Pseudo-Likelihood Estimator

• Pseudo-Likelihood Estimator is a method that replaces complex full likelihoods with products of local conditional likelihoods to enable tractable inference.
• It achieves strong consistency and, under finite-range conditions, asymptotic normality, ensuring reliable parameter estimation in spatial and network models.
• The approach is effectively applied to models like the Lennard–Jones process, demonstrating its practical utility in analyzing high-dimensional dependencies.

A pseudo-likelihood estimator is a statistical method that replaces the generally intractable or computationally intensive full likelihood with a product or sum of tractable local or conditional likelihood components. It was originally introduced by Besag (1974) and has found wide application in spatial statistics, Markov random fields, network models, graphical models, latent variable models, and numerous high-dimensional inference problems. This approach exploits the local dependency structure of complex systems to yield estimators that are both computationally feasible and, under appropriate conditions, possess desirable asymptotic properties.

1. Mathematical Foundations and Definition

A pseudo-likelihood estimator substitutes the true likelihood—typically intractable due to global dependencies—with a product (or sum of logs) of local conditional likelihoods. For a random object $Y$ with components $(Y_1,\dots, Y_p)$, and parameter vector $\theta$, the log pseudo-likelihood is defined as

$$\mathcal{L}_{PL}(\theta; y) = \sum_{j=1}^p \log P_\theta(y_j | y_{-j}),$$

where $y_{-j}$ denotes all components of $y$ except $y_j$.

In the context of spatial Gibbs point processes, the pseudo-likelihood is constructed using the local energy function $V(x|\varphi; \theta)$ (for configuration $\varphi$ in bounded domain $\Lambda$): $$PL_\Lambda(\varphi; \theta) = \exp\left( - \int_\Lambda e^{-V(x|\varphi; \theta)} dx \right) \prod_{x \in \varphi_\Lambda} e^{-V(x | \varphi \setminus \{x\}; \theta)}$$ and the normalized (contrast) function driving estimation is

$$U_n(\theta) = -\frac{1}{|\Lambda_n|} \ell_{\Lambda_n}(\varphi; \theta).$$

The estimator, termed the maximum pseudo-likelihood estimator (MPLE), is the minimizer: $$\hat{\theta} = \arg\min_{\theta \in \Theta} U_n(\theta).$$

2. Asymptotic Properties: Consistency and Normality

Theoretical analysis of the MPLE requires a set of structural conditions on the local (conditional or energy-related) functions:

• Strong Consistency: Under integrability, identifiability, continuity, and differentiability conditions on $V(x|\varphi; \theta)$ (labeled [C1]–[C4] in (Coeurjolly et al., 2010)), the MPLE converges almost surely to the true parameter value $\theta^\star$ as the observation domain $|\Lambda_n| \rightarrow \infty$:

$$\hat{\theta} \xrightarrow{\text{a.s.}} \theta^\star.$$

• Asymptotic Normality: With additional conditions ([N1]–[N4]) involving higher-order differentiability and moment integrability (ensuring, e.g., positive definiteness of limiting Hessian and covariance), the MPLE satisfies a central limit theorem:

$|\Lambda_n|^{1/2} \ddU(\theta^\star) (\hat{\theta} - \theta^\star) \xrightarrow{d} \mathcal{N}\left(0,\ \Sigma(D, \theta^{\star})\right)$

where the explicit form of $\Sigma(D, \theta^\star)$ involves expectations of blockwise gradient products over the tessellated observation domain.

These results are proven for general Gibbs point processes, removing older restrictions such as local stability or linearity in the parameters.

3. Sufficient Conditions in Terms of Model Structure

The foundational conditions ensuring the above asymptotic properties are expressed explicitly in terms of the local energy function $V(x|\varphi; \theta)$:

• Integrability ([C1]): $e^{-V(0|\Phi; \theta)}$ and its product with $|V(0|\Phi; \theta)|$ must have finite expected value.
• Identifiability ([C2]): There must exist a collection of $\ell \geq p$ events (where $p$ is the number of parameters) such that a certain contrast difference $D(0 | \varphi)$ vanishing on these events implies $\theta = \theta^\star$.
• Continuity ([C3]): The contrast $U_n(\varphi;\theta)$ is almost everywhere continuous in $\theta$.
• Differentiability ([C4]): $V(x|\varphi; \theta)$ is differentiable in $\theta$ and derivatives weighted by $e^{-V(0|\Phi; \theta)}$ have finite second moments.

For asymptotic normality, these are augmented by additional differentiability and higher-moment integrability requirements to guarantee existence and regularity of the Hessian and variance matrices that control the limiting distribution.

4. Applications to the Lennard–Jones Model

The theory is directly applied to the Lennard–Jones (LJ) model in spatial statistics, characterized by the pairwise local energy: $$V^{\mathrm{LJ}}\left( x |\varphi; \theta \right) = \theta_1 + 4 \theta_2 \sum_{y \in \varphi} \left[ \left( \frac{\theta_3}{\|x - y\|} \right)^{12} - \left( \frac{\theta_3}{\|x - y\|} \right)^6 \right]$$ with $\theta_1 \in \mathbb{R}$, $\theta_2, \theta_3 \in \mathbb{R}^+$. The results in (Coeurjolly et al., 2010) demonstrate:

• Strong consistency for the MPLE holds for both the infinite-range and finite-range variants, by verifying the above sufficient conditions.
• Asymptotic normality is established only for the finite-range version, due to a critical locality condition ([Mod–L]) required for controlling global dependencies in the normality proof.

Thus, for estimating parameters in the LJ process, MLPE provides both robustness (avoiding likelihood degeneracies) and a basis for inferential theory, especially under finite-range truncation, as is relevant in empirical applications.

5. Central Mathematical Formulas

The pseudo-likelihood and its key components are formulated as:

• Pseudo-likelihood:

$$PL_{\Lambda}(\varphi;\theta) = \exp\left( - \int_{\Lambda} e^{-V(x|\varphi;\theta)}\,dx \right) \prod_{x\in\varphi_{\Lambda}} e^{-V(x|\varphi\setminus\{x\};\theta)}$$

• Log pseudo-likelihood:

$$\ell_{\Lambda}(\varphi; \theta) = - \int_{\Lambda} e^{-V(x|\varphi;\theta)}\,dx - \sum_{x\in\varphi_\Lambda} V(x|\varphi\setminus \{x\};\theta)$$

• Scaled contrast:

$$U_n(\theta) = -\frac{1}{|\Lambda_n|} \ell_{\Lambda_n}(\varphi;\theta)$$

• Gradient (score) for estimation:

$\left( \Vect{U}^{(1)}_n(\theta) \right)_j = -\frac{1}{|\Lambda_n|} \left[ \int_{\Lambda_n} \frac{\partial V(x|\varphi;\theta)}{\partial \theta_j} e^{-V(x|\varphi;\theta)} dx - \sum_{x\in\varphi_\Lambda} \frac{\partial V(x|\varphi\setminus\{x\};\theta)}{\partial \theta_j} \right]$

6. Broader Implications for Spatial Statistics and Beyond

The results in (Coeurjolly et al., 2010) relax key limitations of earlier work on pseudo-likelihood methods in point process inference. They show that:

• Strong consistency can be established without the need for local stability or linearity in the local energy.
• Asymptotic normality, enabling classical inference (such as Wald-type confidence sets), can be rigorously established under finite-range interactions.
• The explicit, verifiable sufficient conditions guide practitioners in checking when a particular spatial model is amenable to pseudo-likelihood inference.
• For physically motivated models such as the Lennard–Jones process—ubiquitous in statistical physics and material science—pseudo-likelihood estimation is theoretically justified even with strong, nonlinear interactions.

These contributions make the MPLE not only computationally pragmatic in complex models but also inferentially robust. Standard errors and hypothesis tests can be constructed using formulas for the asymptotic variance derived from derivatives of the local energy. The framework provides a rigorous statistical foundation for analyzing high-dimensional and spatially complex processes using tractable, local inference tools.

7. Summary Table: Asymptotic Properties and Model Scope

Property	Model Requirements	Implication
Strong Consistency	[C1]–[C4]: Integrability, identifiability, etc.	MPLE converges to true value
Asymptotic Normality	[C1]–[C4], [N1]–[N4]: + differentiability, moments	MPLE is asymptotically normal
Lennard–Jones (Infinite Range)	[C1]–[C4]	MPLE consistent only
Lennard–Jones (Finite Range)	[C1]–[C4], [N1]–[N4]	MPLE consistent and normal

This table summarizes how the combined sufficient conditions enable both strong consistency and normality, with the key distinction that finite-range models guarantee full inferential validity.

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In conclusion, the pseudo-likelihood estimator is a rigorously tractable substitute for the full likelihood in spatial and other high-dimensional dependent data models. With explicit sufficient conditions, it provides reliable and efficient inference for a wide class of models—including nonlinear and non-stable energies—that are otherwise challenging for classical likelihood-based approaches (Coeurjolly et al., 2010).