Bayesian Continual Learning

• Bayesian Continual Learning is an online paradigm that incrementally updates a model’s probabilistic belief state to integrate new tasks without forgetting past information.
• It employs recursive Bayesian updates and methods like variational inference, regularization, replay, and adaptive architectures to balance stability and plasticity.
• This approach is key for robust uncertainty estimation and overcoming catastrophic forgetting in dynamic, nonstationary real-world environments.

Bayesian continual learning is an online learning paradigm in which a model incrementally acquires knowledge from a sequence of tasks or datasets, with the explicit aim of updating its internal belief state—encoded as a probability distribution over parameters—without forgetting information gained from previous tasks. This paradigm formalizes the idea that as new data arrive, the model should update its prior beliefs using Bayes’ theorem, naturally integrating past experiences while remaining adaptable to novel information. The approach finds deep connections to the sequential nature of human learning and is increasingly influential in efforts to overcome catastrophic forgetting, a core barrier to deploying deep models in real-world, evolving environments (Adel, 11 Jul 2025).

1. Bayesian Foundations of Continual Learning

The foundational principle of Bayesian continual learning (BCL) is the recursive application of Bayes’ rule to incrementally construct the posterior distribution over model parameters. After observing m tasks, the Bayesian posterior is given by

$$p(\theta|D_{1:m}) \propto p(\theta) \prod_{t=1}^{m} \prod_{n=1}^{N_t} p(y^n_t|\theta, x^n_t)$$

where $p(\theta)$ is the prior and $p(y^n_t|\theta, x^n_t)$ denotes the likelihood for each input-output pair in each task (Adel, 11 Jul 2025). Upon encountering a new task $D_{t}$, this framework updates the prior to the previous posterior and incorporates likelihood terms from the new data:

$$p(\theta|D_{1:t}) \propto p(\theta|D_{1:t-1}) p(D_t|\theta)$$

Prediction is then done by Bayesian model averaging:

$$p(y^*|x^*, D_{1:m}) = \int p(y^*|x^*, \theta) p(\theta|D_{1:m}) d\theta$$

This perspective underpins a broad taxonomy of approaches, including regularization-based, variational inference-based, replay-based, and architecture-based methods (Adel, 11 Jul 2025).

2. Main Algorithmic Paradigms and Methodologies

Bayesian continual learning encompasses several methodological families, each leveraging probabilistic reasoning to address forgetting and enable knowledge transfer:

Regularization-Based Methods.

Such methods, including Elastic Weight Consolidation (EWC) and Synaptic Intelligence (SI), impose a MAP-style penalty constraining parameter drift from prior-optimal values (Adel, 11 Jul 2025). Typically, the loss for task $t$ is augmented by a quadratic regularizer encoding the Fisher information or importance metric:

$$L_t^{\text{MAP}}(\theta) = L_t(\theta) + \frac{1}{2} (\theta - \theta^*_{t-1})^\top F_{t-1} (\theta - \theta^*_{t-1})$$

where $F_{t-1}$ is the (often diagonal) Fisher information matrix from previous data (Li et al., 3 Apr 2025).

Variational Inference Approaches.

These approaches use approximations to intractable posteriors—most notably, Gaussian mean-field approximations—transferring the previous approximate posterior $q_{t-1}(\theta)$ as a new prior and minimizing the ELBO:

$$\mathcal{L}^{(t)}_{\mathrm{VCL}}(q_t(\theta)) = \sum_{n} \mathbb{E}_{\theta \sim q_t(\theta)} [\log p(y_t^{(n)}|\theta, x_t^{(n)})] - \mathrm{KL}[q_t(\theta) \| q_{t-1}(\theta)]$$

This variational approach can be extended by introducing scalars to tune the regularization strength (Servia-Rodriguez et al., 2021), leveraging natural gradients for improved geometry (Chen et al., 2019), or combining with replay mechanisms (Farquhar et al., 2019).

Replay and Likelihood-Focused Methods.

To address the loss of past data, replay-based strategies use generative models or buffers to generate or store samples from previous tasks (Farquhar et al., 2019). For example, the Variational Generative Replay (VGR) method introduces synthetic data from past tasks directly in the ELBO’s likelihood terms, forming so-called "hybrid" objectives:

$$\mathcal{L}^{(t)}_{\mathrm{Hybrid}}(q_t(\omega)) = \sum_{t'=1}^{t} \int \log p(y|x, \omega) p_{t'}(x, y) q_t(\omega) dx dy d\omega - \mathrm{KL}[q_t(\omega) \| q_{t-1}(\omega)]$$

(Farquhar et al., 2019).

Architecture-Based Methods.

Some BCL algorithms employ Bayesian nonparametric priors (such as the Indian Buffet Process) to adaptively select active network structure per task and enable parameter sharing among related tasks (Kumar et al., 2019). This can avoid both parameter explosion and excessive interference, as masked priors allow flexible adaptation while retaining forward transfer.

3. Uncertainty Quantification and Task Adaptivity

An advantage of Bayesian continual learning is principled uncertainty estimation, essential for reliable deployment and for distinguishing in-distribution from out-of-distribution data (Servia-Rodriguez et al., 2021, Bonnet et al., 18 Apr 2025). Models compute measures such as predictive entropy

$$\mathcal{H}[y^*|x^*, D] = - \sum_{c} p(y^*=c|x^*,D) \log p(y^*=c|x^*,D)$$

and mutual information between predictions and parameter uncertainty, often via Monte Carlo sampling from learned posteriors. Several frameworks (e.g., MESU (Bonnet et al., 18 Apr 2025), Bayesian SNNs (Skatchkovsky et al., 2022)) specifically utilize uncertainty for adaptive learning rates and metaplasticity, with parameters of high uncertainty being more plastic and those with low uncertainty acting as consolidated memory.

Moreover, frameworks may include explicit mechanisms to modulate the trade-off between remembering and learning (e.g., a scalar $\beta$ scaling the KL term in the variational bound (Servia-Rodriguez et al., 2021)), making it possible to prioritize stability or plasticity in deployment.

4. Practical Implementations and Experimental Evidence

Bayesian continual learning algorithms have been evaluated on a range of benchmarks, from Permuted and Split MNIST to CIFAR-100 and time-series data (Li et al., 2019, Kumar et al., 2019, Gong et al., 2022). Empirical findings consistently show that:

• Hybrid objectives combining prior- and likelihood-focused terms improve uncertainty calibration and retention of prior/task knowledge (Farquhar et al., 2019).
• Structural adaptation methods, such as those employing Bayesian nonparametrics, outperform static architectures in scenarios requiring dynamic network growth (Kumar et al., 2019).
• Explicitly modeling uncertainty (MESU (Bonnet et al., 18 Apr 2025), SNNs (Skatchkovsky et al., 2022)) supports sustained accuracy over hundreds of sequential tasks and enables robust out-of-distribution detection.
• Recent model merging techniques derive closed-form solutions for balancing stability and plasticity, yielding superior performance on continual learning benchmarks with provable properties (Li et al., 3 Apr 2025).

Calibration metrics (expected calibration error, ECE) and ablation studies further substantiate the advantages of Bayesian frameworks over frequentist or memoryless baselines, particularly in retaining knowledge as data distributions shift (Milasheuski et al., 21 Apr 2025).

5. Extensions and Connections to Related Areas

Bayesian continual learning has strong ties to several related machine learning subfields:

• Transfer Learning: By reusing learned posteriors as priors, knowledge transfer from source to target domains is formalized.
• Domain Adaptation: Bayesian models are well-suited for adapting to nonstationary data distributions, as priors can encode robust invariances.
• Meta-Learning: In some frameworks, neural networks are meta-learned to output sufficient statistics for exponential family update rules, unifying meta-learning and Bayesian continual learning (Lee et al., 29 May 2024).
• Developmental Psychology: The analogy with scaffolding and cognitive flexibility/stability highlights the suitability of Bayesian continual learning for modeling aspects of human learning, such as adaptive forgetting and transfer (Adel, 11 Jul 2025).

Federated, task-free, and class-incremental settings have recently drawn attention, with federated Bayesian continual learning providing rigorous means to maintain reliability and calibration across distributed, temporally shifting datasets (Milasheuski et al., 21 Apr 2025). Spiking neural networks and biologically inspired metaplasticity rules further underscore the breadth of BCL methodologies (Bonnet et al., 18 Apr 2025, Skatchkovsky et al., 2022).

6. Current Challenges and Open Research Questions

Key challenges in Bayesian continual learning include:

• Efficient Posterior Approximation: The repeated use of variational inference or Laplace approximations (often in diagonal form) may degrade with numerous tasks, impacting posterior fidelity and computational efficiency (Adel, 11 Jul 2025).
• Scalability: Second-order information (e.g., full Fisher or Hessian) is expensive to compute for large models; efficient and scalable approximations remain a priority.
• Robustness to Model Misspecification: Even with exact sequential Bayesian inference, models can suffer from catastrophic forgetting when the functional forms are mismatched to task heterogeneity (Kessler et al., 2023).
• Task-Free and Class-Incremental Learning: Most methods still assume explicit task boundaries; real-world scenarios require approaches robust to smoothly changing or unknown task identities.
• Negative Transfer and Interference: When transferring inappropriate knowledge, Bayesian learning by itself may exacerbate interference. More sophisticated resource allocation or adaptive priors may be required (Adel, 11 Jul 2025).
• Benchmarks and Evaluation: The need for standardized benchmarks that include realistic, nonstationary sequences and controlled evaluation of uncertainty and calibration is frequently noted.

Future directions include meta-learned BCL architectures, adaptive structural priors, advanced uncertainty modeling especially for task-free settings, and further biological integration—such as synaptic metaplasticity and online adaptation principles (Lee et al., 29 May 2024, Bonnet et al., 18 Apr 2025).

Summary Table: Main Bayesian Continual Learning Frameworks

Approach Type	Representative Methods	Key Features
Regularization-Based	EWC, SI, MAP-Laplace	Quadratic parameter penalties via task importance
Variational Inference	VCL, VGR, ProtoCL	Posterior approximation, prior-to-posterior transfer, replay
Replay-Based	VGR, coresets, GPs, FER	Generative or memory-based likelihood replay
Architecture/Structure	IBP, CLAW, sparse networks	Adaptive structure; allocation via Bayesian nonparametrics
Meta-Learning/Hybrid	SB-MCL, BECAME	Neural meta-learners + exact Bayesian updates; model merging for plasticity—stability trade-off

(Adel, 11 Jul 2025, Farquhar et al., 2019, Kumar et al., 2019, Lee et al., 29 May 2024, Li et al., 3 Apr 2025, Bonnet et al., 18 Apr 2025, Kessler et al., 2023, Gong et al., 2022, Servia-Rodriguez et al., 2021, Yang et al., 2022, Skatchkovsky et al., 2022, Li et al., 2019, Pyla et al., 2023, Chen et al., 2019, Xu et al., 2019, Luo et al., 2019, Kapoor et al., 2020, Foster et al., 2023, Milasheuski et al., 21 Apr 2025)

Conclusion

Bayesian continual learning provides a principled probabilistic framework for online knowledge acquisition, balancing memory retention and adaptability. Through recursive update rules, uncertainty modeling, and a spectrum of algorithmic strategies, BCL addresses the central challenge of catastrophic forgetting and supports robust knowledge transfer in dynamic environments. Ongoing research is directed toward more scalable, adaptive, and biologically inspired variants, as well as broader applicability to realistic continuous learning settings.