Probabilistic Completion Models

• Probabilistic completion models are generative frameworks that infer unobserved data from partial inputs by modeling conditional likelihoods, capturing inherent ambiguity and multimodality.
• They employ techniques such as autoregressive networks, variational autoencoders, and diffusion models to offer calibrated uncertainty and diversity in applications like code synthesis, matrix, and tensor completion.
• These models have broad applications, from 3D scene and semantic shape reconstruction to knowledge graph and image completion, underpinned by rigorous Bayesian and likelihood-based inference.

Probabilistic Completion Models

Probabilistic completion models are generative frameworks that infer the full, unobserved data (or structure) from partial observations by explicitly modeling the conditional distribution over all possible completions. Unlike deterministic completion approaches, which produce a single best-guess solution, probabilistic formulations capture the inherent ambiguity and multimodality of the completion problem. These models assign calibrated probabilities to possible completions, enabling uncertainty quantification, diversity sampling, and rigorous Bayesian or likelihood-based inference. Probabilistic completion is central to practical applications ranging from code synthesis and 3D scene reconstruction to knowledge graph and matrix completion.

1. Probabilistic Formulation of Completion

At their core, probabilistic completion models specify or approximate the conditional likelihood $P(\text{completed} \mid \text{partial})$. The classical chain-rule factorization is ubiquitous: for sequence or token-level data such as autocompletion, this becomes

$$P(S) = \prod_{t=1}^n P(w_t \mid w_1, ..., w_{t-1}).$$

When task structure allows, this is extended to higher-level units (e.g., lines of code or semantic clusters). In code completion, the focus shifts from single-token prediction to the probability of an entire code line, conditioned on surrounding context (Wang et al., 2020): $$P(s_1...s_T \mid \text{context}) = \prod_{t=1}^T P(s_t \mid \text{context}, s_1, ..., s_{t-1}).$$ In matrix and tensor completion, the probabilistic approach models the unknown entries as latent variables and fits a low-dimensional factorization under a probabilistic loss (e.g., Gaussian, multinomial) (Lafond et al., 2014, Zhao et al., 2015, Léger et al., 2010). For knowledge graphs, entities and relations are assigned probabilistic embeddings, scores, or circuit-based semantics with distributions over possible missing facts (Fan et al., 2015, Wang et al., 18 May 2025, Li et al., 24 Nov 2024). In 3D scene and semantic completion, models often learn the distribution of plausible geometric or semantic completions conditioned on sparse, noisy, or occluded input (Zhang et al., 2022, Zhou et al., 2021, Cao et al., 26 Sep 2024).

Table 1 illustrates representative probabilistic formulations.

Application Domain	Conditional Distribution	Example Reference
Code completion	$P(\text{line}\mid\text{context})$	(Wang et al., 2020)
Matrix completion	$P(\text{matrix} \mid \text{observed})$	(Lafond et al., 2014, Léger et al., 2010)
Shape completion	$P(\mathbf{X} \mid \mathbf{X_\text{obs}})$	(Zhou et al., 2021, Jiang et al., 2022)
Knowledge graphs	$P(\text{triple}\mid\text{KG})$	(Fan et al., 2015, Wang et al., 18 May 2025)

2. Model Architectures and Inference Methods

The choice of probabilistic model and inference procedure is dictated by domain structure and computational tractability. Major paradigms include:

1. Autoregressive Language and Transformer Models: For structured or sequential domains (text/code), neural LMs (RNNs, Transformers) model $P(\text{completion}\mid\text{context})$ by left-to-right or masked decoding, optimized via negative log-likelihood (Wang et al., 2020). Improvements include subword/BPE tokenization, syntax masking (e.g., action-masked decoding for code ASTs), and beam search for diverse outputs.

2. Latent Variable and VAE Models: For problems with latent structure (e.g., matrix/tensor, knowledge graphs, 3D shapes), variational autoencoders (VAEs) introduce continuous or discrete latent codes, optimizing the ELBO: $$\log p_\theta(\mathbf{x}) \geq \mathbb{E}_{q(z\mid \mathbf{x})} [ \log p_\theta(\mathbf{x}\mid z)] - D_{KL}(q(z\mid \mathbf{x}) || p(z)).$$ Hierarchical or relational VAEs are used for high-dimensional shapes (Jiang et al., 2022, Pan et al., 2021), with canonical factor (CP/Tucker) decompositions in tensors (Zhao et al., 2015, Jiang et al., 2022).

3. Denoising Diffusion Probabilistic Models: Generative diffusion models have become state-of-the-art for multimodal 3D and semantic completion, employing learned Markov chains to denoise from pure noise towards a plausible full sample, conditioned on partial data (Zhou et al., 2021, Cao et al., 26 Sep 2024, Nakashima et al., 2023).

4. Probabilistic Circuits and Rule Context Mixtures: In symbolic domains (e.g., rule-based knowledge graphs), probabilistic circuits over contexts efficiently express highly correlated or non-independent rule application, enabling explainable, tractable, and uncertainty-aware inference (Patil et al., 8 Aug 2025). Each circuit node defines a probability over a context or subset of active rules.

5. Count-based, Nonparametric Case Reasoning: For knowledge graphs, nonparametric models estimate probabilities of completions by aggregating statistics of relational paths among similar entities, with all probabilities estimated via closed-form count ratios and simple priors (Das et al., 2020).

6. Mixture-of-Experts and GMM Models: In pluralistic image completion, Gaussian mixture models over latent codes parameterize the multimodal conditional distribution $q(z_c\mid z_m)$, allowing direct control of sample diversity (Xia et al., 2022).

Model training typically requires maximizing variational lower bounds (ELBO), negative log-likelihood, or tractable surrogates (e.g., score-matching in diffusion models), and inference is conducted via sampling, beam search, or circuit marginalization.

3. Datasets, Evaluation Metrics, and Experimental Protocols

Probabilistic completion models are assessed using both classical and distributional metrics, often requiring tailored datasets with rich annotation of plausible completions:

• Exact Match, MRR, Top-k Accuracy: Common in code (Wang et al., 2020), knowledge graph (Wang et al., 18 May 2025, Fan et al., 2015), and matrix completion (Lafond et al., 2014) as standard supervised metrics.
• Distributional Divergence (KL, JS): For tasks with diverse valid completions—e.g., commonsense frame completion (Cheng et al., 6 Jun 2024), pluralistic image completion (Xia et al., 2022), and multimodal 3D shape completion (Jiang et al., 2022, Zhou et al., 2021)—probabilistic metrics using KL divergence between the empirical human and model distributions are used.
• BLEU, Edit Similarity, Chamfer, and EMD: Structure- and geometry-aware metrics for code (Wang et al., 2020), shapes (Zhou et al., 2021), and images.
• Uncertainty Calibration: Coverage, uncertainty region area, and correct/incorrect free-space volumes for depth and LiDAR completion (Popovic et al., 2020, Cao et al., 26 Sep 2024, Nakashima et al., 2023).
• Community Interpretability and Latent Structure: For knowledge graphs, models such as DSLFM-KGC evaluate the interpretability and sparsity of inferred community clusters and provide empirical analysis of latent feature structure (Li et al., 24 Nov 2024).

Data regimes span large code corpora, 3D shape and scan repositories, synthetic and real-world LiDAR datasets, standard KG benchmarks (WN18RR, FB15K-237, etc.), and massive survey-annotated commonsense datasets (Cheng et al., 6 Jun 2024).

4. Representative Applications

Code and Sequence Completion

Transformer LMs, trained on code tokens or grammar actions, yield plausible whole-line completions with calibrated uncertainty over predictions, handling OOV/token fragmentation via subword models and enhancing semantic validity through syntax-awareness. Inference produces diverse top-k completions and enables identifier-agnostic assessment (Wang et al., 2020).

3D Scene, Semantic, and Shape Completion

Probabilistic completion is essential for shape completion from partial, noisy scans:

• Diffusion models (e.g., cGCA, PVD, DiffSSC) capture multimodality and generate high-fidelity, diverse geometric and semantic completions (Zhang et al., 2022, Zhou et al., 2021, Cao et al., 26 Sep 2024).
• VAE-based models, often with hierarchical stochastic layers, estimate distributions over local canonical factors or global shape latents and yield state-of-the-art UHD and TMD metrics for completion (Jiang et al., 2022, Pan et al., 2021).
• Depth completion pipelines for mapping in robotics explicitly propagate per-pixel or per-voxel uncertainty, leading to more accurate maps and avoidance of spurious structure (Popovic et al., 2020, Nakashima et al., 2023).

Knowledge Graph Completion

Multiple probabilistic paradigms are supported:

• Embedding and interaction models define explicit probability distributions over triples and support maximum likelihood or variational EM training (Fan et al., 2015, Kim et al., 2016, Wang et al., 18 May 2025, Li et al., 24 Nov 2024).
• Probabilistic circuits or logic program mixtures capture dependencies among rules, yielding tractable, high-accuracy completion with reduced rule sets (Patil et al., 8 Aug 2025).
• Nonparametric path-based models efficiently marginalize over reasoning templates derived from similar entities, supporting open-world, streaming updates without retraining (Das et al., 2020).

5. Theoretical Guarantees and Algorithmic Properties

Many frameworks derive non-asymptotic recovery guarantees and rigorous Bayesian quantification:

• Nuclear-norm penalized estimators yield KL and Frobenius norm error bounds scaling as $O(\frac{(\text{rank}\cdot\log N)}{n})$ for multinomial/binary matrix completion under nonuniform sampling (Lafond et al., 2014).
• Bayesian sparse Tucker models with group-sparsity priors automatically infer optimal multilinear rank, recovering uncertainty over missing tensor entries via explicit posterior computation (Zhao et al., 2015).
• Probabilistic circuits and their marginal queries require only linear time in the size of the rule context and circuit, with exact/approximate inference guarantees (Patil et al., 8 Aug 2025).
• For log-linear models, results on the number and uniqueness of completions, and geometric boundary characterization, are provided (Cai et al., 2023).

Limitations include computational scalability for high-dimensional factors, combinatorial explosion of compositional paths, or underestimation of epistemic uncertainty in MAP approximations without full variational inference.

6. Implications, Limitations, and Future Directions

Probabilistic completion models provide calibrated uncertainty, sample diversity, and explainability superior to deterministic approaches. However, challenges remain:

• Balancing fluency/diversity against precision, especially in open-ended outputs or high-dimensional spaces.
• Efficiently capturing rare or long-tail completions (e.g., rare identifier or object completions) without excessive fragmentation (Wang et al., 2020).
• Incorporating richer structural, semantic, or context cues (e.g., type constraints in code, community structure in KGs, or global clustering for shapes) (Li et al., 24 Nov 2024, Jiang et al., 2022).
• Bridging the gap between probabilistic generative training objectives and practical diversity in sampling and evaluation (e.g., KL-based training for cluster-diverse answer sets (Cheng et al., 6 Jun 2024)).
• Unifying symbolic and neural models for knowledge completion, especially by leveraging probabilistic circuits and hybrid approaches (Patil et al., 8 Aug 2025, Wang et al., 18 May 2025).

Continued integration of scalable variational inference, diffusion-based generative modeling, and structured probabilistic logic is poised to further advance the state of probabilistic completion across domains.