[Paper Review] Deep Probabilistic Spatial Modeling for Multivariate Mixed-Type Responses

Yeseul Jeon, Kyeong Eun Lee+1 more|arXiv (Cornell University)|Mar 7, 2026

Soil Geostatistics and Mapping0 citations

TL;DR

We propose MultiDeepGP, a scalable framework that jointly models multivariate mixed-type spatial outcomes using a shared latent spatial component and Monte Carlo dropout to enable coherent uncertainty quantification.

ABSTRACT

Many scientific applications involve mixed spatially indexed outcomes of heterogeneous types that are driven by shared latent mechanisms. Modeling such data is challenging due to complex, nonlinear, and potentially nonstationary spatial dependence, as well as the need for coherent joint inference across mixed outcome distributions. Existing multivariate mixed outcome models often rely on restrictive linear assumptions, while recent deep learning approaches emphasize predictive flexibility but typically lack coherent joint modeling and uncertainty quantification for spatial data. We develop MultiDeepGP, a scalable and statistically principled framework for joint modeling of multivariate mixed outcomes in spatial settings. The proposed approach introduces a shared latent spatial component that governs cross-outcome dependence while allowing outcome-specific distributions. Spatial dependence and nonlinear structure are captured through a deep latent representation, and uncertainty quantification is enabled via an efficient Monte Carlo-based inference strategy. This construction balances modeling flexibility with probabilistic interpretability and computational feasibility. The proposed method is evaluated through simulation studies designed to reflect key challenges in mixed outcome spatial modeling, as well as an application to georeferenced environmental and public health data from the African Great Lakes region. The results demonstrate that the proposed framework provides accurate joint prediction and reliable uncertainty quantification in complex spatial settings.

Motivation & Objective

• Motivate the need to jointly model mixed-type spatial data with shared latent structure.
• Develop a flexible, nonlinear shared representation that captures complex spatial dependencies.
• Allow outcome-specific distributions while linking them through a common latent spatial process.
• Provide scalable uncertainty quantification for joint predictions across multiple outcomes.

Proposed method

• Introduce a shared latent spatial process H(s) that governs cross-outcome dependence with outcome-specific likelihoods.
• Parameterize H(s) via a deep neural network to capture nonlinear and nonstationary spatial structure.
• Link shared representation to outcome-specific natural parameters through outcome-specific heads with appropriate links.
• Interpret the neural network as a deep Gaussian process to enable probabilistic inference.
• Use Monte Carlo dropout as an approximate variational inference strategy to compute predictive distributions and uncertainties.
• Train by minimizing regularized negative log-likelihood with layer-wise weight decay and then perform MC dropout for prediction and uncertainty quantification.

Experimental results

Research questions

• RQ1 How can a shared latent spatial component induce cross-outcome dependence in multivariate mixed-type spatial data?
• RQ2 Can a deep latent representation capture nonlinear and nonstationary spatial dependencies while allowing outcome-specific distributions?
• RQ3 Does MC dropout provide reliable, scalable uncertainty quantification for joint predictions across heterogeneous outcomes?
• RQ4 How does MultiDeepGP compare to deterministic DNNs and traditional kriging in terms of predictive performance and calibrated uncertainty?

Key findings

• The framework enables accurate joint prediction across mixed-type spatial outcomes.
• Uncertainty quantification via MC dropout yields well-calibrated predictive distributions in complex spatial settings.
• Simulation studies demonstrate robustness to nonlinear and nonstationary spatial structure, and cross-outcome dependence.
• Application to georeferenced environmental and public health data from the African Great Lakes region shows practical utility and coherent joint inference.