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Analytically deriving Partial Information Decomposition for affine systems of stable and convolution-closed distributions
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Derives analytical bivariate partial information decomposition (BROJA-PID) for affine stable, convolution-closed, and exponential-family systems by constructing Markov-chain couplings, plus an analytical upper bound for convolution-closed systems checked numerically on Poisson.
Problem
Computing bivariate partial information decomposition (PID) requires optimization over distributions and has few analytical solutions; only the jointly Gaussian case was known analytically.
Approach
Affine dependence structures in stable and convolution-closed families allow constructing a Markov chain M->X->Y (or M->Y->X) within the constraint set by adding independent noise (or via data thinning/fission), which forces one unique information term to zero and yields all PID terms analytically; for non-affine convolution-closed systems an upper bound on the BROJA objective is analytically mini
Key results
- Analytical upper bound for Poisson systems is within <1% of numerical ground truth for 16 of 20 non-linear function pairs.
- Analytical PID is derived for stable, convolution-closed, and exponential-family systems (e.g., gamma, Poisson, beta, binomial, negative binomial, Cauchy-stable, exponential, uniform), where one UI term is zero.
- Generalizes Barrett's Gaussian PID result to continuous and discrete stable distributions, including fat-tailed Cauchy and Lévy-stable cases.
Surprising findings
- Exponential and uniform systems yield both unique information terms equal to zero.
Limitations
- Most analytical results require P(X|M) and P(Y|M) to depend on affine functions of M; non-affine cases remain open.
- Upper bound tightness is only tested in simulation on Poisson and requires further refinement and testing.
- Univariate M, X, Y in main results; multivariate extensions are partial.
Experimental setup
Benchmarks: numerical BROJA-PID ground truth on discretized Poisson systems
Metrics: mutual information difference between analytical upper-bound
Baselines: BROJA-PID numerical solver (Venkatesh code)
Builds on: BROJA-PID Barrett's Gaussian PID result data thinning (Neufeld et al.) data fission (Leiner et al.)
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