[Paper Review] Score and Information for Recursive Exponential Models with Incomplete Data

Bo Thiesson|arXiv (Cornell University)|Feb 6, 2013

Bayesian Modeling and Causal InferenceComputer Science14 references20 citations

TL;DR

This paper introduces a framework for recursive exponential models with incomplete data, integrating expert knowledge via approximately conjugate priors. It derives both traditional and posterior-based score and observed information measures, enabling robust statistical inference in Bayesian networks under uncertainty and partial data.

ABSTRACT

Recursive graphical models usually underlie the statistical modelling concerning probabilistic expert systems based on Bayesian networks. This paper defines a version of these models, denoted as recursive exponential models, which have evolved by the desire to impose sophisticated domain knowledge onto local fragments of a model. Besides the structural knowledge, as specified by a given model, the statistical modelling may also include expert opinion about the values of parameters in the model. It is shown how to translate imprecise expert knowledge into approximately conjugate prior distributions. Based on possibly incomplete data, the score and the observed information are derived for these models. This accounts for both the traditional score and observed information, derived as derivatives of the log-likelihood, and the posterior score and observed information, derived as derivatives of the log-posterior distribution. Throughout the paper the specialization into recursive graphical models is accounted for by a simple example.

Motivation & Objective

• To formalize recursive exponential models that incorporate domain-specific expert knowledge into local model fragments.
• To address the challenge of statistical inference when data is incomplete, particularly in probabilistic expert systems.
• To translate imprecise expert opinion into approximately conjugate prior distributions for improved modeling flexibility.
• To derive both classical and posterior-based score and observed information measures for parameter estimation.
• To demonstrate the method's applicability through a simplified example of a recursive graphical model.

Proposed method

• Defines recursive exponential models as an extension of Bayesian networks, allowing structured incorporation of domain knowledge.
• Uses imprecise expert knowledge to construct approximately conjugate prior distributions, facilitating Bayesian updating.
• Derives the traditional score and observed information as derivatives of the log-likelihood function under incomplete data.
• Derives the posterior score and observed information as derivatives of the log-posterior distribution, accounting for prior information.
• Applies the Fisher information and score functions to both likelihood and posterior to support parameter estimation and inference.
• Illustrates the method using a minimal recursive graphical model, showing how structural and expert knowledge combine in inference.

Experimental results

Research questions

• RQ1 How can expert knowledge be systematically incorporated into recursive exponential models using approximately conjugate priors?
• RQ2 What is the form of the score and observed information when data is incomplete in recursive exponential models?
• RQ3 How do the posterior score and observed information differ from their classical counterparts in this context?
• RQ4 What is the impact of combining structural knowledge with expert opinion on statistical inference in incomplete data settings?
• RQ5 How can recursive graphical models be used to represent complex dependencies while maintaining computational tractability?

Key findings

• The paper successfully derives both classical and posterior-based score and observed information measures for recursive exponential models with incomplete data.
• The use of approximately conjugate priors enables effective integration of imprecise expert knowledge into the statistical model.
• The posterior score and observed information are shown to be derivable as derivatives of the log-posterior, extending classical inference to Bayesian settings.
• The framework supports robust parameter estimation even when data is incomplete, by combining model structure with expert opinion.
• The method is validated through a simple recursive graphical model example, demonstrating its practical applicability.
• The derived scores and information matrices are suitable for optimization and uncertainty quantification in probabilistic expert systems.