Modeling wildfire activity is crucial for informing science-based risk management and understanding the spatiotemporal dynamics of fire-prone ecosystems worldwide. Models help disentangle the relative influences of different factors, understand wildfire predictability, and provide insights into specific events. Here, we develop Firelihood, a two-component, Bayesian, hierarchically structured, probabilistic model of daily fire activity, which is modeled as the outcome of a marked point process: individual fires are the points (occurrence component), and fire sizes are the marks (size component). The space-time Poisson model for occurrence is adjusted to gridded fire counts using the integrated nested Laplace approximation (INLA) combined with the stochastic partial differential equation (SPDE) approach. The size model is based on piecewise-estimated Pareto and generalized Pareto distributions, adjusted with INLA. The Fire Weather Index (FWI) and forest area are the main explanatory variables. Temporal and spatial residuals are included to improve the consistency of the relationship between weather and fire occurrence. The posterior distribution of the Bayesian model provided 1,000 replications of fire activity that were compared with observations at various temporal and spatial scales in Mediterranean France. The number of fires larger than 1 ha across the region was coarsely reproduced at the daily scale, and was more accurately predicted on a weekly basis or longer. The regional weekly total number of larger fires (10-100 ha) was predicted as well, but the accuracy degraded with size, as the model uncertainty increased with event rareness. Local predictions of fire numbers or burned areas also required a longer aggregation period to maintain model accuracy. The estimation of fires larger than 1 ha was also consistent with observations during the extreme fire season of the 2003 unprecedented heat wave, but the model systematically underrepresented large fires and burned areas, which suggests that the FWI does not consistently rate the actual danger of large fire occurrence during heat waves. Firelihood enabled a novel analysis of the stochasticity underlying fire hazard, and offers a variety of applications, including fire hazard predictions for management and projections in the context of climate change.