (0.6%)

    In many real-world scenarios it is crucial to be able to reliably and efficiently reason under uncertainty while capturing complex relationships in data. Probabilistic circuits (PCs), a prominent family of tractable probabilistic models, offer a remedy to this challenge by composing simple, tractable distributions into a high-dimensional probability distribution. However, learning PCs on heterogeneous data is challenging and densities of some parametric distributions are not available in closed form, limiting their potential use. We introduce characteristic circuits (CCs), a family of tractable probabilistic models providing a unified formalization of distributions over heterogeneous data in the spectral domain. The one-to-one relationship between characteristic functions and probability measures enables us to learn high-dimensional distributions on heterogeneous data domains and facilitates efficient probabilistic inference even when no closed-form density function is available. We show that the structure and parameters of CCs can be learned efficiently from the data and find that CCs outperform state-of-the-art density estimators for heterogeneous data domains on common benchmark data sets.

    probabilistic circuits characteristic functions probabilistic deep learning
  2. Transport with support: Data-conditional diffusion bridges

    Tamir, Ella and Trapp, Martin and Solin, Arno

    Transactions in Machine Learning Research TMLR

    The dynamic Schrödinger bridge problem provides an appealing setting for solving constrained time-series data generation tasks posed as optimal transport problems. It consists of learning non-linear diffusion processes using efficient iterative solvers. Recent works have demonstrated state-of-the-art results (eg. in modelling single-cell embryo RNA sequences or sampling from complex posteriors) but are limited to learning bridges with only initial and terminal constraints. Our work extends this paradigm by proposing the Iterative Smoothing Bridge (ISB). We integrate Bayesian filtering and optimal control into learning the diffusion process, enabling the generation of constrained stochastic processes governed by sparse observations at intermediate stages and terminal constraints. We assess the effectiveness of our method on synthetic and real-world data generation tasks and we show that the ISB generalises well to high-dimensional data, is computationally efficient, and provides accurate estimates of the marginals at intermediate and terminal times.

2022

  1. Uncertainty-Guided Source-Free Domain Adaptation

    Roy, Subhankar and Trapp, Martin and Pilzer, Andrea and Kannala, Juho and Sebe, Nicu and Ricci, Elisa and Solin, Arno

    17th European Conference on Computer Vision ECCV

    Source-free domain adaptation (SFDA) aims to adapt a classifier to an unlabelled target data set by only using a pre-trained source model. However, the absence of the source data and the domain shift makes the predictions on the target data unreliable. We propose quantifying the uncertainty in the source model predictions and utilizing it to guide the target adaptation. For this, we construct a probabilistic source model by incorporating priors on the network parameters inducing a distribution over the model predictions. Uncertainties are estimated by employing a Laplace approximation and incorporated to identify target data points that do not lie in the source manifold and to down-weight them when maximizing the mutual information on the target data. Unlike recent works, our probabilistic treatment is computationally lightweight, decouples source training and target adaptation, and requires no specialized source training or changes of the model architecture. We show the advantages of uncertainty-guided SFDA over traditional SFDA in the closed-set and open-set settings and provide empirical evidence that our approach is more robust to strong domain shifts even without tuning.

    domain adaptation Bayesian deep learning uncertainty quantification

2021

  1. Leveraging probabilistic circuits for nonparametric multi-output regression

    Yu, Zhongjie and Zhu, Mingye and Trapp, Martin and Skryagin, Arseny and Kersting, Kristian

    37th Conference on Uncertainty in Artificial Intelligence UAI

    Inspired by recent advances in the field of expert-based approximations of Gaussian processes (GPs), we present an expert-based approach to large-scale multi-output regression using single-output GP experts. Employing a deeply structured mixture of single-output GPs encoded via a probabilistic circuit allows us to capture correlations between multiple output dimensions accurately. By recursively partitioning the covariate space and the output space, posterior inference in our model reduces to inference on single-output GP experts, which only need to be conditioned on a small subset of the observations. We show that inference can be performed exactly and efficiently in our model, that it can capture correlations between output dimensions and, hence, often outperforms approaches that do not incorporate inter-output correlations, as demonstrated on several data sets in terms of the negative log predictive density.

    probabilistic circuits multi-output regression Gaussian processes
  2. Periodic Activation Functions Induce Stationarity

    Meronen, Lassi and Trapp, Martin and Solin, Arno

    34th Conference on Neural Information Processing Systems NeurIPS

    Neural network models are known to reinforce hidden data biases, making them unreliable and difficult to interpret. We seek to build models that ‘know what they do not know’ by introducing inductive biases in the function space. We show that periodic activation functions in Bayesian neural networks establish a connection between the prior on the network weights and translation-invariant, stationary Gaussian process priors. Furthermore, we show that this link goes beyond sinusoidal (Fourier) activations by also covering triangular wave and periodic ReLU activation functions. In a series of experiments, we show that periodic activation functions obtain comparable performance for in-domain data and capture sensitivity to perturbed inputs in deep neural networks for out-of-domain detection.

    Bayesian deep learning activation functions function space Gaussian processes

2020

  1. Random Sum-Product Networks: A Simple and Effective Approach to Probabilistic Deep Learning

    Peharz, Robert and Vergari, Antonio and Stelzner, Karl and Molina, Alejandro and Shao, Xiaoting and Trapp, Martin and Kersting, Kristian and Ghahramani, Zoubin

    35th Conference on Uncertainty in Artificial Intelligence UAI

    Sum-product networks (SPNs) are expressive probabilistic models with a rich set of exact and efficient inference routines. However, in order to guarantee exact inference, they require specific structural constraints, which complicate learning SPNs from data. Thereby, most SPN structure learners proposed so far are tedious to tune, do not scale easily, and are not easily integrated with deep learning frameworks. In this paper, we follow a simple “deep learning” approach, by generating unspecialized random structures, scalable to millions of parameters, and subsequently applying GPU-based optimization. Somewhat surprisingly, our models often perform on par with state-of-the-art SPN structure learners and deep neural networks on a diverse range of generative and discriminative scenarios. At the same time, our models yield well-calibrated uncertainties, and stand out among most deep generative and discriminative models in being robust to missing features and being able to detect anomalies.

    probabilistic circuits sum-product networks probabilistic deep learning deep generative models
  2. Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits

    Peharz, Robert and Lang, Steven and Vergari, Antonio and Stelzner, Karl and Molina, Alejandro and Trapp, Martin and Van Den Broeck, Guy and Kersting, Kristian and Ghahramani, Zoubin

    37th International Conference on Machine Learning ICML

    Probabilistic circuits (PCs) are a promising avenue for probabilistic modeling, as they permit a wide range of exact and efficient inference routines. Recent “deep-learning-style” implementations of PCs strive for a better scalability, but are still difficult to train on real-world data, due to their sparsely connected computational graphs. In this paper, we propose Einsum Networks (EiNets), a novel implementation design for PCs, improving prior art in several regards. At their core, EiNets combine a large number of arithmetic operations in a single monolithic einsum-operation, leading to speedups and memory savings of up to two orders of magnitude, in comparison to previous implementations. As an algorithmic contribution, we show that the implementation of Expectation-Maximization (EM) can be simplified for PCs, by leveraging automatic differentiation. Furthermore, we demonstrate that EiNets scale well to datasets which were previously out of reach, such as SVHN and CelebA, and that they can be used as faithful generative image models.

    probabilistic circuits scalable probabilistic inference probabilistic deep learning
  3. Deep Structured Mixtures of Gaussian Processes

    Trapp, Martin and Peharz, Robert and Pernkopf, Franz and Rasmussen, Carl Edward

    23rd International Conference on Artificial Intelligence and Statistics AISTATS

    Gaussian Processes (GPs) are powerful non-parametric Bayesian regression models that allow exact posterior inference, but exhibit high computational and memory costs. In order to improve scalability of GPs, approximate posterior inference is frequently employed, where a prominent class of approximation techniques is based on local GP experts. However, local-expert techniques proposed so far are either not well-principled, come with limited approximation guarantees, or lead to intractable models. In this paper, we introduce deep structured mixtures of GP experts, a stochastic process model which i) allows exact posterior inference, ii) has attractive computational and memory costs, and iii) when used as GP approximation, captures predictive uncertainties consistently better than previous expert-based approximations. In a variety of experiments, we show that deep structured mixtures have a low approximation error and often perform competitive or outperform prior work.

    probabilistic circuits Gaussian processes scalable probabilistic inference

2019

  1. Bayesian Learning of Sum-Product Networks

    Trapp, Martin and Peharz, Robert and Ge, Hong and Pernkopf, Franz and Ghahramani, Zoubin

    32nd Conference on Neural Information Processing Systems NeurIPS

    Sum-product networks (SPNs) are flexible density estimators and have received significant attention due to their attractive inference properties. While parameter learning in SPNs is well developed, structure learning leaves something to be desired: Even though there is a plethora of SPN structure learners, most of them are somewhat ad-hoc and based on intuition rather than a clear learning principle. In this paper, we introduce a well-principled Bayesian framework for SPN structure learning. First, we decompose the problem into i) laying out a computational graph, and ii) learning the so-called scope function over the graph. The first is rather unproblematic and akin to neural network architecture validation. The second represents the effective structure of the SPN and needs to respect the usual structural constraints in SPN, i.e. completeness and decomposability. While representing and learning the scope function is somewhat involved in general, in this paper, we propose a natural parametrisation for an important and widely used special case of SPNs. These structural parameters are incorporated into a Bayesian model, such that simultaneous structure and parameter learning is cast into monolithic Bayesian posterior inference. In various experiments, our Bayesian SPNs often improve test likelihoods over greedy SPN learners. Further, since the Bayesian framework protects against overfitting, we can evaluate hyper-parameters directly on the Bayesian model score, waiving the need for a separate validation set, which is especially beneficial in low data regimes. Bayesian SPNs can be applied to heterogeneous domains and can easily be extended to nonparametric formulations. Moreover, our Bayesian approach is the first, which consistently and robustly learns SPN structures under missing data.

    probabilistic circuits sum-product networks Bayesian learning

2017

  1. Safe Semi-Supervised Learning of Sum-Product Networks

    Trapp, Martin and Madl, Tamas and Peharz, Robert and Pernkopf, Franz and Trappl, Robert

    33rd Conference on Uncertainty in Artificial Intelligence UAI

    In several domains obtaining class annotations is expensive while at the same time unlabelled data are abundant. While most semi-supervised approaches enforce restrictive assumptions on the data distribution, recent work has managed to learn semi-supervised models in a non-restrictive regime. However, so far such approaches have only been proposed for linear models. In this work, we introduce semi-supervised parameter learning for Sum-Product Networks (SPNs). SPNs are deep probabilistic models admitting inference in linear time in number of network edges. Our approach has several advantages, as it (1) allows generative and discriminative semi-supervised learning, (2) guarantees that adding unlabelled data can increase, but not degrade, the performance (safe), and (3) is computationally efficient and does not enforce restrictive assumptions on the data distribution. We show on a variety of data sets that safe semi-supervised learning with SPNs is competitive compared to state-of-the-art and can lead to a better generative and discriminative objective value than a purely supervised approach.

    probabilistic circuits sum-product networks semi-supervised learning
  2. One Million Posts: A Data Set of German Online Discussions

    Schabus, Dietmar and Skowron, Marcin and Trapp, Martin

    40th International ACM SIGIR Conference on Research and Development in Information Retrieval SIGIR

    In this paper we introduce a new data set consisting of user comments posted to the website of a German-language Austrian newspaper. Professional forum moderators have annotated 11,773 posts according to seven categories they considered crucial for the efficient moderation of online discussions in the context of news articles. In addition to this taxonomy and annotated posts, the data set contains one million unlabeled posts. Our experimental results using six methods establish a first baseline for predicting these categories. The data and our code are available for research purposes from https://ofai.github.io/million-post-corpus.