[Paper Review] A Holonomic Ideal Annihilating the Fisher-Bingham Integral

Tamio Koyama|arXiv (Cornell University)|Apr 7, 2011

Stochastic processes and financial applications7 references7 citations

TL;DR

This paper proves that the set of differential operators conjectured in [9] to annihilate the Fisher–Bingham integral indeed generates a holonomic ideal for any dimension $ n $, by computing the integration ideal of the annihilating ideal of the integrand and showing it matches the proposed operators. The result confirms the holonomic gradient descent framework's applicability to Fisher–Bingham likelihood estimation in directional statistics.

ABSTRACT

We calculate the integration ideal of annihilating differential operators of the non-normalized Fisher-Bingham distribution and show that the ideal agrees with the set of operators for the Fisher-Bingham integral given in "Holonomic Gradient Descent and its Application to the Fisher-Bingham Integral". They conjectured that the set generates a holonomic ideal and we prove their conjecture.

Motivation & Objective

• To prove the conjecture in [9] that the set of differential operators annihilating the Fisher–Bingham integral forms a holonomic ideal for all $ n $.
• To compute the integration ideal of the annihilating ideal of the Fisher–Bingham integrand in the ring of differential operators with polynomial coefficients.
• To establish that this integration ideal is generated precisely by the operators proposed in [9], thereby confirming their holonomicity.
• To provide a general proof for arbitrary $ n $, extending prior computer-verified results for $ n=1,2 $.

Proposed method

• Uses the ring $ D_d $ of differential operators with polynomial coefficients to model the Fisher–Bingham integral and its annihilating operators.
• Applies the concept of integration ideals: $ (I + ext{terms in } t) igcap D_{d'} $, where $ D_{d'} $ contains only $ x, y $ variables.
• Employs Oaku's algorithm for computing integration ideals, adapted for general $ n $ via symbolic computation and Gröbner basis techniques under a weighted term order.
• Utilizes the fact that the Haar measure on $ S^n(r) $ is annihilated by specific differential operators, including $ t_i^2 + \cdots + t_{n+1}^2 - r^2 $ and $ t_i\partial_{t_j} - t_j\partial_{t_i} $.
• Applies the fundamental theorem of algebraic analysis to verify holonomicity by checking the Krull dimension of the characteristic ideal.
• Uses a change of variables and module-theoretic arguments to show that the integration ideal is generated by the operators in (1.3), by reducing modulo $ t_i - \partial_{y_i} $.

Experimental results

Research questions

• RQ1 Does the set of differential operators proposed in [9] to annihilate the Fisher–Bingham integral generate a holonomic ideal for all $ n $?
• RQ2 Can the integration ideal of the annihilating ideal of the Fisher–Bingham integrand be explicitly computed and shown to match the proposed operators?
• RQ3 Is the holonomic gradient descent method applicable to the Fisher–Bingham distribution beyond small dimensions?
• RQ4 What is the precise structure of the annihilating ideal of the integrand $ \exp(\sum x_{ij}t_i t_j + \sum y_i t_i) |dt| $?

Key findings

• The integration ideal of the annihilating ideal of the Fisher–Bingham integrand is generated by the operators listed in (1.3), proving their completeness.
• The set of operators in (1.3) generates a holonomic ideal in $ D_{d'} $ for any $ n $, confirming the conjecture in [9].
• The proof establishes that the holonomic gradient descent method is theoretically valid for Fisher–Bingham likelihood estimation in arbitrary dimensions.
• The result holds for general $ n $, extending prior computer-verified cases for $ n=1,2 $ to all natural numbers.
• The integration ideal computation relies on Gröbner basis techniques under a weighted term order, with initial ideal generated by $ \{t_i\} $, leading to the conclusion that only the zero operator satisfies the divisibility condition.