Dalia Chakrabarty

Kane Warrior, Dalia Chakrabarty

In probabilstic supervised learning of an input-output relationship - as a sample function of a Gaussian Process (GP) - priors are typically specified for the hyperparameters of the kernel that parametrises the covariance function of the GP, where the induced covariance matrix of the (resulting multivariate Normal) likelihood, governs the learning and prediction. When the sought function is highly multivariate, multiple lengthscale parameters must be learnt simultaneously, making inference difficult. We develop a ``self-assembled'' Wishart prior for the covariance matrix, while undertaking Bayesian inference on the kernel hyperparameters using MCMC. The construction uses a look-back window over recent MCMC iterations to define a time-step dependent scale matrix, thereby introducing adaptiveness to the chain. Results suggest that direct prior specification on the covariance matrix can be useful for diagnosing weakly informative inputs within the GP-based learning paradigm. We support our prior development with two distinct empirical illustrations - one on synthetic data, and another on a real-world dataset.

Kane Warrior, Dalia Chakrabarty

Many three-dimensional spatial fields are anisotropic, with directions of rapid and slow variation that need not align with the coordinate axes. Standard Gaussian process kernels with Automatic Relevance Determination (ARD) capture only axis-aligned anisotropy, while generic full symmetric positive definite (SPD) metrics can represent rotated anisotropy but do not parameterise principal length-scales and directions directly. We introduce an interpretable rotationally anisotropic GP kernel that parameterises a three-dimensional SPD covariance metric using three principal length-scales and an explicit SO(3) rotation. The rotation is represented by an axis-angle vector and mapped to SO(3) via the Lie-algebra exponential map, giving unconstrained Euclidean coordinates for inference while always inducing a valid SPD metric. The construction spans the same family of three-dimensional SPD covariance metrics as a generic full-SPD parameterisation, but exposes the geometry differently: length-scales and orientation are explicit, interpretable, and directly available for prior specification and posterior summaries. We perform Bayesian inference on these quantities using Markov Chain Monte Carlo (MCMC), and characterise the resulting symmetries and weakly identified regimes. On synthetic data with rotated anisotropy, the posterior recovers the generating metric and improves prediction relative to an axis-aligned ARD baseline, while matching the predictive performance of a generic full SPD baseline. When the ground truth is axis-aligned, posterior mass concentrates near the identity rotation and predictive performance matches ARD. On a material-density dataset from a laboratory-fabricated nano-brick, the inferred metric reveals rotated anisotropy that is not captured by axis-aligned kernels.

Dalia Chakrabarty, Kane Warrior, Chuqiao Zhang et al.

We advance a new probabilistic supervised learning approach that permits reliable, automated, and early individualised prediction of the severity with which a disease will develop in a prospective patient. The prediction capacity is illustrated via the pre-transplant prediction of the score of severity of Veno Occlusive Disease (or VOD) in the digital twin (DT) of the considered prospective patient, where this score parametrises the severity with which VOD will develop in this patient, after they undergo their Bone Marrow Transplant. The learning of the relationship between the pre-transplant variables, and a severity score variable is undertaken by modelling this relationship as a (random) function that is treated as a sample function of an adequately-chosen stochastic process. The parameters of this underlying process are learnt using a training dataset that is generated using the real-time evolution of retrospective patients in a cohort, with this training dataset subsequently augmented in size by a probabilistic inverse learning of the score of prospective patients. The augmented training set, then permits the learning of the function that capacitates - at the pre-transplant stage - automated prediction of the score of the severity of VOD that characterises the DT of a physical patient in their unique pre-transplant state. This score is subsequently fed back to the real prospective patient as the severity with which VOD will develop in them, after this patient undergoes their transplant. Such a score then permits the treating Haematologist-Oncologists to decide on the treatment regimen, which in this illustration reduces to deciding on treating the patient with Defibrotide. An AI facility is developed to undertake such automated prediction, with the physician inputting the data on the pre-transplant state that characterises the DT of the prospective patient under consideration.

Gargi Roy, Dalia Chakrabarty

We introduce parametrisation of that property of the available training dataset, that necessitates an inhomogeneous correlation structure for the function that is learnt as a model of the relationship between the pair of variables, observations of which comprise the considered training data. We refer to a parametrisation of this property of a given training set, as its ``inhomogeneity parameter''. It is easy to compute this parameter for small-to-large datasets, and we demonstrate such computation on multiple publicly-available datasets, while also demonstrating that conventional ``non-stationarity'' of data does not imply a non-zero inhomogeneity parameter of the dataset. We prove that - within the probabilistic Gaussian Process-based learning approach - a training set with a non-zero inhomogeneity parameter renders it imperative, that the process that is invoked to model the sought function, be non-stationary. Following the learning of a real-world multivariate function with such a Process, quality and reliability of predictions at test inputs, are demonstrated to be affected by the inhomogeneity parameter of the training data.

Chuqiao Zhang, Sarath Chandra Dantu, Debarghya Mitra et al.

Identification of critical residues of a protein is actively pursued, since such residues are essential for protein function. We present three ways of recognising critical residues of an example protein, the evolution of which is tracked via molecular dynamical simulations. Our methods are based on learning a Random Geometric Graph (RGG) variable, where the state variable of each of 156 residues, is attached to a node of this graph, with the RGG learnt using the matrix of correlations between state variables of each residue-pair. Given the categorical nature of the state variable, correlation between a residue pair is computed using Cramer's V. We advance an organic thresholding to learn an RGG, and compare results against extant thresholding techniques, when parametrising criticality as the nodal degree in the learnt RGG. Secondly, we develop a criticality measure by ranking the computed differences between the posterior probability of the full graph variable defined on all 156 residues, and that of the graph with all but one residue omitted. A third parametrisation of criticality informs on the dynamical variation of nodal degrees as the protein evolves during the simulation. Finally, we compare results obtained with the three distinct criticality parameters, against experimentally-ascertained critical residues.

Chuqiao Zhang, Crina Grosan, Dalia Chakrabarty

Patients who are undergoing physical rehabilitation, benefit from feedback that follows from reliable assessment of their cumulative performance attained at a given time. In this paper, we provide a method for the learning of the recovery trajectory of an individual patient, as they undertake exercises as part of their physical therapy towards recovery of their loss of movement ability, following a critical illness. The difference between the Movement Recovery Scores (MRSs) attained by a patient, when undertaking a given exercise routine on successive instances, is given by a statistical distance/divergence between the (posterior) probabilities of random graphs that are Bayesianly learnt using time series data on locations of 20 of the patient's joints, recorded on an e-platform as the patient exercises. This allows for the computation of the MRS on every occasion the patient undertakes this exercise, using which, the recovery trajectory is drawn. We learn each graph as a Random Geometric Graph drawn in a probabilistic metric space, and identify the closed-form marginal posterior of any edge of the graph, given the correlation structure of the multivariate time series data on joint locations. On the basis of our recovery learning, we offer recommendations on the optimal exercise routines for patients with given level of mobility impairment.

Gargi Roy, Dalia Chakrabarty

We present a new strategy for learning the functional relation between a pair of variables, while addressing inhomogeneities in the correlation structure of the available data, by modelling the sought function as a sample function of a non-stationary Gaussian Process (GP), that nests within itself multiple other GPs, each of which we prove can be stationary, thereby establishing sufficiency of two GP layers. In fact, a non-stationary kernel is envisaged, with each hyperparameter set as dependent on the sample function drawn from the outer non-stationary GP, such that a new sample function is drawn at every pair of input values at which the kernel is computed. However, such a model cannot be implemented, and we substitute this by recalling that the average effect of drawing different sample functions from a given GP is equivalent to that of drawing a sample function from each of a set of GPs that are rendered different, as updated during the equilibrium stage of the undertaken inference (via MCMC). The kernel is fully non-parametric, and it suffices to learn one hyperparameter per layer of GP, for each dimension of the input variable. We illustrate this new learning strategy on a real dataset.