Meta-learning aims to leverage prior knowledge from related tasks to enable a base learner to quickly adapt to new tasks with limited labeled samples. However, traditional meta-learning methods have limitations as they provide an optimal initialization for all new tasks, disregarding the inherent uncertainty induced by few-shot tasks and impeding task-specific self-adaptation initialization. In response to this challenge, this article proposes a novel probabilistic meta-learning approach called prototype Bayesian meta-learning (PBML). PBML focuses on meta-learning variational posteriors within a Bayesian framework, guided by prototype-conditioned prior information. Specifically, to capture model uncertainty, PBML treats both meta- and task-specific parameters as random variables and integrates their posterior estimates into hierarchical Bayesian modeling through variational inference (VI). During model inference, PBML employs Laplacian estimation to approximate the integral term over the likelihood loss, deriving a rigorous upper-bound for generalization errors. To enhance the model's expressiveness and enable task-specific adaptive initialization, PBML proposes a data-driven approach to model the task-specific variational posteriors. This is achieved by designing a generative model structure that incorporates prototype-conditioned task-dependent priors into the random generation of task-specific variational posteriors. Additionally, by performing latent embedding optimization, PBML decouples the gradient-based meta-learning from the high-dimensional variational parameter space. Experimental results on benchmark datasets for few-shot image classification illustrate that PBML attains state-of-the-art or competitive performance when compared to other related works. Versatility studies demonstrate the adaptability and applicability of PBML in addressing diverse and challenging few-shot tasks. Furthermore, ablation studies validate the performance gains attributed to the inference and model components.