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Bayesian inference gets its name fromBayes’s theorem, expressing posterior probabilities for hypotheses about a data generating process as the (normalized) product of prior probabilities and a likelihood function. But Bayesian inference uses all of probability theory, not just Bayes’s theorem. Many hypotheses of scientific interest arecomposite hypotheses, with the strength of evidence for the hypothesis dependent on knowledge about auxiliary factors, such as the values of nuisance parameters (e.g., uncertain background rates or calibration factors). Many important capabilities of Bayesian methods arise from use of the law of total probability, which instructs analysts to compute probabilities for composite hypotheses bymarginalizationover auxiliary factors. This tutorial targets relative newcomers to Bayesian inference, aiming to complement tutorials that focus on Bayes’s theorem and how priors modulate likelihoods. The emphasis here is on marginalization over parameter spaces—both how it is the foundation for important capabilities, and how it may motivate caution when parameter spaces are large. Topics covered include the difference between likelihood and probability, understanding the impact of priors beyond merely shifting the maximum likelihood estimate, and the role of marginalization in accounting for uncertainty in nuisance parameters, systematic error, and model misspecification. 

Abstract Aluto volcano (Central Ethiopia) displays a complex, hybrid topography, combining elements typical of caldera systems (e.g., a central, flat caldera floor) and stratovolcanoes (e.g., relatively high and steep, radial flanks, related to eruptions occurring clustered in space). The most recent known eruptions at Aluto have commonly generated column‐collapse pyroclastic density currents (PDCs), a hazardous phenomenon that can pose a significant risk to inhabited areas on and around the volcano. In order to analyze and quantify the role that Aluto's complex topography has on PDC hazard, we apply a versatile probabilistic strategy, which merges the TITAN2D model for PDCs with a novel zero‐censored Gaussian Process (zGP) emulator, enabling robust uncertainty quantification at tractable computational costs. Results from our analyses indicate a critical role of the eruptive vent location, but also highlight a complex interplay between the topography and PDC volume and mobility. The relative importance of each factor reciprocally depends on the other factors. Thus, large PDCs (≥0.1–0.5 km³) can diminish the influence of topography over proximal regions of flow propagation, but PDCs respond to large‐ and small‐scale topographic features over medial to distal areas, and the zGP captures processes like PDC channelization and overbanking. The novel zGP can be applied to other PDC models and can enable specific investigations of PDC dynamics, topographic interactions, and PDC hazard at many volcanic systems worldwide. Potentially, it could also be used during volcanic crises, when time constraints usually only permit computation of scenario‐based hazard assessments. 

Effective volcanic hazard management in regions where populations live in close proximity to persistent volcanic activity involves understanding the dynamic nature of hazards, and associated risk. Emphasis until now has been placed on identification and forecasting of the escalation phase of activity, in order to provide adequate warning of what might be to come. However, understanding eruption hiatus and post-eruption unrest hazards, or how to quantify residual hazard after the end of an eruption, is also important and often key to timely post-eruption recovery. Unfortunately, in many cases when the level of activity lessens, the hazards, although reduced, do not necessarily cease altogether. This is due to both the imprecise nature of determination of the “end” of an eruptive phase as well as to the possibility that post-eruption hazardous processes may continue to occur. An example of the latter is continued dome collapse hazard from lava domes which have ceased to grow, or sector collapse of parts of volcanic edifices, including lava dome complexes. We present a new probabilistic model for forecasting pyroclastic density currents (PDCs) from lava dome collapse that takes into account the heavy-tailed distribution of the lengths of eruptive phases, the periods of quiescence, and the forecast window of interest. In the hazard analysis, we also consider probabilistic scenario models describing the flow’s volume and initial direction. Further, with the use of statistical emulators, we combine these models with physics-based simulations of PDCs at Soufrière Hills Volcano to produce a series of probabilistic hazard maps for flow inundation over 5, 10, and 20 year periods. The development and application of this assessment approach is the first of its kind for the quantification of periods of diminished volcanic activity. As such, it offers evidence-based guidance for dome collapse hazards that can be used to inform decision-making around provisions of access and reoccupation in areas around volcanoes that are becoming less active over time.