#
Geometric Shrinkage Priors for Kählerian Signal Filters^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Backgrounds

#### 2.1. Kählerian Filters

**ξ**is characterized by a transfer function h(w;

**ξ**) in the frequency domain w with

**ξ**) is defined as the absolute square of the transfer function

**ξ**) under the stability condition, minimum phase, and

_{r}is an impulse response function. The Z-transformed power spectrum is also defined in the similar way. In this case, the conditions on the transfer function for constructing information geometry are identical to the spectral density function representation except for

_{ij}and ${g}_{i\overline{j}}$, respectively.

_{r}is the coefficient of z

^{−r}in the series expansion of the logarithmic transfer function, also known as a complex cepstrum coefficient [15]. It is obvious that η

_{0}= log h

_{0}.

_{0}in the unilateral transfer function case, is a constant in model parameters

**ξ**. In this paper, for simplicity, we only consider unilateral transfer functions with non-zero h

_{0}and the Kähler manifolds with the explicit Hermitian conditions on the metric tensors because complex manifolds are always Hermitian manifolds [16]. In this case, the necessary and sufficient condition for being a Kähler manifold is that h

_{0}(

**ξ**) is a constant in

**ξ**[7].

^{2}-norm) of the logarithmic transfer function (or the square of the complex cepstrum norm) on the unit disk ⅅ

#### 2.2. Superharmonic Priors

**ξ**) based on given samples x of size N, one of the best approaches is using Bayesian predictive density p

_{π}(y|x

^{(}

^{N}

^{)}) with a prior π(

**ξ**):

_{I}are derived from the difference between two risk functions with respect to the true probability density, one from the Jeffreys prior and another from the superharmonic prior:

_{KL}is the Kullback–Leibler divergence and π

_{J}is the Jeffreys prior which is the volume form of the statistical manifold. Each risk function indicates how far a given Bayesian predictive density is from the true distribution in the Kullback–Leibler divergence in average. Sine better priors are obtained from smaller risk functions, the priors outperforming the Jeffreys prior make the above expression greater than zero. Since the first term on the right-hand side is non-negative, the risk function of the Komaki prior is decreased with respect to the risk function of the Jeffreys prior if a prior function ψ = π

_{I}/π

_{J}is superharmonic. If a superharmonic prior function ψ can be found, it is possible to do better Bayesian prediction in the viewpoint of information theory. In the same paper, Komaki also pointed out that shrinkage priors are information-theoretically more improved in prediction than the Jeffreys prior if and only if the square root of a prior function is superharmonic.

^{i}is a pole of the transfer function. Tanaka [6] generalized the two-dimensional case to superharmonic priors for the AR model in an arbitrary dimension p. The shrinkage prior function for the AR(p) model is in the form of

^{i}is a pole of the AR transfer function.

^{i}. However, its generalization to any arbitrary dimensions has been unknown. Moreover, the Komaki priors for the ARMA models and the ARFIMA models are not reported yet.

## 3. Geometric Shrinkage Priors

^{*}is a constant in $\mathit{\xi}=\left({\mathit{\xi}}^{1},\phantom{\rule{0.2em}{0ex}}{\mathit{\xi}}^{2},\cdots ,{\mathit{\xi}}^{n}\right)$ and its complex conjugate $\overline{\mathit{\xi}}$. The following lemma is worthwhile when the algorithm for the prior functions is constructed.

**Lemma 1.**On a Kähler manifold, a function$\psi (\mathit{\xi},\overline{\mathit{\xi}})$ is superharmonic if$\psi (\mathit{\xi},\overline{\mathit{\xi}})$ is in the form of$\psi (\mathit{\xi},\overline{\mathit{\xi}})=\mathrm{\Psi}\left({u}^{*}-\kappa (\mathit{\xi},\overline{\mathit{\xi}})\right)$ such that κ is subharmonic (or harmonic) and Ψ′(τ) > 0, Ψ″(τ) ≤ 0 (or Ψ′(τ) > 0, Ψ″(τ) < 0).

**Proof.**The Laplace–Beltrami operator on ψ is given by

**Theorem 1.**On a Kähler manifold, a positive function ψ = Ψ(u

^{*}−κ) is a superharmonic prior function if κ is subharmonic (or harmonic) and Ψ′(τ) > 0, Ψ″(τ) ≤ 0 (or Ψ′(τ) > 0, Ψ″(τ) < 0).

**Proof.**Since this is a special case of Lemma 1, the proof is obvious. □

**Example 1.**Given subharmonic (or harmonic) κ and positive τ, i.e., upper-bounded κ, the following functions are candidates for Ψ

**Proof.**We only cover a subharmonic case for κ here and it is also straightforward for the harmonic case. First of all, Ψ

_{1}and Ψ

_{2}are all positive. For Ψ

_{1}, it is easy to verify the followings:

_{2}:

_{1}and Ψ

_{2}satisfy the conditions for Ψ in Lemma 1.□

**Example 2.**For positive real numbers a

_{r}and b

_{i}, the following subharmonic functions are candidates for κ in the cases that those are upper-bounded:

**Proof.**Let us assume that the ansätze are upper-bounded in given domains. For κ

_{1}, it is easy to show that the Kähler potential K is subharmonic:

_{2}is as follows:

_{3}is tested by

## 4. Example: ARFIMA Models

_{i}, λ

_{i}, σ are a pole, a root, and a gain in the ARMA model, respectively. It is noteworthy that the transfer function of the ARFIMA model is decomposed into the ARMA model part and the fractionally integration part. Additionally, every poles and roots of the linear system are located inside the unit disk, i.e., |λ

_{i}| < 1 for i = 1, ⋯, p and |μ

_{i}| < 1 for i = 1, ⋯, q.

_{0}= 1 up to the gain of the signal filter. We will work on this submanifold.

_{2}and κ

_{3}are also utilized for the superharmonic prior function ansätze in the ARFIMA models because the both functions are upper-bounded on the ARFIMA manifold. Moreover, if we set d = 0 for κ

_{2}or b

_{0}= 0 for κ

_{3}, the ansätze for the ARFIMA models are reducible to the Komaki priors of the ARMA models.

## 5. Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Choi, J.; Mullhaupt, A.P.
Geometric Shrinkage Priors for Kählerian Signal Filters. *Entropy* **2015**, *17*, 1347-1357.
https://doi.org/10.3390/e17031347

**AMA Style**

Choi J, Mullhaupt AP.
Geometric Shrinkage Priors for Kählerian Signal Filters. *Entropy*. 2015; 17(3):1347-1357.
https://doi.org/10.3390/e17031347

**Chicago/Turabian Style**

Choi, Jaehyung, and Andrew P. Mullhaupt.
2015. "Geometric Shrinkage Priors for Kählerian Signal Filters" *Entropy* 17, no. 3: 1347-1357.
https://doi.org/10.3390/e17031347