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Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.

Availability of insufficient data is a frequent issue resulting in the inaccurate probabilistic characterization of properties and, finally the inaccurate reliability estimates of rock structures. This study presents a Bayesian multi-model inference methodology which couples multi-model inference with traditional Bayesian approach to characterize uncertainties in both—(1) probability models, and (2) model parameters of rock properties arising due to insufficient data, and to estimate the reliability of rock slopes and tunnels considering their effect. Further, this methodology was coupled with Sobol’s sensitivity, metropolis–hastings Markov chain Monte Carlo sampling and moving least square-response surface method to improve the computational efficiency and applicability for problems with implicit performance functions (PFs). Methodology is demonstrated for a Himalayan rock slope (implicit PF) prone to stress-controlled failure in India. Analysis is also performed using recently developed limited data reliability methods, i.e., traditional Bayesian (considers uncertainty in model parameters only) and bootstrap-based re-sampling reliability methods (considers uncertainties in model types and parameters). Proposed methodology is concluded to be superior to other methods due to its capability of considering uncertainties in both model types and parameters, and to include the prior information in the analysis.

For residues maintained on the soil surface, microbial colonization of the substrate is slower initially due to the microbial population's adaptation phase to the substrate. Subsequently, decomposition becomes more intense due to easily mineralizable matter, and as the process progresses, there is a predominance of more resistant materials that can reduce microbial attack. Thus, the maximum rate of CO2 release occurs in the early days of decomposition, and this process is described by the Gompertz model, which is a nonlinear sigmoidal regression model. The theory for regression models is valid for sufficiently large samples, and generally, in research with carbon mineralization data, few observations are used, and parameter estimation should preferably be done using Bayesian methodology since prior information is incorporated, reducing the effect of having few observations. One way to determine objective priors is through maximum entropy prior distributions. This study aims to fit the Gompertz model to the release of carbon dioxide over time from leguminous species using a Bayesian approach with maximum entropy priors for the model parameters. The treatments (leguminous species) evaluated were Arachis pintoi, Calopogonium mucunoides, Stylosanthes guianensis, and Stizolobium aterrium. Eight observations of carbon released over time up to 480 hours from the start of incubation were made. In the soil with the addition of legumes, the abscissa of the inflection point was estimated between 4 and 5 days, meaning this was the time the microorganisms needed to reach the maximum decomposition rate. The species A. pintoi showed the average estimate of 481 mg CO2 of potentially mineralizable carbon, being the species that released the most carbon.

Plastid matK gene sequences for 374 genera representing all angiosperm orders and 12 genera of gymnosperms were analyzed using parsimony (MP) and Bayesian inference (BI) approaches. Traditionally, slowly evolving genomic regions have been preferred for deep-level phylogenetic inference in angiosperms. The matK gene evolves approximately three times faster than the widely used plastid genes rbcL and atpB. The MP and BI trees are highly congruent. The robustness of the strict consensus tree supercedes all individual gene analyses and is comparable only to multigene-based phylogenies. Of the 385 nodes resolved, 79% are supported by high jackknife values, averaging 88%. Amborella is sister to the remaining angiosperms, followed by a grade of Nymphaeaceae and Austrobaileyales. Bayesian inference resolves Amborella + Nymphaeaceae as sister to the rest, but with weak (0.42) posterior probability. The MP analysis shows a trichotomy sister to the Austrobaileyales representing eumagnoliids, monocots + Chloranthales, and Ceratophyllum + eudicots. The matK gene produces the highest internal support yet for basal eudicots and, within core eudicots, resolves a crown group comprising Berberidopsidaceae/Aextoxicaceae, Santalales, and Caryophyllales + asterids. Moreover, matK sequences provide good resolution within many angiosperm orders. Combined analyses of matK and other rapidly evolving DNA regions with available multigene data sets have strong potential to enhance resolution and internal support in deep level angiosperm phylogenetics and provide additional insights into angiosperm evolution.

The dynamics of organic waste decomposition in the soil can be described by nonlinear regression models, however, the theory for regression models is valid for sufficiently large samples, and in general, in small samples, these properties are unknown. One of the methods for data analysis that has been widely used to overcome this problem is the bayesian inference, as it has the advantage of being able to work with small samples, in addition to allowing the incorporation of information from previous studies, and even having a probability distribution for the parameters, consequently, to present a direct interpretation for the credibility interval. However, criticism has been made because of the effect that a prior subjective distribution can have on posterior distribution. One way of determining objective prior is through of maximum entropy prior distributions. For data of organic waste decomposition in the soil, little is known about the probability distributions of the parameters. The present study aimed to use of maximum entropy prior distributions to the parameters of the Stanford & Smith nonlinear model. In addition, using simulated data, to understand the effect that hyperparameters of prior distribution has on the posterior curve, and also to apply the methodology in the description of CO2 mineralization data from swine manure applied to the soil surface. Data analyzed came from an experiment conducted in a laboratory that evaluated the carbon mineralization of swine manure on the soil surface over time. The posterior distributions were obtained, so the bayesian methodology with maximum entropy prior was efficient in the study of the Stanford & Smith nonlinear model to the data of carbon mineralization of swine manure on the soil surface.

On the Moon, in the near infrared wavelength range, spectral diagnostic features such as the 1-μm and 2-μm absorption bands can be used to estimate abundances of the constituent minerals. However, there are several factors that can darken the overall spectrum and dampen the absorption bands. Namely, (1) space weathering, (2) grain size, (3) porosity, and (4) mineral darkening agents such as ilmenite have similar effects on the measured spectrum. This makes spectral unmixing on the Moon a particularly challenging task. Here, we try to model the influence of space weathering and mineral darkening agents and infer the uncertainties introduced by these factors using a Markov Chain Monte Carlo method. Laboratory and synthetic mixtures can successfully be characterized by this approach. We find that the abundance of ilmenite, plagioclase, clino-pyroxenes and olivine cannot be inferred accurately without additional knowledge for very mature spectra. The Bayesian approach to spectral unmixing enables us to include prior knowledge in the problem without imposing hard constraints. Other data sources, such as gamma-ray spectroscopy, can contribute valuable information about the elemental abundances. We here find that setting a prior on TiO2 and Al2O3 can mitigate many of the uncertainties, but large uncertainties still remain for dark mature lunar spectra. This illustrates that spectral unmixing on the Moon is an ill posed problem and that probabilistic methods are important tools that provide information about the uncertainties, that, in turn, help to interpret the results and their reliability.

Spatial cluster detection is one of the focus areas of spatial analysis, whose objective is the identification of clusters from spatial distributions of point events aggregated in districts with small areas. Choi et al. (2018) formulated cluster detection as a parameter estimation problem to leverage the parameter selection capability of the sparse modeling method called the generalized fused lasso. Although this work is superior to conventional methods for detecting multiple clusters, its estimation results are limited to point estimates. This study therefore extended the above work as a Bayesian cluster detection method to describe the probabilistic variations of clustering results. The proposed method combines multiple sparsity-inducing priors and encourages sparse solutions induced by the generalized fused lasso. Evaluations were performed with simulated and real-world distributions of point events to demonstrate that the proposed method provides new information on the quantified reliabilities of clustering results at the district level while achieving comparable detection performances to that of the previous work.

Zinc uptake is essential for crop development; thus, knowledge about soil zinc availability is fundamental for fertilization in periods of higher crop demand. A nonlinear first-order kinetic model has been employed to evaluate zinc availability. Studies usually employ few observations; however, inference in nonlinear models is only valid for sufficiently large samples. An alternative is the Bayesian method, where inferences are made in terms of probability, which is effective even with small samples. The aim of this study was to use Bayesian methodology to evaluate the fitness of a nonlinear first-order kinetic model to describe zinc extraction from soil with sewage sludge using seven different extraction solutions. The analysed data were obtained from an experiment using a completely randomized design and three replicates. Fifteen zinc extractions were evaluated for each extraction solution. Posterior distributions of a study that evaluated the nonlinear first-order kinetic model were used as prior distributions in the present study. Using the full conditionals, samples of posterior marginal distributions were generated using the Gibbs sampler and Metropolis-Hastings algorithms and implemented in R. The Bayesian method allowed the use of posterior distributions of another study that evaluated the model used as prior distributions for parameters  in the present study. The posterior full conditional distributions for the parameters  were normal distributions and gamma distributions, respectively. The Bayesian method was efficient for the study of the first-order kinetic model to describe zinc extraction from soil with sewage sludge using seven extraction solutions.

The last decade witnessed a growing interest in Bayesian learning. Yet, the technicality of the topic and the multitude of ingredients involved therein, besides the complexity of turning theory into practical implementations, limit the use of the Bayesian learning paradigm, preventing its widespread adoption across different fields and applications. This self-contained survey engages and introduces readers to the principles and algorithms of Bayesian Learning for Neural Networks. It provides an introduction to the topic from an accessible, practical-algorithmic perspective. Upon providing a general introduction to Bayesian Neural Networks, we discuss and present both standard and recent approaches for Bayesian inference, with an emphasis on solutions relying on Variational Inference and the use of Natural gradients. We also discuss the use of manifold optimization as a state-of-the-art approach to Bayesian learning. We examine the characteristic properties of all the discussed methods, and provide pseudo-codes for their implementation, paying attention to practical aspects, such as the computation of the gradients.

The Lévy walk is a pattern that is often seen in the movement of living organisms; it has both ballistic and random features and is a behavior that has been recognized in various animals and unicellular organisms, such as amoebae, in recent years. We proposed an amoeba locomotion model that implements Bayesian and inverse Bayesian inference as a Lévy walk algorithm that balances exploration and exploitation, and through a comparison with general random walks, we confirmed its effectiveness. While Bayesian inference is expressed only by P(h) = P(h|d), we introduce inverse Bayesian inference expressed as P(d|h) = P(d) in a symmetry fashion. That symmetry contributes to balancing contracting and expanding the probability space. Additionally, the conditions of various environments were set, and experimental results were obtained that corresponded to changes in gait patterns with respect to changes in the conditions of actual metastatic cancer cells.