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Epistemic artificial intelligence

From EverybodyWiki Bios & Wiki






Uncertainty quantification in artificial intelligence

There is growing recognition in artificial intelligence of the need for being able to quantify how certain AI models are when they make a prediction, and estimate or a decision.[1][2].

Limitations of current AI

Typically, AI/machine learning models (for instance, neural networks) are trained to produce an output (prediction) in response to an input data point - however, there is no indication of the likelihood of this output to be correct. This issue is known in machine learning as calibration. This is a serious stumbling block towards the deployment of AI systems in mission-critical situations[3], such as, e.g., clinical data analysis[4], autonomous driving[5], drug discovery[6] or engineering[7].

Recent advances in large language models (LLMs) and image and video generation (e.g., diffusion models), in particular, have exposed fundamental limitations of even cutting-edge AI models[8]. How much are we supposed to believe the response to a query from an LLM?[9] Hallucinations[10], i.e., situations in which language models simply make up answers which are false or inaccurate, is possibly the central weakness of this class of models and or chatbots using them as an engine[11]. Related to this, generative models (including, for instance, diffusion models[12]) are not very creative, in the sense that the variety of their outputs is typically much lower than that of the training set.

More in general, modern machine learning systems often make inaccurate and/or overconfident predictions when fed data which does not fit their original training distribution, including out-of-domain samples[13] or adversarial attacks[14].

Aleatoric vs epistemic uncertainty

Researchers in this field often make a distinction between aleatoric uncertainty, which arises from the intrinsic, irreducible variability of the data, and epistemic uncertainty, induced by limited knowledge of the data-generating process[15]. This distinction reflects a similar taxonomy in field such as economics, for instance, where (epistemic, or Knightian) uncertainty is considered distinct from risk (aleatoric probability)[16][17].

A simple example can help. Consider a lottery in which balls (red or white) are extracted from a bag. If one knows the exact percentage of red and white balls in the bag, the lottery can be described by a simple probability distribution (e.g., Prob(Red) = 0.7, Prob(White) = 0.3). In this case, the uncertainty involved is purely aleatoric. However, if one does not know the distribution of balls inside the bag, the uncertainty is on the true probability distribution driving the lottery itself: this uncertainty is called "epistemic" or "reducible",[18] for it stems from lack of knowledge or data about the process at hand. A classical example of epistemic uncertainty is associated with Ellberg's paradox[19].

The discussion around the difference between aleatoric and epistemic uncertainty in artificial intelligence, or even whether such a distinction makes sense at all, is currently attracting significant interest in the field[20], but also in other disciplines such as structural safety[21], but also management science[22].

Importance of uncertainty quantification

As we have seen, uncertainty quantification is key to several aspects of artificial intelligence: its reliability, creativity, truthfulness, dependability in critical real-world problems[23]. These can be classed into four sub-areas: robustness, adaptation, calibration and decision making.

Adversarial robustness

As mentioned, traditional neural networks (and machine learning models more in general) often suffer from overconfidence, i.e., they are prone to output wrong predictions with very high confidence scores[24], especially when test inputs are outside their training distribution. For instance, a neural network may claim an input image portraits a flower whereas it contains an animal, with a confidence of 99% (this is quite a common occurrence, in fact[25]).

This can be exploited by adversaries to manipulate ML models, in particular by feeding them subtly altered data to cause unintended, incorrect behaviour[26]. These alterations can be very subtle: for instance, it may suffice to add to an image another image representing low-intensity noise of a suitable kind, to throw the network/model entirely off course. Lack of robustness to adversarial attacked can render an AI model all but unusable[27].

Domain adaptation

Possibly the most important open challenge in machine learning is giving models the ability to adapt to test data outside the domain distribution of their training set (domain adaptation)[28], a subject of study closely related to statistical learning theory.

A classical situation is one in which a model (for instance, a computer vision object/person detector) is trained on a corpus of data collected in a given situation (e.g., surveillance videos from a specific bank in a specific city), to then being deployed in an entire different context (e.g., a supermarket in a different city or country). Collecting data (and annotating it, if one is an a supervised learning setting) is expensive and can take a very long time, making it difficult if not impossible to be able to train a model for every possible occurrence.

Whereas a variety of different mathematical techniques have been proposed to tackle the domain adaptation problem[29] (from minimax learning[30] to the use of custom loss functions[31], to adversarial strategies, to kernel methods[32]), models still falter under significant training-testing shifts. Uncertainty quantification can provide a principled way to addressing this issue[33][34].

Calibration

As we pointed out above, neural networks are typically uncalibrated[35], i.e., their predicted confidence rarely matches their observed empirical accuracy. The ideal situation, instead, is one in which the network/model is aware of the uncertainty of its prediction, so that it can transparently admit it does not know enough about it. Among the general public, too many users take AI predictions at face value, without realizing how dangerous this can be[36].

Techniques exist for dealing with calibration problems: for instance, Bayesian binning[37] (assigning predicted scores to a number of "bins"), Platt scaling (a different way of transforming the outputs of a classification model into a probability distribution over classes), temperature scaling, implicit regularization[38], data augmentation can improve this[39]. The most common metric for calibration[40] is the Expected Calibration Error (ECE), which quantifies the discrepancy between predicted probabilities and observed frequencies by dividing the prediction space into several bins based on the predicted probabilities. Other metrics exist, such as the Brier score.

Sequential decision-making

Uncertainty also affects sequential decision making settings (i.e., situations in which a machine needs to make a series of decisions to achieve its long-term goal), for instance reinforcement learning (RL, where the purpose is to maximize the expected future cumulative reward an environment generates in response to the agent's actions). Indeed, especially at the beginning of the process, the environment surrounding the RL agent, the reward the agent receives in response to its actions/decisions, and most importantly the expected cumulative reward of its sequence of decisions are subject to high uncertainty, both epistemic and aleatoric[41].

Generally speaking, epistemic uncertainty is significant when having to make predictions or decisions in the very long terms, for instance for climate change prediction[42] and adaptation[43].

Techniques have been proposed to deal with uncertainty quantification in reinforcement learning[44], for instance based on Bayesian Bellman formulations[45] or ensembles[46]. Still, a proper mathematical formulation of epistemic, second-order uncertainty in RL[47], starting from classical partially-observable Markov diffusion processes (POMDPs), is wanting.

Approaches to uncertainty quantification in AI

In response to the fundamental challenge of how uncertainty about AI models and their predictions should be mathematically modelled, a variety of different approaches have been proposed. The matter has been surveyed in a number of very interesting recent papers. The following classes of methods can be highlighted: traditional models, deep deterministic uncertainty approaches, ensemble methods, Bayesian deep learning, conformal prediction, interval and evidential techniques.

Traditional models, including classical neural networks and deep networks, do not quantify uncertainty at all, but make deterministic predictions assuming exact input-output relations[48].

Deep Deterministic Uncertainty (DDU)[49] and similar approaches[50] attempt to do so without changing the way a network is trained, but by estimating epistemic uncertainty via latent representation analysis or distance-sensitive functions rather than soft-max probabilities.

Bayesian Deep Learning (BDL)[51], instead, models network parameters as probability distributions[52]. Rather than learning a deterministic weight for each network connection, as in traditional models, the training of a Bayesian deep network results in a (joint) probability distribution over the weight. This does two things: boosts the expressive power of the network, and acknowledges the epistemic uncertainty associated with the learning process. BDL requires a number of steps, such as defining a suitable prior for the distribution of the weights[53] (which is often not obvious), designing a likelihood model[54], and generating from the trained posterior a predictive distribution over the output space of the network[55] (often by sampling from an approximated posterior). These design choices introduce arbitrary elements into the process. Also, Bayesian deep networks can be computationally expensive to train[56].

Bayesian models are sensitive to prior mis-specification[57] (which risks biasing the entire estimation process), and incur heavy computational overhead, whereas Bayesian Model Averaging (BMA, i.e., the idea of generating a prediction by sampling from the learnt posterior and then taking the mean)[58] runs the risk of diluting useful predictive information.

Ensemble methods such as Deep Ensembles (DE)[59] and Epistemic Neural Networks (ENN)[60], on the other hand, estimate uncertainty by aggregating predictions from multiple independently trained models/networks (often trained using a different set of hyperparameter values, rather than different slices of the data). Ensemble methods are also computationally demanding, and their theoretical justification remains weak, despite research advances[61].

Conformal prediction[62] is an elegant technique which outputs, for a given prediction problem, prediction sets in the output space, without an attached probability distribution. Thus, the outputs of conformal prediction are sets, rather than probabilities. This happens by computing empirical cumulative distributions of "nonconformity" scores (which measure how unusual or atypical a data point is compared to other examples) and applying a form of hypothesis testing. Unfortunately, conformal prediction[63] primarily captures aleatoric uncertainty, as it relies on applying frequentist hypothesis testing and on the assumption data exchangeability[64] (i.e., predictions need to be invariant to the ordering of the data). Nevertheless, extensions of conformal predictions to epistemic uncertainty[65][66] or which relax the exchangeability assumption[67] have recently been proposed.

Finally, evidential methods[68] predict second-order Dirichlet distribution parameters in the output space of the network, rather than point probabilities. A Dirichlet distribution is a family of probability distributions which, by virtue of its shape, is suitable to model "second-order" probability, and therefore uncertainty on the predicted probability vector. They have been applied to neural networks, decision trees (classifiers which output a class by traversing a suitable tree learned from the data), K-nearest neighbors frameworks[69], and evidential deep learning classifiers[70] in which feature vectors are converted into belief functions and aggregated using Dempster’s rule in a Dempster-Shafer (DS) layer. Evidential approaches intended in the narrow sense (i.e., those predicting Dirichlet second-order distributions), however, appear to violate asymptotic conditions and struggle with out-of-distribution data[71].

The figure below illustrates the major approaches to uncertainty in AI. The diagram on the right illustrates the fact that traditional, ensemble, Bayesian, evidential and epistemic predictions can all be seen as predicting a credal set (i.e., a convex set of probability distributions) in the output space of the model. This opens the way to comparing their performance homogenously as generators of credal sets, and to the concept of epistemic AI.

Major uncertainty quantification approaches in AI.

Epistemic artificial intelligence

A common limitation of most of the above approaches is that epistemic uncertainty is either approximated heuristically or collapsed into point-valued predictions, rather than represented explicitly as uncertainty about the model itself. Indeed, epistemic uncertainty can be defined either as uncertainty about the "true" probability distribution generating the data the model is trained on[72], or as uncertainty in the hypothesis space (the class of models, e.g., the class of convolutional neural networks with a set number of layers and neurons for each layer) chosen[73]. As a result, existing methods struggle to faithfully encode ignorance under data scarcity, distribution shift, and open-world conditions.

Formal definition

Epistemic AI maintains that an AI system must be designed not only to learn from the data it observes, but also to be prepared for data it has not yet encountered, in a Socratic stance ("know that you do not know"). This is possibly the paramount problem in machine learning, where a mapping from inputs to outputs needs to be learn from data that is forcibly scarce[74], or not representative of the entirety of the problem[75].

The problem can be formalized as one of learning a mapping (epistemic model) from input data points to predictions in the form of a second-order uncertainty measure, either on the target space (e.g., in classification, the collection of possible classes) or on the parameter space of the model itself (e.g., for neural networks, the set of connection weights and biases): see Figure below. Later, this prediction may be updated in the light of new data, in a continual learning setting.

When the epistemic prediction is a credal set, as in most cases[76] (as shown in the above Section), a probability (‘pignistic’) estimate can be computed as its center of mass to provide a "precise" probabilistic prediction, which can then be evaluated in the traditional way. It is well known that second-order uncertainty measures subsume classical probabilistic approaches as special cases[77] (in particular, a standard probability distribution is a trivial credal set containing a single point; it is also a special case of finite random sets or belief functions[78]; more general classes of uncertainty measures, such as capacities[79] or lower previsions[80], also exist).

Epistemic AI can therefore be understood as a generalization of existing Bayesian and ensemble methods to second-order uncertainty.

Principle of epistemic learning.

Formally, one can define Epistemic AI as a class of learning algorithms that: (i) employ second-order uncertainty measures (e.g., credal sets, probability intervals, random sets or belief functions), rather than single probability distributions, as a mathematical formalization of the uncertainty of the learning process, and (ii) preserve and propagate ignorance under data scarcity, distribution shift, or open-world conditions, instead of collapsing it into point estimates.

Formally, one can define Epistemic AI as a class of learning algorithms that: (i) employ second-order uncertainty measures (e.g., credal sets, probability intervals[81][82], random sets[83][84] or belief functions), rather than single probability distributions, as a mathematical formalization of the uncertainty of the learning process, and (ii) preserve and propagate ignorance under data scarcity, distribution shift, or open-world conditions, instead of collapsing it into point estimates. As the field is still evolving, the best way of formalizing this concept is still open for discussion.

Limitations of first-order probabilistic representations

Existing models fundamentally struggle to capture epistemic uncertainty. Some authors have argued this is primarily because a single probability distribution cannot fully express ignorance about the data-generating process[77].

Bayesian methods, in particular, though widely used, particularly falter in data-sparse or ambiguous settings because they must assign fixed belief mass even when knowledge is lacking. Uninformative priors such as Jeffreys' (1998), which attempt to assume the least about the prior distribution of the data, for instance by assigning equal probability to all eventualities, are unfortunately not invariant under a reparameterization of the sample space[85] and can be improper[86], violating objectivity and the strong likelihood principle[87]. Moreover, priors must be specified even for systems without past data, leading to arbitrary modeling choices that can bias the learning process for a long time (Bernstein-von Mises theorem). Bayesian posteriors may appear similar whether we have no knowledge or weak evidence, conflating ignorance with imprecise belief and potentially causing misleading overconfidence.

Model selection and prior choice lack objective criteria[88], and prior sensitivity worsens with scarce data[89]. Further, Bayesian models cannot naturally represent set-valued or propositional evidence[90], because the additivity of probability forces allocation to individual outcomes, even when evidence supports sets of hypotheses. Random-sets, in opposition, can naturally model missing data[77] without the need for imputation techniques such as missing at random. For a simple introduction to the topic, the reader can consult. Bayes’ rule also assumes that new evidence is sharp and definitive (although this can be overcome in non-standard frameworks, such as Popper's probability[91]), which is unrealistic in many real-world cases. Hierarchical Bayesian models, which place priors over priors, can model epistemic uncertainty and potentially address some of these issues, but are very computationally expensive in high-dimensional or open-world settings.

Moreover, Bayesian inference tends to smooth out epistemic uncertainty by averaging over models, collapsing diverse possibilities into a single estimate and failing to distinguish knowns from unknowns[92]. Computationally, Bayesian models also often suffer from slow convergence and large inference time[93], limiting their suitability for real-time safety-critical systems.

Target vs parameter representations

Epistemic models have recently demonstrated the ability to outperform competitor methods in areas such as classification, out-of-distribution detection[94], robustness to adversarial attacks, by employing either credal[95] or random-set[96] representations.

Uncertainty, however, can be modelled at two levels: (i) a target level, where the network/model outputs an uncertainty measure on the target space, while its parameters (weights) remain deterministic; (ii) a parameter level, where uncertainty is modelled on the parameter space (i.e., the neural network's weights). Epistemic AI models developed so far have only considered a target-space approach[97].

Results and ongoing work

Numerous methods based on credal set representations have been brought forward, especially in 2024-26.

For instance, Credal-Set Interval Neural Networks[94], based on Interval Neural Networks[98], predict probability intervals for classes (e.g., intervals of the kind P_(c)≤P(c)≤P‾(c)for each class c). As any probability interval determines a credal set, the result model can be thought of as outputting a credal set.

In a slightly different approach, Credal Deep Ensembles[99][100] use ensembles of credal networks (each outputting a separate credal set) to provide upper and lower probability bounds; trained with a distributionally-robust optimization (DRO)-inspired loss[101][102], credal deep ensembles do outperform traditional ones in terms of out-of-distribution detection and general quality of uncertainty quantification. Credal Wrapper[103] also improves uncertainty estimation by "wrapping" both Bayesian and ensemble predictions as credal sets with upper/lower bounds per class, using the intersection probability transformation[104], devised within the framework of the geometric approach to uncertainty, to map a credal set to a single distribution. Credal deep evidential classification[105] leverages a credal set and an interval of evidential predictive distributions to avoid overfitting to the training data and to systematically assess both epistemic (reducible) and aleatoric (irreducible) uncertainties. Credal prediction based on relative likelihood has also been recently proposed in[106]. A framework for credal ensembling in multi-class classification can be found in[107], where probabilistic ensemble members are aggregated into either a representative classifier or a credal classifier, and various decision tasks based on this uncertainty quantification are performed. A weakly supervised framework for credal learning is proposed in[108], where credal learning is studied from the learning-theoretic and complexity-theoretic perspectives. Credal graph neural networks have been formulated[109], where the output layer of a GNN is modified to produce a credal set for classification purposes. A generalization of statistical learning theory under the assumption that a series of training sets are generated each by a different probability distribution within a finitely-generated credal set is the focus of[110], where generalization bounds which generalize classical ones are derived.

Random-set representations have also been advanced. Random-Set Neural Networks[111], in particular, efficiently predict belief values for sets of classes, addressing ambiguity and incomplete data, using a budgeting method levering the unsupervised clustering of the training set to reduce complexity. RS-NN demonstrate superior performance over its competitors, in terms of accuracy, uncertainty estimation and out-of-distribution (OoD) detection on multiple benchmark datasets, including CIFAR-10, MNIST, ImageNet. Extensions of the random-set approach to graph neural networks[112] and large language models[113] have recently been proposed. A possible random-set treatment of statistical learning is explored in[114][115].

A key future research direction is thus extending Epistemic AI from target-level to parameter-level uncertainty representations, with the aim to fully generalize Bayesian (deep) learning, e.g., by transforming Bayesian Neural Network posteriors into random-set posteriors using an epistemic wrapper[116] without retraining via Shafer's likelihood transform[117]. Credal Bayesian Deep Learning[118], in opposition, introduces sets of posteriors over parameters, deriving predictive distributions at inference time that distinguish and quantify aleatoric and epistemic uncertainty, yielding either a set of outputs with guarantees or a single best prediction. Another promising direction is to employ Smets’ Generalized Bayes Theorem[119] to generate belief functions, i.e., finite random-sets, over a model's parameters from a generalized likelihood and observations, under conditional cognitive independence (a generalization of the i.i.d. assumption), to directly learn random-set parametric representations from a training set.

Natural extensions to regression can also be envisaged, as done for instance in CREDO[120], which first builds an interpretable credal envelope that widens when local evidence is weak, then applies split conformal calibration on top of this envelope to guarantee marginal coverage without further assumptions. Two-sample credal tests of epistemic uncertainty have been proposed[121].

Generative AI

The extension to generative AI (GenAI), the part of AI which concerns itself with the generation of new data, is also a paramount open direction of research.

Large Language Models (LLMs) perform very strongly in natural language processing (NLP) tasks such as answering questions, reasoning[122], mathematical problem-solving[123], and code generation[124]. Pre-trained on large text corpora via next-token prediction, LLMs are fine-tuned for specific applications. Despite their success, they face challenges like hallucinations[125]. Mechanisms to enhance their truthfulness (calibration) and quantify uncertainty may improve their reliability. Bayesian approaches such as Laplace-LORA[126], BLoB[127] and Monte-Carlo Dropout[128] have been applied to LLMs for uncertainty quantification. ENN-LLM[129] uses Epistemic Neural Network-inspired ensembles, while other approaches leverage hidden states, softmax entropy[130] or semantic entropy[131]. These methods, however, often trade performance for inference efficiency. The use of knowledge distillation techniques has also been proposed[132]. The epistemic uncertainty of visual tokens for object hallucinations in Large Vision-Language Models is considered in[133].

A good recent survey on uncertainty quantification and confidence calibration in large language models is[134]

An epistemic AI take on generative AI (both in the context of LLMs and beyond) requires us to address the all-important challenge of how to elicit second-order representations from "traditional" ground truth datasets, such as question-answer pairs. How do we teach a model that the examples it sees are only samples from an incredibly rich set of possibilities? Developing appropriate evaluation methods for uncertainty-aware LLMs (building on initial efforts in a classification context[135]) is another challenge that needs to be addressed before such models can be effectively trained and deployed. In the context of GenAI, Epistemic AI can teach generative models the range of possible outputs they could produce from a limited training set, capturing the epistemic uncertainty of the generative process itself[136]. Some results are coming in: credal sets have been shown to be able to expose calibration gaps in language models[137], while attempts to design credal transformers[138] have also been made.

Challenges

A number of open challenges can be identified, which are likely to drive further work in this area.

Complexity of second-order representations

Challenges that are shared across UQ may be exacerbated when using second-order uncertainty measures[139], owing to their higher expressiveness and complexity. Working with sets of distributions (e.g., credal or random-sets) may involve costly sampling and inference procedures[140], particularly for decision-making[141]. Recent work, however, has seemingly addressed this by employing set budgeting techniques to efficiently constrain the complexity of using random-sets[111]. However, further research is needed to expand this to other second-order representations, potentially more general, such as imprecise probability, lower previsions or monotone capacities.

Scaling up

Most evidential approaches[142] struggle with scalability beyond medium-sized datasets. The clustering approach proposed within the random-set approach[96] has unlocked the potential of random-set representations to large datasets such as ImageNet and architectures including vision transformers, with future extensions possibly incorporating Dirichlet mixture models[143] and dynamic clustering for continual learning. A key challenge remains: can epistemic representations scale to foundation model-size and massive datasets? While efficient belief function/random-set representations have been explored[144], further work is needed. Quantum approaches show some promise, with recent work on belief representation[145], combination, and integration into quantum circuits[146]

Continual learning

Continual learning is a more faithful representation of life-long real-world learning processes, especially in contexts in which models are continually updated in the light of streaming data whose distribution, however, may vary over time in unknown ways. Despite recent efforts[147][148], a unified framework linking uncertainty modeling and continual learning remains an entirely open challenge, not just for Epistemic AI but for uncertainty quantification in artificial intelligence more in general.

Statistical guarantees

Most current Epistemic AI methods do not provide statistical coverage guarantees on their predictions[149], albeit they can do so in combination with classical conformal learning. Already mentioned efforts to generalize conformal learning certainly go in this direction. Recent studies have been looking at extending the notion of confidence interval to belief functions, under the name of confidence structures[150], which generalize standard confidence distributions and generate "frequency-calibrated" belief functions. Also in the random-set setting, Inferential Models (IMs) can produce belief functions with well-defined frequentist properties[151]. An alternative approach relies on the notion of predictive belief function[152], which, under repeated sampling, is less committed than the true probability distribution of interest with some prescribed probability.

References

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  115. ↑ Benjamin Dupuis, Paul Viallard, George Deligiannidis, Umut Simsekli (2024). "Uniform Generalization Bounds on Data-Dependent Hypothesis Sets via PAC-Bayesian Theory on Random Sets". Journal of Machine Learning Research (JMLR). 25 (409): 1−55.CS1 maint: Multiple names: authors list (link)
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  118. ↑ Caprio, Michele; Dutta, Souradeep; Jang, Kuk Jin; Lin, Vivian; Ivanov, Radoslav; Sokolsky, Oleg; Lee, Insup (2024-10-22), Credal Bayesian Deep Learning, arXiv:2302.09656
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  135. ↑ Shireen Kudukkil Manchingal, Muhammad Mubashar, Kaizheng Wang, Fabio Cuzzolin (2025). "A Unified Evaluation Framework for Epistemic Predictions". Proceedings of the 28th International Conference on Artificial Intelligence and Statistics. PMLR 258: 2017–2025.CS1 maint: Multiple names: authors list (link)
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