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Finite dependence is not finite conditional memory

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Finite dependence is not finite conditional memory

This is a mathematical safeguard found during the detailed source reread, not a compression breakthrough or a claim that the entire source is invalid.

In Unser, Tafti, Amini and Kirshner (2014), Part II, printed page 3044 / PDF page 9, Property 4 states a finite-past conditional-density identity for the generalized increment process. Its proof invokes independence of samples whose B-spline windows do not overlap. The non-overlap argument establishes finite-range marginal dependence; it does not establish the displayed finite-order Markov identity. This distinction matters even for Gaussian innovations. The displayed equation was checked in the rendered source, not inferred from faulty text extraction.

Counterexample within the source's own model class

Take L=D^2 and unit Gaussian white noise. The generalized increment kernel is the cardinal triangular B-spline beta(t), supported on [0,2], with unit slope on [0,1] and negative unit slope on [1,2]. Define u_k=<w,beta(k-.)>. Its covariance is

Cov(u_k,u_k)=integral beta(t)^{2} dt=2/3, Cov(u_k,u_(k-1))=integral beta(t) beta(t-1) dt=1/6, Cov(u_k,u_(k-2))=0.

The last equality gives marginal independence at lag two because the vector is jointly Gaussian. But Gaussian conditioning gives

Cov(u_k,u_(k-2) | u_(k-1)) = 0 - (1/6)(2/3)^(-1)(1/6) = -1/24,

which is nonzero. Equivalently, the optimal prediction using the last two samples is (4/15)u_(k-1)-(1/15)u_(k-2), while using only the last sample gives (1/4)u_(k-1). Therefore this order-two increment process is not first-order Markov, contrary to the conditional identity as written for N=2.

The covariance is that of a Gaussian MA(1), with invertible coefficient 2-sqrt(3) and innovation variance (2+sqrt(3))/6. A generic MA(1) has short covariance range but an infinite autoregressive representation. This is an elementary counterexample, not a new probabilistic phenomenon.

Consequences for this campaign

Compact exponential B-splines still localize increments, and the covariance, characteristic-functional and spectral-factorization machinery remains useful. One must not infer that N-1 raw increments suffice for every future conditional likelihood. Exact finite-dimensional latent filtering state at fixed parameters is a different statement from finite observed-history Markov order or a parameter-revisable sufficient statistic.

Study 03 uses an explicitly APPROXIMATE finite-conditioning likelihood and an information-budget calculation; it does not rely on the source's disputed conditional identity. Its validity is not altered by this finding. Likewise, Property 5's separate warning that decorrelation is not general non-Gaussian independence is important: Gaussian quadratic memories cannot silently be promoted to sufficient memories for sparse Levy innovations.

A targeted search for an erratum did not find one; that is not evidence that no correction exists. No author contact, external issue or public correction submission was made. This note records the independently checkable local reasoning and preserves the original source unchanged.

Source: A Unified Formulation of Gaussian Versus Sparse Stochastic Processes - Part II: Discrete-Domain Theory, IEEE Transactions on Information Theory 60(5), 3036-3051, May 2014, DOI 10.1109/TIT.2014.2311903. https://bigwww.epfl.ch/publications/unser1402.pdf

Original: research/compression_physical_memory_20260915/DEPENDENCE_CAVEAT.md · Raw source file

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# Finite dependence is not finite conditional memory

This is a mathematical safeguard found during the detailed source reread,
not a compression breakthrough or a claim that the entire source is invalid.

In Unser, Tafti, Amini and Kirshner (2014), Part II, printed page 3044 / PDF
page 9, Property 4 states a finite-past conditional-density identity for the
generalized increment process. Its proof invokes independence of samples
whose B-spline windows do not overlap. The non-overlap argument establishes
finite-range marginal dependence; it does not establish the displayed
finite-order Markov identity. This distinction matters even for Gaussian
innovations. The displayed equation was checked in the rendered source, not
inferred from faulty text extraction.

## Counterexample within the source's own model class

Take L=D^2 and unit Gaussian white noise. The generalized increment kernel is
the cardinal triangular B-spline beta(t), supported on [0,2], with unit slope
on [0,1] and negative unit slope on [1,2]. Define u_k=<w,beta(k-.)>.
Its covariance is

    Cov(u_k,u_k)=integral beta(t)^2 dt=2/3,
    Cov(u_k,u_(k-1))=integral beta(t) beta(t-1) dt=1/6,
    Cov(u_k,u_(k-2))=0.

The last equality gives marginal independence at lag two because the vector
is jointly Gaussian. But Gaussian conditioning gives

    Cov(u_k,u_(k-2) | u_(k-1))
       = 0 - (1/6)*(2/3)^(-1)*(1/6) = -1/24,

which is nonzero. Equivalently, the optimal prediction using the last two
samples is (4/15)u_(k-1)-(1/15)u_(k-2), while using only the last sample gives
(1/4)u_(k-1). Therefore this order-two increment process is not first-order
Markov, contrary to the conditional identity as written for N=2.

The covariance is that of a Gaussian MA(1), with invertible coefficient
2-sqrt(3) and innovation variance (2+sqrt(3))/6. A generic MA(1) has short
covariance range but an infinite autoregressive representation. This is an
elementary counterexample, not a new probabilistic phenomenon.

## Consequences for this campaign

Compact exponential B-splines still localize increments, and the covariance,
characteristic-functional and spectral-factorization machinery remains useful.
One must not infer that N-1 raw increments suffice for every future conditional
likelihood. Exact finite-dimensional latent filtering state at fixed parameters
is a different statement from finite observed-history Markov order or a
parameter-revisable sufficient statistic.

Study 03 uses an explicitly APPROXIMATE finite-conditioning likelihood and an
information-budget calculation; it does not rely on the source's disputed
conditional identity. Its validity is not altered by this finding. Likewise,
Property 5's separate warning that decorrelation is not general non-Gaussian
independence is important: Gaussian quadratic memories cannot silently be
promoted to sufficient memories for sparse Levy innovations.

A targeted search for an erratum did not find one; that is not evidence that
no correction exists. No author contact, external issue or public correction
submission was made. This note records the independently checkable local
reasoning and preserves the original source unchanged.

Source: A Unified Formulation of Gaussian Versus Sparse Stochastic Processes -
Part II: Discrete-Domain Theory, IEEE Transactions on Information Theory 60(5),
3036-3051, May 2014, DOI 10.1109/TIT.2014.2311903.
https://bigwww.epfl.ch/publications/unser1402.pdf