Posterior Distribution Of Parameters Research Articles

Estimating the Hawkes process from a discretely observed sample path

Estimating the Hawkes process from a discretely observed sample path is challenging due to the intractability of the likelihood in such cases. To overcome this, we employ a state-space representation of the incomplete data problem and use the sequential Monte Carlo (SMC, aka particle filters) to approximate the likelihood function. The resulting estimator of the likelihood function is unbiased and, therefore, can be used along with the Metropolis-Hastings algorithm to construct Markov Chains to approximate the likelihood distribution and, more generally, the posterior distribution of model parameters. The performance of our methodology is assessed using simulation experiments and compared with other recently published methods. The proposed estimator exhibits a smaller mean square error compared to two benchmark estimators. Furthermore, an advantage of our method compared to existing methods is that confidence intervals for the parameters are readily computable. Finally, we apply the proposed estimator to the analysis of weekly count data on measles cases in Tokyo, Japan, and compare the results to one of the benchmark methods. The online supplementary materials contain a Julia package that implements our methodology, along with the technical proofs for two propositions.

Advanced statistical inference of myocardial stiffness: A time series Gaussian process approach of emulating cardiac mechanics for real-time clinical decision support.

Cardiac mechanics modelling promises to revolutionize personalized health care; however, inferring patient-specific biophysical parameters, which are critical for understanding myocardial functions and performance, poses substantial methodological challenges. Our work is primarily motivated to determine the passive stiffness of the myocardium from the measurement of the left ventricle (LV) volume at various time points, which is crucial for diagnosing cardiac physiological conditions. Although there have been significant advancements in cardiac mechanics modelling, the tasks of inference and uncertainty quantification of myocardial stiffness remain challenging, with high computational costs preventing real-time decision support. We adapt Gaussian processes to construct a statistical surrogate model for emulating LV cavity volume during diastolic filling to overcome this challenge. As the LV volumes, obtained at different time points in diastole, constitute a time series, we apply the Kronecker product trick to decompose the complex covariance matrix of the whole system into two separate covariance matrices, one for time and the other for biophysical parameters. To proceed towards personalized health care, we further integrate patient-specific LV geometries into the Gaussian process emulator using principal component analysis (PCA). Utilizing a deep learning neural network for extracting time-series left ventricle volumes from magnetic resonance images, Bayesian inference is applied to determine the posterior probability distribution of critical cardiac mechanics parameters. Tests on real-patient data illustrate the potential for real-time estimation of myocardial properties for clinical decision-making. These advancements constitute a crucial step towards clinical impact, offering valuable insights into posterior uncertainty quantification for complex cardiac mechanics models.

Improved representation of soil moisture processes through incorporation of cosmic-ray neutron count measurements in a large-scale hydrologic model

Abstract. Profound knowledge of soil moisture and its variability plays a crucial role in hydrological modelling to support agricultural management, flood and drought monitoring and forecasting, and groundwater recharge estimation. Cosmic-ray neutron sensing (CRNS) has been recognised as a promising tool for soil moisture monitoring due to its hectare-scale footprint and decimetre-scale measurement depth. But since CRNS provides an integral measurement over several soil horizons, a direct comparison of observed and simulated soil moisture products is not possible. This study establishes a framework to assess the accuracy of soil moisture simulated by the mesoscale Hydrologic Model (mHM) by generating simulated neutron counts and comparing these with observed neutron measurements for the first time. We included three different approaches to estimate CRNS neutron counts in the mHM as a function of the simulated soil moisture profiles: two methods based on the Desilets equation and one based on the forward operator COSMIC (COsmic-ray Soil Moisture Interaction Code). For the Desilets method, we tested two different approaches to average the vertical soil moisture profiles: a uniform vs. a non-uniform weighting scheme depending on the CRNS measurement depth. The methods were tested at two agricultural sites, namely one pasture site and one forest site, in Germany. To explore the prior and posterior distributions of the mHM parameters when constrained by CRNS observations, we used a Monte Carlo method based on Latin hypercube sampling with a large sample size (S = 100 000). We found that all three methods performed well, with a Kling–Gupta efficiency > 0.75 and a percent bias < ± 10 % across the majority of investigated sites and for the best 1 % of parameter sets. The performance of the neutron forward models varied slightly across different land cover types. The non-uniform approach generally showed good performance, particularly at the agricultural sites. On the other hand, the COSMIC method performed slightly better at the forest site. The uniform approach showed slightly better results at the grassland site. We also demonstrated for the first time that the incorporation of CRNS measurements into the mHM could improve both the soil moisture and the evapotranspiration products of the mHM. This suggests that CRNS is capable of improving the model parameter space in general and adds a broader perspective on the potential of CRNS to support large-scale hydrological and land surface models.

Parameter uncertainties for imperfect surrogate models in the low-noise regime

Abstract Bayesian regression determines model parameters by minimizing the expected loss, an upper bound to the true generalization error. However, this loss ignores model form error, or misspecification, meaning parameter uncertainties are significantly underestimated and vanish in the large data limit. As misspecification is the main source of uncertainty for surrogate models of low-noise calculations, such as those arising in atomistic simulation, predictive uncertainties are systematically underestimated. 
We analyze the true generalization error of misspecified, near-deterministic surrogate models, a regime of broad relevance in science and engineering. We show that posterior parameter distributions must cover every training point to avoid a divergence in the generalization error and design a compatible \textit{ansatz} which incurs minimal overhead for linear models. The approach is demonstrated on model problems before application to thousand-dimensional datasets in atomistic machine learning. Our efficient misspecification-aware scheme gives accurate prediction and bounding of test errors in terms of parameter uncertainties, allowing this important source of uncertainty to be incorporated in multi-scale computational workflows.

Physics-Informed Bayesian Neural Networks for Solving Phonon Boltzmann Transport Equation in Forward and Inverse Problems With Sparse and Noisy Data

Abstract Nondiffusive phonon transport presents significant challenges in micro/nanoscale thermal characterization, compounded by the limitations of experimental-numerical techniques and the presence of measurement noise. Additionally, inverse modeling and uncertainty quantification (UQ) for submicron thermal transport remain under-explored. In this study, we introduce a physics-informed Bayesian deep learning framework designed to address phonon Boltzmann transport equation (BTE)-based forward and inverse problems leveraging limited and noisy data. Our approach combines Bayesian neural networks with a nonparametric variational inference method, formulating the BTE-constrained training in a Bayesian manner. This enables the estimation of the posterior distribution of neural network parameters and unknown equation parameters based on a likelihood function that incorporates uncertainties from both the measurement data and the BTE model. Through numerical experiments on various phonon transport scenarios, we demonstrate that our method can accurately reconstruct temperature and heat flux profiles, infer critical quantities of interest (e.g., Knudsen number), and provide robust uncertainty quantification, even when data is sparse and noisy. This framework enhances our capability to conduct nondiffusive thermal simulations and inverse modeling with quantified uncertainty, offering a powerful tool for advancing thermal transport research and optimization in micro/nanoscale devices.

Implementing a Bayesian approach using Stan with Torsten: Population pharmacokinetics analysis of somatrogon.

Fully Bayesian approaches are not commonly implemented for population pharmacokinetic (PK) modeling. In this paper, we evaluate the use of Stan with R and Torsten for population PK modeling of somatrogon, a recombinant long-acting growth hormone approved for the treatment of growth hormone deficiency. As a software for Bayesian inference, Stan provides an easy way to conduct MCMC sampling for a wide range of models with efficient sampling algorithms, and there are several diagnostic tools to evaluate the MCMC convergence and other potential issues. Three different sets of priors were evaluated for estimation and prediction: a weakly informative uniform set, a moderately informative set, and a very informative set of priors. All three prior sets showed good performance and all chains mixed well. There were some minor differences in the final parameter posterior distributions while implementing different prior sets, but the posterior predictions covered the observations nicely, not only for the individuals included in posterior sampling but also for new individuals. The impact of a centered versus non-centered parameterization were evaluated, with the non-centered approach improving the estimation time, but it was still computationally intensive. Computational resources had the biggest impact on sampling time. Stan took approximately 2.5 h total for the MCMC sampling on a high-performance computing platform (6 cores) and may be reduced further with additional computational resources. The model and comparisons presented show that with adequate computational resources, the Bayesian approaches using Stan and Torsten are useful for population PK analysis, especially for the analysis of special populations, small sample datasets, and when complex model structures are needed.

New constraints on the central mass contents of Omega Centauri from combined stellar kinematics and pulsar timing

We performed a combined analysis of stellar kinematics with line-of-sight accelerations of millisecond pulsars (MSPs) to probe the mass content of Omega Centauri (OC ). Our mass model includes the stellar mass distribution, a more concentrated mass component linked to the observed MSP population, a generic cluster of stellar remnants (assumed to be more concentrated than the stars and MSPs), and an intermediate-mass black hole (IMBH), allowing us to determine which of these is statistically preferred to account for these observations. We mass-modeled OC using the package GravSphere to solve the Jeans equations, including constraints in the form of proper motions, line-of-sight velocities, the surface density profile of the stars, the spatial distribution of MSPs, and the recently measured line-of-sight accelerations of a subset of these MSPs, self-consistently modeling their intrinsic spin-down. We explore the impact of different assumed centers of OC on our results and we infer the posterior distributions of the model parameters from the combined likelihood using the nested sampling package dynesty Our analysis favors an extended central mass of $ over an IMBH, setting a 3sigma upper limit on the IMBH mass of $6 We find that pulsar timing observations are an important additional constraint, favoring a central mass distribution that is $ 20<!PCT!>$ more massive and extended than implied by models that are constrained by the stellar kinematics alone. Finally, we find a 3sigma confidence level (CL) upper bound of $6 on the total mass traced by the MSPs, with the density profile following $ p (r) star (r)^ with $ 0.3,$ where $ star (r)$ is the stellar mass density and $ is the stellar velocity dispersion profile. This favors models in which MSPs form via stellar encounters, as in the leading paradigm whereby MSPs are the progeny of low-mass X-ray binaries. Our analysis demonstrates how combining stellar kinematics with MSP accelerations produces new constraints on mass models, shedding light on the presence or absence of IMBHs at the centers of globular clusters. Further, we provide the first validation of its kind where MSP positions are linked to their place of formation in globular clusters, which is in excellent agreement with the expectations of stellar encounter models of MSP formation. This sets a promising precedent amid the rapid growth in the number of observations and discoveries currently taking place in this field.

Stratified distributional analysis-a novel perspective on RT distributions.

Response times and their distributions serve as a powerful lens into cognitive processes. We present a novel statistical methodology called stratified distributional analysis (SDA) to quantitatively assess how key determinants of response times (word frequency and length) shape their distributions. Taking advantage of the availability of millions of lexical decision response times in the English Lexicon Project and the British Lexicon Project, we made important advances into the theoretical issue of linking response times and word frequency by analysing RT distributions as a function of word frequency and word length. We tested these distributions against the lognormal, Wald, and gamma distributions and three measures of word occurrence (word form frequencies obtained from subtitles and contextual diversity as operationalised as discourse contextual diversity and user contextual diversity). We found that the RT distributions were best described by a lognormal distribution across both megastudies when word occurrence was quantified by a contextual diversity measure. The link between the lognormal distribution and its generative process highlights the power of SDA in elucidating mechanisms that govern the generation of RTs through the fitting of probability distributions. Using a hierarchical Bayesian framework, SDA yielded posterior distributions for the distributional parameters at the single-participant level, enabling probabilistic predictions of response times as a function of word frequency and word length, which has the potential to serve as a diagnostic tool to uncover idiosyncratic features of word processing. Crucially, while we applied our parsimonious methodology to lexical decision response times, it is applicable to a variety of tasks such as word-naming and eye-tracking data.

Spatial and Temporal Bayesian Hierarchical Model Over Large Domains With Application to Holocene Sea Surface Temperature Reconstruction in the Equatorial Pacific

AbstractWe present a novel space‐time Bayesian hierarchical model (BHM) to reconstruct annual Sea Surface Temperature (SST) over a large domain based on SST at limited proxy (i.e., sediment core) locations. The model is tested in the equatorial Pacific. The BHM leverages Principal Component Analysis to identify dominant space‐time modes of contemporary variability of the SST field at the proxy locations and employs these modes in a Gaussian process framework to estimate SSTs across the entire domain. The BHM allows us to model the mean field and covariance, varying in space and time in the process layers of the hierarchy. Using the Markov Chain Monte Carlo (MCMC) method and suitable priors on the model parameters, posterior distributions of the model parameters and, consequently, posterior distributions of the SST fields and the attendant uncertainties are obtained for any desired year. The BHM is calibrated and validated in the contemporary period (1854–2014) and subsequently applied to reconstruct SST fields during the Holocene (0–10 ka). Results are consistent with prior inferences of La Niña‐like conditions during the Holocene. This modeling framework opens exciting prospects for modeling and reconstruction of other fields, such as precipitation, drought indices, and vegetation.

Inferring stellar parameters and their uncertainties from high-resolution spectroscopy using invertible neural networks

Context. New spectroscopic surveys will increase the number of astronomical objects in need of characterisation by more than an order of magnitude. Machine learning tools are required to address this data deluge in a fast and accurate fashion. Most machine learning algorithms cannot directly estimate error, making them unsuitable for reliable science. Aims. We aim to train a supervised deep-learning algorithm tailored for high-resolution observational stellar spectra. This algorithm accurately infers precise estimates while providing coherent estimates of uncertainties by leveraging information from both the neural network and the spectra. Methods. We trained a conditional invertible neural network (cINN) on observational spectroscopic data obtained from the GIRAFFE spectrograph (HR 10 and HR 21 setups) within the Gaia-ESO survey. A key feature of cINN is its ability to produce the Bayesian posterior distribution of parameters for each spectrum. By analysing this distribution, we inferred stellar parameters and their corresponding uncertainties. We carried out several tests to investigate how parameters are inferred and errors are estimated. Results. We achieved an accuracy of 28 K in Teff, 0.06 dex in log ɡ, 0.03 dex in [Fe/H], and between 0.05 dex and 0.17 dex for the other abundances for high-quality spectra. Accuracy remains stable with low signal-to-noise ratio (between 5 and 25) spectra, with an accuracy of 39 K in Teff, 0.08 dex in log ɡ, and 0.05 dex in [Fe/H]. The uncertainties obtained are well within the same order of magnitude. The network accurately reproduces astrophysical relationships both on the scale of the Milky Way and within smaller star clusters. We created a table containing the new parameters generated by our cINN. Conclusions. This neural network represents a compelling proposition for future astronomical surveys. These derived uncertainties are coherent and can therefore be reused in future works as Bayesian priors.

Dark Matter Halos of Luminous Active Galactic Nuclei from Galaxy–Galaxy Lensing with the HSC Subaru Strategic Program

We assess the dark matter halo masses of luminous active galactic nuclei (AGNs) over the redshift range 0.2–1.2 using galaxy–galaxy lensing based on imaging data from the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP). We measure the weak lensing signal of a sample of 48,907 AGNs constructed using HSC and Wide-field Infrared Survey Explorer photometry. As expected, we find that the lensing mass profile of total AGN sample is consistent with that of massive galaxies ( log(M*/h−2M⊙)∼ 10.61). Surprisingly, the lensing signal remains unchanged when the AGN sample is split into four host galaxy stellar mass bins. Specifically, we find that the excess surface density of AGNs residing in galaxies with high stellar masses significantly differs from that of the control sample. We further fit a halo occupation distribution model to the data to infer the posterior distribution of parameters including the average halo mass. We find that the characteristic halo mass of the full AGN population lies near the knee ( log(Mh/h−1M⊙)=12.0 ) of the stellar-to-halo mass relation (SHMR). Illustrative of the results given above, the halo masses of AGNs residing in host galaxies with high stellar masses (i.e., above the knee of the SHMR) fall below the calibrated SHMR while the halo masses of the low stellar mass sample are more consistent with the established SHMR. These results indicate that massive halos with a higher clustering bias tend to suppress AGN activity, probably due to the lack of available gas.

Analysis of Upstream, Downstream, and Common Firm Shocks Using a Large Factor‐Augmented Vector Autoregressive Approach

ABSTRACTWe evaluate the roles of upstream (supplier‐to‐user), downstream (user‐to‐supplier), and common factor shock transmission across firms by deriving interfirm networks and common factors from US equities over 1989–2017. We overcome the econometric challenges of estimating the large factor‐augmented vector autoregressive (FAVAR) system by developing two alternative approaches: one prioritizing computational efficiency and the other providing the full posterior distribution of all model parameters and factors. We find that (i) common factors drive an increasing variance share of returns, (ii) supplier shocks are more evident in equity price movements than downstream exposures, and (iii) removing the impact of common factors is increasingly important for revealing interfirm connections.

Introducing µGUIDE for quantitative imaging via generalized uncertainty-driven inference using deep learning

This work proposes µGUIDE: a general Bayesian framework to estimate posterior distributions of tissue microstructure parameters from any given biophysical model or signal representation, with exemplar demonstration in diffusion-weighted magnetic resonance imaging. Harnessing a new deep learning architecture for automatic signal feature selection combined with simulation-based inference and efficient sampling of the posterior distributions, µGUIDE bypasses the high computational and time cost of conventional Bayesian approaches and does not rely on acquisition constraints to define model-specific summary statistics. The obtained posterior distributions allow to highlight degeneracies present in the model definition and quantify the uncertainty and ambiguity of the estimated parameters.

Introducing µGUIDE for quantitative imaging via generalized uncertainty-driven inference using deep learning.

This work proposes µGUIDE: a general Bayesian framework to estimate posterior distributions of tissue microstructure parameters from any given biophysical model or signal representation, with exemplar demonstration in diffusion-weighted magnetic resonance imaging. Harnessing a new deep learning architecture for automatic signal feature selection combined with simulation-based inference and efficient sampling of the posterior distributions, µGUIDE bypasses the high computational and time cost of conventional Bayesian approaches and does not rely on acquisition constraints to define model-specific summary statistics. The obtained posterior distributions allow to highlight degeneracies present in the model definition and quantify the uncertainty and ambiguity of the estimated parameters.

Parameter estimation for allometric trophic network models: A variational Bayesian inverse problem approach

Abstract Differential equation models are powerful tools for predicting biological systems, capable of projecting far into the future and incorporating data recorded at arbitrary times. However, estimating these models' parameters from observations can be challenging because numerical methods are often required to approximate their solution. An example of such a model is the allometric trophic network model, for which studies considering its inverse problem are limited, particularly in the Bayesian framework. Here we develop a variational Bayesian method for parameter inference of the allometric trophic network model and explore how accurately we can recover its parameter values. We represent the model as a Bayesian neural network, which combines an artificial neural network with Bayesian inference, using a surrogate for the posterior distribution of model parameters, and train this model by evolutionary optimization to avoid potentially costly computation of the gradient with respect to the model parameters. Using synthetic data, we compare the accuracy of this variational inference to ordinary least squares estimation. To reduce the number of estimated parameters, we focus on the inference of functional response parameters. Our variational Bayesian method yields parameter estimates that are comparable to the ordinary least squares results in terms of accuracy. The method provides a promising approach for including uncertainty quantification in parameter estimation, which the simple ordinary least squares approach as it is does not address. Regardless of the method, potential multimodality of the inference problem is nonetheless important to keep in mind. The present study suggests a technique for parameter inference of ordinary differential equation models in the Bayesian context. We propose the method especially for validation of the allometric trophic network model against empirical data.

Stochastic nonlinear model updating in structural dynamics using a novel likelihood function within the Bayesian-MCMC framework

The study presents a novel approach for stochastic nonlinear model updating in structural dynamics, employing a Bayesian framework integrated with Markov Chain Monte Carlo (MCMC) sampling for parameter estimation by using an approximated likelihood function. The proposed methodology is applied to both numerical and experimental cases. The paper commences by introducing Bayesian inference and its constituents: the likelihood function, prior distribution, and posterior distribution. The resonant decay method is employed to extract backbone curves, which capture the non-linear behaviour of the system. A mathematical model based on a single degree of freedom (SDOF) system is formulated, and backbone curves are obtained from time response data. Subsequently, MCMC sampling is employed to estimate the parameters using both numerical and experimental data. The obtained results demonstrate the convergence of the Markov chain, present parameter trace plots, and provide estimates of posterior distributions of updated parameters along with their uncertainties. Experimental validation is performed on a cantilever beam system equipped with permanent magnets and electromagnets. The proposed methodology demonstrates promising results in estimating parameters of stochastic non-linear dynamical systems. Through the use of the proposed likelihood functions using backbone curves, the probability distributions of both linear and non-linear parameters are simultaneously identified. Based on this view, the necessity to segregate stochastic linear and non-linear model updating is eliminated.

Euclid preparation

LENSMC is a weak lensing shear measurement method developed for Euclid and Stage-IV surveys. It is based on forward modelling in order to deal with convolution by a point spread function (PSF) with comparable size to many galaxies, sampling the posterior distribution of galaxy parameters via Markov chain Monte Carlo, and marginalisation over nuisance parameters for each of the 1.5 billion galaxies observed by Euclid. We quantified the scientific performance through high-fidelity images based on the Euclid Flagship simulations and emulation of the Euclid VIS images, realistic clustering with a mean surface number density of 250 arcmin−2 (IE < 29.5) for galaxies, and 6 arcmin−2 (IE < 26) for stars, and a diffraction-limited chromatic PSF with a full width at half maximum of 0′.′2 and spatial variation across the field of view. LENSMC measured objects with a density of 90 arcmin−2 (IE < 26.5) in 4500 deg2. The total shear bias was broken down into measurement (our main focus here) and selection effects (which will be addressed in future work). We found measurement multiplicative and additive biases of m1 = (−3.6 ± 0.2) × 10−3, m2 = (−4.3 ± 0.2) × 10−3, c1 = (−1.78 ± 0.03) × 10−4, and c2 = (0.09 ± 0.03) × 10−4; a large detection bias with a multiplicative component of 1.2 × 10−2 and an additive component of −3 × 10−4; and a measurement PSF leakage of α1 = (−9 ± 3) × 10−4 and α2 = (2 ± 3) × 10−4. When model bias is suppressed, the obtained measurement biases are close to Euclid requirement and largely dominated by undetected faint galaxies (−5 × 10−3). Although significant, model bias will be straightforward to calibrate given its weak sensitivity on galaxy morphology parameters. LENSMC is publicly available at gitlab.com/gcongedo/LensMC.

Fast likelihood-free inference in the LSS Stage IV era

Forthcoming large-scale structure (LSS) Stage IV surveys will provide us with unprecedented data to probe the nature of dark matter and dark energy.However, analysing these data with conventional Markov Chain Monte Carlo (MCMC) methods will be challenging, due to the increase in the number of nuisance parameters and the presence of intractable likelihoods. In light of this, we present the first application of Marginal Neural Ratio Estimation (MNRE) (a recent approach in simulation-based inference) to LSS photometric probes: weak lensing, galaxy clustering and the cross-correlation power spectra. In order to analyse the hundreds of spectra simultaneously, we find that a pre-compression of data using principal component analysis, as well as parameter-specific data summaries lead to highly accurate results. Using expected Stage IV experimental noise, we are able to recover the posterior distribution for the cosmological parameters with a speedup factor of ∼ 10-60 compared to classical MCMC methods. To illustrate that the performance of MNRE is not impeded when posteriors are significantly non-Gaussian, we test a scenario of two-body decaying dark matter, finding that Stage IV surveys can improve current bounds on the model by up to one order of magnitude. This result supports that MNRE is a powerful framework to constrain the standard cosmological model and its extensions with next-generation LSS surveys.

Can the young water fraction reduce predictive uncertainty in water transit time estimations?

Transit time distributions (TTDs) of streamflow are informative descriptors of catchment hydrological functioning and solute transport mechanisms. Conventional methods for estimating TTDs generally require model calibration against extensive tracer data time series, which are often limited to well-studied experimental catchments. We challenge this limitation and propose an alternative approach that uses the young water fraction (Fywobs), an increasingly used water age metric which represents the proportion of streamflow with a transit time younger than 2–3 months, and that can be robustly estimated with sparsely measured tracer data. To this end, we conducted a proof of concept study by modeling TTDs using StorAge Selection (SAS) functions with oxygen isotopes (δ18O) measurements for 23 diverse catchments in Germany. In a Monte-Carlo approach, we computed the (averaged) marginal TTDs of a prior parameter distribution and derived a model-based Fyw (Fywsim). We compared Fywsim with Fywobs, obtained from δ18O measurements, and constrained the prior SAS parameters distribution. Subsequently, we derived a posterior distribution of parameters and resulting model simulations. Our findings showed that using Fywobs to constrain the model effectively reduced parameter equifinality and simulation uncertainty. However, the value of Fywobs on reducing model uncertainty varied across sites, with larger values (Fywobs≥0.10) leading to simulations with a narrower uncertainty band and higher model efficiency, whilst smaller values (Fywobs≤0.05) had limited influence on reducing model output uncertainty. We discussed the potential and limitations of combining SAS functions with Fywobs, and considered broader implications of this approach for enhancing our understanding of catchment functioning and water quality status.

Bayesian estimation of stochastic volatility jump diffusion model parameters using S&P 500 and VIX data

The Stochastic Volatility Jump Diffusion (SVJD) model is a common tool used for option pricing. Existing literature has employed many techniques in order to estimate the parameters of this model, including both Bayesian and frequentist approaches. In 2010, Duan and Yeh [Jump and volatility risk premiums implied by VIX. J Econ Dyn Control. 2010;34(11):2232–2244] used a frequentist approach with maximum likelihood estimation that included the VIX, the Chicago Board Options Exchange (CBOE) Volatility Index, which carries real time S&P 500 option price information in addition to the S&P 500 data. In this paper, we combine the previous Bayesian approach in Cape et al. [Estimating Heston's and Bates' models parameters using Markov Chain Monte Carlo simulation. J Stat Comput Simul. 2015;85(11):2295–2314] with the inclusion of VIX by Duan and Yeh. We provide the derivations of the conditional posterior distributions of the SVJD model's parameters and a guide to a Markov Chain Monte Carlo (MCMC) algorithm used to estimate the parameters. We apply the algorithm to the historical S&P 500 and VIX data and provide the subsequent parameter estimates. We offer these tools for others performing similar research with our R code available upon request.