Cette thèse présente des contributions à la littérature des méthodes de Monte Carlo utilisé dans l'analyse des modèles complexes en statistique Bayésienne; l'accent est mis à la fois sur la complexité des modèles et sur les difficultés de calcul.Le premier chapitre élargit Delayed Acceptance, une variante computationellement efficace du Metropolis--Hastings, et agrandit son cadre théorique fournissant une justification adéquate pour la méthode, des limits pour sa variance asymptotique par rapport au Metropolis--Hastings et des idées pour le réglage optimal de sa distribution instrumentale.Nous allons ensuite développer une méthode Bayésienne pour analyser les processus environnementaux non stationnaires, appelées Expansion Dimension, qui considère le processus observé comme une projection depuis une dimension supérieure, où l'hypothèse de stationnarité pourrait etre acceptée. Le dernier chapitre sera finalement consacrée à l'étude des structures de dépendances conditionnelles par une formulation entièrement Bayésienne du modèle de Copule Gaussien graphique. / This thesis presents contributions to the Monte Carlo literature aimed toward the analysis of complex models in Bayesian Statistics; the focus is on both complexity related to complicate models and computational difficulties.We will first expand Delayed Acceptance, a computationally efficient variant ofMetropolis--Hastings, to a multi-step procedure and enlarge its theoretical background, providing proper justification for the method, asymptotic variance bounds relative to its parent MH kernel and optimal tuning for the scale of its proposal.We will then develop a flexible Bayesian method to analyse nonlinear environmentalprocesses, called Dimension Expansion, that essentially consider the observed process as a projection from a higher dimension, where the assumption of stationarity could hold.The last chapter will finally be dedicated to the investigation of conditional (in)dependence structures via a fully Bayesian formulation of the Gaussian Copula graphical model.

Filtering for Stochastic Evolution Equations Vít Kubelka Doctoral thesis Abstract Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain and to delayed SPDEs driven by white noise. Subsequently, the continuous dependence of the filter and observation error on parameters which may be present both in the signal and the obser- vation process is proved. These results are applied to signals governed by stochastic heat equations driven by distributed or pointwise fractional noise. The observation process may be a noisy observation of the signal at given points in the domain, the position of which may depend on the parameter. 1

Ce mémoire par article part de la question suivante: pouvons-nous utiliser des prothèses neurales afin d’activer artificiellement certain muscles dans le but d’accélérer la guérison et le réapprentissage du contrôle moteur après un AVC ou un traumatisme cervical ? Cette question touche plus de 15 millions de personnes chaque année à travers le monde, et est au coeur de la recherche de Numa Dancause et Marco Bonizzato, nos collaborateurs dans le département de Neuroscience de l’Université de Montréal. Il est maintenant possible d’implanter des électrodes à grande capacité dans le cortex dans le but d’acheminer des signaux électriques, mais encore difficile de prédire l’effet de stimulations sur le cerveau et le reste du corps. Cependant, des résultats préliminaires prometteurs sur des rats et singes démontrent qu’une récupération motrice non-négligeable est observée après stimulation de régions encore fonctionnelles du cortex moteur. Les difficultés rattachées à l’implémentation optimale de stimulation motocorticale consistent donc à trouver une de ces régions, ainsi qu’un protocole de stimulation efficace à la récupération. Bien que cette optimisation a été jusqu’à présent faite à la main, l’émergence d’implants capables de livrer des signaux sur plusieurs sites et avec plusieurs patrons spatio-temporels rendent l’exploration manuelle et exhaustive impossible. Une approche prometteuse afin d’automatiser et optimiser ce processus est d’utiliser un algorithme d’exploration bayésienne. Mon travail a été de déveloper et de raffiner ces techniques avec comme objectif de répondre aux deux questions scientifiques importantes suivantes: (1) comment évoquer des mouvements complexes en enchainant des microstimulations corticales ?, et (2) peuvent-elles avoir des effets plus significatifs que des stimulations simples sur la récupération motrice? Nous présentons dans l’article de ce mémoire notre approche hiérarchique utilisant des processus gaussiens pour exploiter les propriétés connues du cerveau afin d’accélérer la recherche, ainsi que nos premiers résultats répondant à la question 1. Nous laissons pour des travaux futur une réponse définitive à la deuxième question. / The idea for this thesis by article sprung from the following question: can we use neural prostheses to stimulate specific muscles in order to help recovery of motor control after stroke or cervical injury? This question is of crucial importance to 15 million people each year around the globe, and is at the heart of Numa Dancause and Marco Bonizzato’s research, our collaborators in the Neuroscience department at the University of Montreal. It is now possible to implant large capacity electrodes for electrical stimulation in cortex, but still difficult to predict their effect on the brain and the rest of the body. Nevertheless, preliminary but promising results on rats and monkeys have shown that a non-negligible motor recovery is obtained after stimulation of regions of motor cortex that are still functional. The difficulties related to optimal microcortical stimulation hence consist in finding both one of these regions, and a stimulation protocol with optimal recovery efficacy. This search has up to present day been performed by hand, but recent and upcoming large scale stimulation technologies permitting delivery of spatio-temporal signals are making such exhaustive searches impossible.A promising approach to automating and optimizing this discovery is the use of Bayesian optimization. My work has consisted in developing and refining such techniques with two scientific questions in mind: (1) how can we evoke complex movements by chaining cortical microstimulations?, and (2) can these outperform single channel stimulations in terms of recovery efficacy? We present in the main article of this thesis our hierarchical Bayesian optimization approach which uses gaussian processes to exploit known properties of the brain to speed up the search, as well as first results answering question 1. We leave to future work a definitive answer to the second question.

Electrocardiography (ECG) is a non-invasive method used in medicine to track the electrical pulses sent by the heart. The time between two subsequent electrical impulses and hence the heartbeat of a subject, is referred to as an RR interval. Previous studies show that RR intervals can be used for identifying sleep patterns and cardiovascular diseases. Additional research indicates that RR intervals can be used to predict the cardiovascular age of a subject. This thesis investigates, if this assumption is true, based on two different datasets as well as simulated data based on Gaussian Processes. The datasets used are Holter recordings provided by the University of Gdańsk as well as a dataset provided by Physionet. The former represents a balanced dataset of recordings during nocturnal sleep of healthy subjects whereas the latter one describes an imbalanced dataset of records of a whole day of subjects that suffered from myocardial infarction. Feature-based models as well as a deep learning architecture called DeepSleep, based on a paper for sleep stage detection, are trained. The results show, that the prediction of a subject's age, only based in RR intervals, is difficult. For the first dataset, the highest obtained test accuracy is 37.84 per cent, with a baseline of 18.23 per cent. For the second dataset, the highest obtained accuracy is 42.58 per cent with a baseline of 39.14 per cent. Furthermore, data is simulated by fitting Gaussian Processes to the first dataset and following a Bayesian approach by assuming a distribution for all hyperparameters of the kernel function in use. The distributions for the hyperparameters are continuously updated by fitting a Gaussian Process to a slices of around 2.5 minutes. Then, samples from the fitted Gaussian Process are taken as simulated data, handling impurity and padding. The results show that the highest accuracy achieved is 31.12 per cent with a baseline of 18.23 per cent. Concludingly, cardiovascular age prediction based on RR intervals is a difficult problem and complex handling of impurity does not necessarily improve the results.

To address inefficiency, waste, and the negative consequences of electricity generation, companies and government entities are looking to behavioural change among residential consumers. To drive behavioural change, consumers need better feedback about their electricity consumption. A monthly or quarterly bill provides the consumer with almost no useful information about the relationship between their behaviours and their electricity consumption. Smart meters are now widely dispersed in developed countries and they are capable of providing electricity consumption readings at an hourly resolution, but this data is mostly used as a basis for billing and not as a tool to assist the consumer in reducing their consumption. One component required to deliver innovative feedback mechanisms is the capability to forecast hourly electricity consumption at the household scale. The work presented by this thesis is an evaluation of the effectiveness of a selection of kernel based machine learning methods at forecasting the hourly aggregate electricity consumption for different sized sets of households. The work of this thesis demonstrates that k-Nearest Neighbour Regression and Gaussian process Regression are the most accurate methods within the constraints of the problem considered. In addition to accuracy, the advantages and disadvantages of each machine learning method are evaluated, and a simple comparison of each algorithms computational performance is made. / För att ta itu med ineffektivitet, avfall, och de negativa konsekvenserna av elproduktion så vill företag och myndigheter se beteendeförändringar bland hushållskonsumenter. För att skapa beteendeförändringar så behöver konsumenterna bättre återkoppling när det gäller deras elförbrukning. Den nuvarande återkopplingen i en månads- eller kvartalsfaktura ger konsumenten nästan ingen användbar information om hur deras beteenden relaterar till deras konsumtion. Smarta mätare finns nu överallt i de utvecklade länderna och de kan ge en mängd information om bostäders konsumtion, men denna data används främst som underlag för fakturering och inte som ett verktyg för att hjälpa konsumenterna att minska sin konsumtion. En komponent som krävs för att leverera innovativa återkopplingsmekanismer är förmågan att förutse elförbrukningen på hushållsskala. Arbetet som presenteras i denna avhandling är en utvärdering av noggrannheten hos ett urval av kärnbaserad maskininlärningsmetoder för att förutse den sammanlagda förbrukningen för olika stora uppsättningar av hushåll. Arbetet i denna avhandling visar att "k-Nearest Neighbour Regression" och "Gaussian Process Regression" är de mest exakta metoder inom problemets begränsningar. Förutom noggrannhet, så görs en utvärdering av fördelar, nackdelar och prestanda hos varje maskininlärningsmetod.

[ES] Esta tesis se enmarca en la intersección entre las técnicas modernas de Machine Learning, como las Redes Neuronales Profundas, y el modelado probabilístico confiable. En muchas aplicaciones, no solo nos importa la predicción hecha por un modelo (por ejemplo esta imagen de pulmón presenta cáncer) sino también la confianza que tiene el modelo para hacer esta predicción (por ejemplo esta imagen de pulmón presenta cáncer con 67% probabilidad). En tales aplicaciones, el modelo ayuda al tomador de decisiones (en este caso un médico) a tomar la decisión final. Como consecuencia, es necesario que las probabilidades proporcionadas por un modelo reflejen las proporciones reales presentes en el conjunto al que se ha asignado dichas probabilidades; de lo contrario, el modelo es inútil en la práctica. Cuando esto sucede, decimos que un modelo está perfectamente calibrado. En esta tesis se exploran tres vias para proveer modelos más calibrados. Primero se muestra como calibrar modelos de manera implicita, que son descalibrados por técnicas de aumentación de datos. Se introduce una función de coste que resuelve esta descalibración tomando como partida las ideas derivadas de la toma de decisiones con la regla de Bayes. Segundo, se muestra como calibrar modelos utilizando una etapa de post calibración implementada con una red neuronal Bayesiana. Finalmente, y en base a las limitaciones estudiadas en la red neuronal Bayesiana, que hipotetizamos que se basan en un prior mispecificado, se introduce un nuevo proceso estocástico que sirve como distribución a priori en un problema de inferencia Bayesiana. / [CA] Aquesta tesi s'emmarca en la intersecció entre les tècniques modernes de Machine Learning, com ara les Xarxes Neuronals Profundes, i el modelatge probabilístic fiable. En moltes aplicacions, no només ens importa la predicció feta per un model (per ejemplem aquesta imatge de pulmó presenta càncer) sinó també la confiança que té el model per fer aquesta predicció (per exemple aquesta imatge de pulmó presenta càncer amb 67% probabilitat). En aquestes aplicacions, el model ajuda el prenedor de decisions (en aquest cas un metge) a prendre la decisió final. Com a conseqüència, cal que les probabilitats proporcionades per un model reflecteixin les proporcions reals presents en el conjunt a què s'han assignat aquestes probabilitats; altrament, el model és inútil a la pràctica. Quan això passa, diem que un model està perfectament calibrat. En aquesta tesi s'exploren tres vies per proveir models més calibrats. Primer es mostra com calibrar models de manera implícita, que són descalibrats per tècniques d'augmentació de dades. S'introdueix una funció de cost que resol aquesta descalibració prenent com a partida les idees derivades de la presa de decisions amb la regla de Bayes. Segon, es mostra com calibrar models utilitzant una etapa de post calibratge implementada amb una xarxa neuronal Bayesiana. Finalment, i segons les limitacions estudiades a la xarxa neuronal Bayesiana, que es basen en un prior mispecificat, s'introdueix un nou procés estocàstic que serveix com a distribució a priori en un problema d'inferència Bayesiana. / [EN] This thesis is framed at the intersection between modern Machine Learning techniques, such as Deep Neural Networks, and reliable probabilistic modeling. In many machine learning applications, we do not only care about the prediction made by a model (e.g. this lung image presents cancer) but also in how confident is the model in making this prediction (e.g. this lung image presents cancer with 67% probability). In such applications, the model assists the decision-maker (in this case a doctor) towards making the final decision. As a consequence, one needs that the probabilities provided by a model reflects the true underlying set of outcomes, otherwise the model is useless in practice. When this happens, we say that a model is perfectly calibrated. In this thesis three ways are explored to provide more calibrated models. First, it is shown how to calibrate models implicitly, which are decalibrated by data augmentation techniques. A cost function is introduced that solves this decalibration taking as a starting point the ideas derived from decision making with Bayes' rule. Second, it shows how to calibrate models using a post-calibration stage implemented with a Bayesian neural network. Finally, and based on the limitations studied in the Bayesian neural network, which we hypothesize that came from a mispecified prior, a new stochastic process is introduced that serves as a priori distribution in a Bayesian inference problem. / Maroñas Molano, J. (2022). Modeling Uncertainty for Reliable Probabilistic Modeling in Deep Learning and Beyond [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/181582 / TESIS

Latente Gauß-Prozess-Modelle (latent Gaussian process models) werden von Wissenschaftlern benutzt, um verborgenen Muster in Daten zu er- kennen, Expertenwissen in probabilistische Modelle einfließen zu lassen und um Vorhersagen über die Zukunft zu treffen. Diese Modelle wurden erfolgreich in vielen Gebieten wie Robotik, Geologie, Genetik und Medizin angewendet. Gauß-Prozesse definieren Verteilungen über Funktionen und können als flexible Bausteine verwendet werden, um aussagekräftige probabilistische Modelle zu entwickeln. Dabei ist die größte Herausforderung, eine geeignete Inferenzmethode zu implementieren. Inferenz in probabilistischen Modellen bedeutet die A-Posteriori-Verteilung der latenten Variablen, gegeben der Daten, zu berechnen. Die meisten interessanten latenten Gauß-Prozess-Modelle haben zurzeit nur begrenzte Anwendungsmöglichkeiten auf großen Datensätzen. In dieser Doktorarbeit stellen wir eine neue effiziente Inferenzmethode für latente Gauß-Prozess-Modelle vor. Unser neuer Ansatz, den wir augmented variational inference nennen, basiert auf der Idee, eine erweiterte (augmented) Version des Gauß-Prozess-Modells zu betrachten, welche bedingt konjugiert (conditionally conjugate) ist. Wir zeigen, dass Inferenz in dem erweiterten Modell effektiver ist und dass alle Schritte des variational inference Algorithmus in geschlossener Form berechnet werden können, was mit früheren Ansätzen nicht möglich war. Unser neues Inferenzkonzept ermöglicht es, neue latente Gauß-Prozess- Modelle zu studieren, die zu innovativen Ergebnissen im Bereich der Sprachmodellierung, genetischen Assoziationsstudien und Quantifizierung der Unsicherheit in Klassifikationsproblemen führen. / Latent Gaussian process (GP) models help scientists to uncover hidden structure in data, express domain knowledge and form predictions about the future. These models have been successfully applied in many domains including robotics, geology, genetics and medicine. A GP defines a distribution over functions and can be used as a flexible building block to develop expressive probabilistic models. The main computational challenge of these models is to make inference about the unobserved latent random variables, that is, computing the posterior distribution given the data. Currently, most interesting Gaussian process models have limited applicability to big data. This thesis develops a new efficient inference approach for latent GP models. Our new inference framework, which we call augmented variational inference, is based on the idea of considering an augmented version of the intractable GP model that renders the model conditionally conjugate. We show that inference in the augmented model is more efficient and, unlike in previous approaches, all updates can be computed in closed form. The ideas around our inference framework facilitate novel latent GP models that lead to new results in language modeling, genetic association studies and uncertainty quantification in classification tasks.

Les nouvelles voitures sont pourvues d’aides à la conduite qui améliorent le confort et la sécurité. Bien que ces systèmes contribuent à la réduction des accidents de la route, leur déploiement montre que leurs performances sont encore limitées par leur faible compréhension de situation. Cela est principalement lié aux limites des capteurs de perception, et à la non prise en compte du contexte. Ces limites se traduisent par des détections de risques tardives, et donc en assistances sous forme d’alertes ou de freinages automatiques. Cette thèse se concentre sur l’introduction d’informations contextuelles dans le processus de décision des systèmes d’aides à la conduite. Le but est de détecter des risques plus tôt que les systèmes conventionnels, ainsi que d’améliorer la confiance qu’on peut avoir dans les informations générées.Le comportement d’un véhicule dépend de divers éléments tels que le réseau routier, les règles de la circulation, ainsi que de la cohabitation avec d’autres usagers de la route. Ces interactions se traduisent par une interdépendance forte entre chaque élément. De plus, bien que chaque conducteur doive suivre les mêmes règles de circulation, ils peuvent réagir de façon différente à une même situation. Cela implique qu’un même comportement peut être considéré comme sûr ou risqué, selon le conducteur. Ces informations doivent être prises en compte dans le processus de prise de décision des systèmes. Cette thèse propose un cadre qui combine les informations a priori contenues dans les cartes de navigation numériques avec l’information temps réel fournie par les capteurs de perception et/ou communications sans fil, pour permettre une meilleure compréhension de situation et ainsi mieux anticiper les risques. Ce principe est comparable aux tâches qu’un copilote doit accomplir. Ces travaux se répartissent en deux principales étapes : la compréhension de situation, et l’estimation des risques.L’étape de compréhension de situation consiste à donner du sens aux différentes observations réalisées par les capteurs de perception, en exploitant des informations a priori. Le but est de comprendre comment les entités perçues interagissent, et comment ces interactions contraignent le comportement du véhicule. Cette étape établit les relations spatio-temporelles entre les entités perçues afin d’évaluer leur pertinence par rapport au véhicule, et ainsi extraire les entités les plus contraignantes. Pour cela, une ontologie contenant des informations a priori sur la façon dont différentes entités de la route interagissent est proposée. Cette première étape a été testée en temps réel, utilisant des données enregistrées sur un véhicule évoluant en environnements contraints.L’étape de détection des risques s’appuie sur la situation perçue, et sur les signes annonciateurs de risques. Le cas d’usage choisi pour cette étude se concentre sur les intersections, puisqu’une grande majorité des accidents de la route y ont lieux. La manière de réagir d’un conducteur lorsqu’il se rapproche d’une intersection est apprise par des Processus Gaussiens. Cette connaissance à priori du conducteur est ensuite exploitée, avec les informations contextuelles, par un réseau Bayésien afin d’estimer si le conducteur semble interagir comme attendu avec l’intersection. L’approche probabiliste qui a été choisie permet de prendre en compte les incertitudes dont souffrent chacune des sources d’information. Des tests ont été réalisés à partir de données enregistrées à bord d’un véhicule afin de valider l’approche. Les résultats montrent qu’en prenant en compte les individualités des conducteurs, leurs actions sur le véhicule, ainsi que l’état du véhicule, il est possible de mieux estimer si le conducteur interagit comme attendu avec l’environnement, et donc d’anticiper les risques. Finalement, il est montré qu’il est possible de générer une assistance plus préventive que les systèmes d’aide à la conduite conventionnels. / Modern vehicles include advanced driving assistance systems for comfort and active safety features. Whilst these systems contribute to the reduction of road accidents, their deployment has shown that performance is constrained by their limited situation understanding capabilities. This is mainly due to perception constraints and by ignoring the context within which these vehicles evolve. It results in last minute risk assessment, and thus in curative assistance in the form of warning alerts or automatic braking. This thesis focuses on the introduction of contextual information into the decision processes of driving assistance systems. The overall purpose is to infer risk earlier than conventional driving assistance systems, as well as to enhance the level of trust on the information provided to drivers.Several factors govern the vehicle behaviour. These include the road network and traffic rules, as well as other road users such as vehicles and pedestrians with which the vehicle interacts. This results in strong interdependencies amongst all entities, which govern their behaviour. Further, whilst traffic rules apply equally to all participants, each driver interacts differently with the immediate environment, leading to different risk level for a given behaviour. This information must be incorporated within the decision-making processes of these systems. In this thesis, a framework is proposed that combines a priori information from digital navigation maps with real time information from on board vehicle sensors and/or external sources via wireless communications links, to infer a better situation understanding, which should enable to anticipate risks. This tenet is similar to the task of a co-pilot when using a priori notated road information. The proposed approach is constrained by using only data from close to production sensors. The framework proposed in this thesis consists of two phases, namely situation understanding and risk assessment.The situation understanding phase consists in performing a high level interpretation of all observations by including a priori information within the framework. The purpose is to understand how the perceived road entities interact, and how the interactions constrain the vehicle behaviour. This phase establishes the spatio-temporal relationships between the perceived entities to determine their relevance with respect to the subject vehicle motion, and then to identify which entities to be tracked. For this purpose, an ontology is proposed. It stores a priori information about the manner how different road entities relate and interact. This initial phase was tested in real time using data recorded on a passenger vehicle evolving in constrained environments.The risk assessment phase then looks into the perceived situation and into the manner how it becomes dangerous. To demonstrate the framework applicability, a use case applied to road intersections was chosen. Intersections are complex parts in the road network where different entities converge and most accidents occur. In order to detect risk situations, the manner how the driver reacts in a given situation is learned through Gaussian Processes. This knowledge about the driver is then used within a context aware Bayesian Network to estimate whether the driver is likely to interact as expected with the relevant entities or not. The probabilistic approach taken allows to take into consideration all uncertainties embedded in the observations. Field trials were performed using a passenger vehicle to validate the proposed approach. The results show that by incorporating drivers’ individualities and their actuations with the observation of the vehicle state, it is possible to better estimate whether the driver interacts as expected with the environment, and thus to anticipate risk. Further, it is shown that it is possible to generate assistance earlier than conventional safety systems.

<p> Generative models form powerful tools for learning data distributions and simulating new samples. Recent years have seen significant advances in the flexibility and applicability of such models, with Bayesian approaches like nonparametric Bayesian models and deep neural network models such as Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs) finding use in a wide range of domains. However, the black-box nature of these models means that they are often hard to interpret, and they often come with modeling implications that are inconsistent with side knowledge resulting from domain knowledge. This thesis studies situations where the modeler has side knowledge represented as probability distributions on functionals of the objects being modeled, and we study methods to incorporate this particular kind of side knowledge into flexible generative models. This dissertation covers three main parts. </p> <p><br></p> <p>The first part focuses on incorporating a special case of the aforementioned side knowledge into flexible nonparametric Bayesian models. Many times, practitioners have additional distributional information about a subset of the coordinates of the observations being modeled. The flexibility of nonparametric Bayesian models usually implies incompatibility with this side information. Such inconsistency triggers the necessity of developing methods to incorporate this side knowledge into flexible nonparametric Bayesian models. We design a specialized generative process to build in this side knowledge and propose a novel sigmoid Gaussian process conditional model. We also develop a corresponding posterior sampling method based on data augmentation to overcome a doubly intractable problem. We illustrate the efficacy of our proposed constrained nonparametric Bayesian model in a variety of real-world scenarios including modeling environmental and earthquake data. </p> <p><br></p> <p>The second part of the dissertation discusses neural network approaches to satisfying the said general side knowledge. Further, the generative models considered in this part broaden into black-box models. We formulate this side knowledge incorporation problem as a constrained divergence minimization problem and propose two scalable neural network approaches as its solution. We demonstrate their practicality using various synthetic and real examples. </p> <p><br></p> <p> The third part of the dissertation concentrates on a specific generative model of individual pixels of the fMRI data constructed from a latent group image. Usually there is two-fold side knowledge about the latent group image: spatial structure and partial activation zones. The former can be captured by modeling the prior for the group image with Markov random fields. The latter, which is often obtained from previous related studies, is left for future research. We propose a novel Bayesian model with Markov random fields and aim to estimate the maximum a posteriori for the group image. We also derive a variational Bayes algorithm to overcome local optima in the optimization.</p>

We build on the method introduced by Ringer, et al., applying it to an 1820 event that happened near South Sulawesi, Indonesia. We utilize other statistical models to aid our Metropolis-Hastings sampler, including a Gaussian process which informs the prior. We apply the method to multiple possible fault zones to determine which fault is the most likely source of the earthquake and tsunami. After collecting nearly 80,000 samples, we find that between the two most likely fault zones, the Walanae fault zone matches the anecdotal accounts much better than Flores. However, to support the anecdotal data, both samplers tend toward powerful earthquakes that may not be supported by the faults in question. This indicates that even further research is warranted. It may indicate that some other type of event took place, such as a multiple-fault rupture or landslide tsunami.