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GPC: An expressive and tractable deep generative model for genetic variation data [1]

['Prateek Anand', 'Department Of Computer Science', 'University Of California', 'Los Angeles', 'California', 'United States Of America', 'Anji Liu', 'National University Of Singapore', 'Singapore', 'Meihua Dang']

Date: 2026-10 

Generative models play an important role in population genetics, being used to generate artificial genomes (AGs) that help benchmark methods, test evolutionary hypotheses, and construct reference panels for imputation while working around data-sharing restrictions. Existing generative models of genetic variation, however, struggle to faithfully express dependencies in the data while retaining tractability and preserving privacy. We introduce Genetic Probabilistic Circuits (GPC), a deep generative model for genetic variation data based on hidden Chow-Liu trees represented as probabilistic circuits. GPC generalizes traditional hidden Markov models by allowing arbitrary tree structures over latent variables, enabling it to capture long-range dependencies among SNPs. GPC is tractable, supporting exact computation of marginal and conditional probabilities that enables both AG generation and direct genotype imputation (avoiding the need for simulating AGs). We show that GPC generates AGs that accurately reproduce population structure and linkage disequilibrium patterns across a range of length scales. Compared to other deep generative approaches, GPC consistently improves imputation accuracy, with particularly strong gains for low-frequency variants and in populations that are not well-represented in public reference panels. Finally, we show that GPC better preserves privacy of the training data, thereby providing a practical framework for AG generation in settings with limited data access.

Human genetic data are essential for studying disease, but they are also deeply personal, and privacy rules often prevent researchers from sharing them. One way around this is to create artificial genomes: synthetic data that reproduce the statistical patterns of a real population without belonging to any real person. To be useful, artificial genomes have to capture how variants across the genome relate to one another, and they must not reveal information about the people whose data was used to build them. Existing methods struggle to do both at once. We describe a model that learns which positions in the genome most strongly influence each other and connects them directly, instead of assuming that only neighboring positions matter. The model is built so that its probabilities can be calculated exactly, which lets it fill in a person’s missing genetic information directly rather than by first simulating artificial data. We find that this improves the accuracy of filling in missing variants, especially rare ones and those in populations that are poorly represented in public reference collections, while revealing less about the individuals used for training.

Funding: The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. PA and SS are supported in part by NIH R35 GM153406 and NSF CAREER 1943497. AL is supported in part by National University of Singapore under its Start-up Grant SUG 251RES250. XW is supported in part by NIH R35 GM150579 and NSF IIS 2435801. GVdB is supported in part by the DARPA ANSR, CODORD, and SAFRON programs under awards FA8750 23 2 0004, HR00112590089, and HR00112530141, NSF IIS 1943641, and gifts from Adobe Research, Cisco Research, Qualcomm, and Amazon. Approved for public release; distribution is unlimited.

Copyright: © 2026 Anand et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

We first compare GPC to baselines on genomic regions chosen from the 1000 Genomes Project (1KG) and UK Biobank (UKBB), demonstrating that GPC achieves higher held-out log-likelihoods than simpler probabilistic models and generates AGs that accurately capture population structure and linkage disequilibrium patterns across all distance ranges. We then evaluate GPC extensively for genotype imputation. Across both datasets, GPC yields significant improvements in imputation accuracy compared to other deep generative models, with substantial improvements in settings where reference genomes from the target population are not publicly available. In such population-specific settings, GPC consistently outperforms other deep generative methods, with particularly strong gains for low-frequency variants. Finally, we find that AGs produced by GPC offer improved privacy properties compared to previous deep generative models. Overall, our results indicate that the increased expressivity and tractability of GPC lead to more accurate and privacy-preserving models of human genetic variation.

GPC, like GANs and RBMs, can generate AGs for use as reference panels with standard imputation tools. Unlike these methods, GPC can also perform imputation directly through efficient computation of conditional probabilities, without the need for generating AGs as an intermediate step. We show that this ability further improves imputation accuracy.

We address this need by introducing GPC (Genetic Probabilistic Circuits), a class of deep generative models that fits genetic variation data with high accuracy while supporting efficient and exact inference. To model the distribution over single nucleotide polymorphisms (SNPs) in an individual, we propose a latent variable model in which each SNP is associated with a hidden variable and the hidden variables form a tree-structured graphical model. This model, termed the hidden Chow-Liu tree (HCLT) [ 35 ], generalizes the HMM. Classical HMMs impose a chain structure where hidden variables corresponding to consecutive SNPs are adjacent in the tree. HCLTs relax this constraint by allowing arbitrary tree structures, which enables the model to capture long-range dependencies, or linkage disequilibrium (LD), across the genome [ 36 ]. This flexible structure allows HCLTs to more faithfully represent the distribution of genetic variation. While HCLTs are more expressive than HMMs, naively training and performing inference in HCLTs at the scale required for modern genomic datasets would be computationally prohibitive. Our approach represents HCLTs as probabilistic circuits (PCs) [ 37 , 38 ], a circuit-based framework that permits tractable inference tasks such as exact likelihood and conditional probability computation. This representation allows us to leverage stochastic gradient-based learning and GPU acceleration to train HCLTs with millions of parameters on large genomic datasets.

Motivated by the success of deep learning, recent work has explored deep generative models as an alternative approach to represent genetic variation [ 23 – 25 ]. Models based on generative adversarial networks (GANs) [ 25 – 28 ], variational autoencoders (VAEs) [ 23 , 29 , 30 ], restricted Boltzmann machines (RBMs) [ 25 ], and more recently diffusion models [ 31 ] can be more expressive than HMMs and often produce AGs that visually resemble real genomes in principal component analysis (PCA) or allele frequency summaries. However, these models have critical limitations for genomic applications. First, GANs do not define a probability distribution over the data, making likelihood-based inference impossible. RBMs define a distribution but require computing an intractable partition function over exponentially many configurations, preventing exact likelihood evaluation. VAEs define a probability distribution but the marginal likelihood requires integrating over a continuous latent space with a nonlinear decoder, which is intractable; only a lower bound (the ELBO) is accessible in practice. Diffusion models can in principle compute exact likelihoods via probability flow ordinary differential equations, but scale poorly to SNP data: existing approaches [ 31 ] require lossy dimensionality reduction (e.g., per-gene PCA) as a preprocessing step before the model operates [ 31 ], and have not been evaluated on imputation tasks. Second, none of these approaches support efficient conditional probability estimation. This limitation makes model comparison difficult and limits their use in downstream tasks such as genotype imputation, which require computing the probability of unobserved variants conditioned on observed variants. A common workaround involves generating AGs from these models, which then serve as a reference panel for imputation methods such as Impute5, Beagle, or Minimac [ 32 – 34 ], though this approach can introduce additional noise in the imputation process. Third, training these models is challenging due to complex optimization objectives and large hyperparameter spaces, and convergence must be assessed through subjective visual inspection rather than principled likelihood monitoring. These difficulties collectively highlight the need for deep generative models that are both expressive and tractable.

Classical population genetic simulators rely on the coalescent model [ 11 ], which generates genomes based on demographic history together with mutation and recombination rates. The coalescent with recombination [ 12 ] connects observed genetic variation to latent genealogies that change along the genome [ 13 ] and is highly expressive, but exact inference under this model is computationally challenging, motivating tractable approximations. One class approximates the coalescent as a Markovian process along the genome [ 14 , 15 ], forming the basis of widely used simulators such as msprime [ 9 ] and its extensions [ 10 ]. A second class dispenses with explicit genealogical modeling and instead directly approximates the data distribution. A prominent example is the product-of-approximate-conditionals (PAC) model [ 16 ], which naturally yields a hidden Markov model (HMM) [ 17 ]. HMM-based models have proven highly successful in applications including haplotype phasing [ 18 , 19 ], genotype imputation [ 3 , 20 ], and ancestry inference [ 21 , 22 ].

Generative models of genetic variation play a central role in population genomics [ 1 , 2 ]. By modeling dependencies across individuals and variants, these models support core analyses such as genotype imputation [ 3 ], haplotype phasing [ 4 ], and ancestry inference [ 5 ]. They also form the foundation of simulators that generate artificial genomes (AGs) [ 6 – 10 ], which are widely used to test evolutionary hypotheses, estimate population genetic parameters, validate empirical results, and benchmark new methods. As the sharing of primary genetic data becomes increasingly restricted due to privacy and consent constraints, accurate simulation tools have become essential for reproducibility and equity. Trained generative models, or data simulated from them, can in principle be shared without requiring access to identifiable genomic data.

2 Results

2.1 Methods overview We present GPC, a deep generative model for genetic variation data. GPC is based on hidden Chow-Liu trees (HCLTs) [35], a class of latent variable models that generalize hidden Markov models (HMMs), represented as probabilistic circuits (PCs) [37,38] for efficient inference and learning. Hidden Chow-Liu trees HCLTs are tree-structured graphical models that represent the probability distribution over observed random variables, . In the context of genetic variation across N SNPs, each observed represents the genotype at SNP n (e.g., for haploid genomes). HCLTs rely on the following key designs: Hidden Variable Mapping: For every observed variable

Chow-Liu Tree Architecture: Each observed variable 39] to capture the strongest pairwise correlations between variables (Fig 1).

39] to capture the strongest pairwise correlations between variables (Fig 1). Parameterization: The entire HCLT is defined by just two sets of conditional probability distributions: the emissions PNG larger image

TIFF original image Download: Fig 1. Generating HCLT structures given genetic data. (a) Genotype data: An example dataset of 6 SNPs ( ), where dashed lines indicate SNPs with strong correlations. The highlighted pair X 2 and X 6 is strongly correlated despite being distant in genomic order. (b) HMM: The chain over latent variables imposed by a hidden Markov model, with each emitting its own . Because Z 2 and Z 6 are not adjacent, the dependence between X 2 and X 6 has no direct edge: it must be routed through Z 3 , Z 4 and Z 5 , and is weakened at each intervening latent variable, since each has a finite number of states. (c) Learned Chow–Liu tree: A Chow–Liu Tree learned from the SNP data. The tree structure places latent variables with stronger dependencies closer together, capturing the dominant correlation structure among the SNPs; the same correlated pair is now adjacent, so the dependence is carried by a single edge. (d) HCLT: The Hidden CLT constructed from the CLT, in which each latent variable is additionally connected to its corresponding observed SNP as a child node. The resulting structure respects the dependencies among SNPs while providing a principled generative model over the observed data. The topology of a Chow–Liu tree learned from genetic data is shown in Fig 5. https://doi.org/10.1371/journal.pgen.1012321.g001 The critical distinction from HMMs is structural. Classical HMMs impose a chain structure where hidden variables corresponding to consecutive SNPs are adjacent in the tree ( ). This chain topology can only propagate information between distant SNPs through all intermediate variables. HCLTs relax this constraint by allowing arbitrary tree structures, so that SNPs with strong long-range correlations (such as X 2 and X 6 being highly correlated without being correlated with intermediate SNPs) can be placed close together in the tree (Fig 1). This flexible structure allows HCLTs to more faithfully capture linkage disequilibrium patterns at all length scales. Fig 1 contrasts the two topologies on the same example. Under the chain (Fig 1b), Z 2 and Z 6 are four edges apart, so the dependence between X 2 and X 6 has no direct edge and must be carried through Z 3 , Z 4 and Z 5 . Each of these latent variables has a limited number of states, so some of the correlation is lost at every step. In the learned Chow–Liu tree (Fig 1c) the same pair is adjacent, and the dependence is represented by a single edge (Fig 2). The topology of an HCLT learned from data is shown in Fig 5. PNG larger image

TIFF original image Download: Fig 2. Experimental design for the imputation evaluations. (a) Training haplotypes are used to fit each generative model. Along the AG route, the fitted model is sampled to produce artificial genomes, which form an Impute5 reference panel, either the AGs alone or the AGs merged with real European genomes, and Impute5 then imputes the masked SNPs of the held-out test haplotypes. Along the direct route, the conditional is computed exactly from the model itself, with no artificial genomes and no Impute5; this route is available only to models that support tractable conditional inference. A separate benchmark supplies real reference genomes to the same Impute5 step, which isolates the contribution of Impute5 from that of the generative model. Both routes end at the same accuracy computation, the squared Pearson correlation r2 between imputed dosages and the true genotypes, binned by minor allele frequency with confidence intervals from 10 bootstrap replicates. (b) The three imputation settings share this pipeline and differ only in the training split, the test split, and which SNPs are masked: one SNP at a time for the general and population-specific settings, and the non-array SNPs jointly for the array-based setting. https://doi.org/10.1371/journal.pgen.1012321.g002 Probabilistic circuits and tractable inference While HCLTs are more expressive than HMMs, naively training and performing inference in HCLTs at the scale required for modern genomic datasets would be computationally prohibitive. Our key insight is to represent HCLTs as probabilistic circuits (PCs) [37,38]. A PC represents a joint probability distribution through a directed acyclic graph (DAG) consisting of input, sum, and product nodes. Product nodes define factorized distributions over their inputs, while sum nodes represent mixtures weighted by learnable parameters. Under structural constraints known as smoothness and decomposability, arbitrary marginal queries can be computed in time linear in the size of the circuit. This tractability is what enables GPC to perform three key operations efficiently: (1) computing likelihoods on held-out data for principled model comparison, (2) generating artificial genomes via ancestral sampling, and (3) performing direct genotype imputation through exact conditional probability computation. By representing HCLTs as equivalent PCs and training them using the PC package PyJuice [40], we can extensively parallelize Expectation-Maximization (EM) updates on GPUs, enabling efficient training even for models with over 88 million parameters (Section 4.2). Generating AGs from GPC is performed via ancestral sampling through the circuit in time linear in the circuit size. For genotype imputation, GPC computes conditional probabilities directly as ratios of marginal queries, both of which can be evaluated via a single feedforward pass through the circuit. This direct imputation capability, unique to GPC among the deep generative approaches we consider, eliminates the need for intermediate AG generation and, as we show, yields consistently higher accuracy (see Section 4.2.2 for full details on sampling and conditional inference).

2.2 Data Our experiments evaluate different aspects of GPC in terms of its ability to generate realistic AGs, its ability to perform genotype imputation, and its privacy properties. We perform the evaluations on three datasets with distinct properties: 1000 Genomes Project Phase 3 (1KG) [41], UK Biobank (UKBB) [42], and high-coverage 1KG (high-coverage 1KG) [43], with the high-coverage 1KG dataset only being used for the array-based imputation experiment (see Section 4.1 for details).

2.3 Evaluation Baselines We compare GPC against two categories of baselines. The first category consists of deep generative models that have been previously proposed for AG generation: (1) generative adversarial networks (WGAN) and (2) restricted Boltzmann machines (RBM) as implemented in [25]. We focus our comparison on these two methods as they are the most established deep generative approaches specifically designed and benchmarked for the local genetic structure and imputation tasks we evaluate here. More recent approaches including VAEs [29,30] and diffusion models [31] have been proposed for related but distinct tasks, and do not support efficient, tractable probabilistic inference; we leave systematic comparison with these methods to future work. For both WGAN and RBM baselines, we use the samples generated by the corresponding authors for comparison (Note that [25] did not perform train/test splits so WGAN and RBM are actually trained on train+test.) in our 1KG experiments, but we also retrain their models on our own training split to ensure a fair comparison. In imputation experiments, we additionally compare against Impute5 [32], a state-of-the-art imputation tool widely used in genetics. Impute5 serves a dual role: it is used with real reference genomes to establish performance benchmarks, and it is used with AG reference panels generated by each method to evaluate their utility for imputation. The second category consists of simpler probabilistic graphical models (PGMs) that support tractable likelihood computation: fully-factorized distributions (Indep), Markov chain models of order 1 (Markov), and non-homogeneous hidden Markov models (HMM). These serve as baselines for density estimation and AG quality comparisons. The HMM is built within the same circuit framework as GPC and differs from it only in the latent graph, a chain rather than a learned tree, so every comparison between the two isolates the contribution of the topology. Full specifications are given in Section 4.4. Evaluation criteria We evaluate these models using the following metrics: (1) log-likelihood on test data to assess the capability of each model as a density estimator; (2) summaries of AGs sampled from each model, including the top principal components and linkage disequilibrium at pairs of SNPs; (3) genotype imputation, which evaluates the models’ utility on an important downstream task; and (4) privacy analysis of the AGs sampled from each model.

2.4 GPC trains efficiently with objective convergence criteria Training GPC requires minimal hyperparameter tuning. We set the number of latent states to 128 (the maximum feasible given memory constraints) and use a small pseudocount (0.005) for smoothing. Models were trained for 2,000–5,000 epochs depending on dataset size. We use the PC package PyJuice [40] to train GPC efficiently on GPUs, achieving EM updates in under 2, 10, and 3 seconds per epoch on the 1KG, UKBB, and high-coverage 1KG datasets, respectively. On a single NVIDIA RTX A5000 GPU with 24GB available memory, GPC trains in roughly 2, 6, and 4 hours on each dataset, respectively. Once the model is loaded, generating a single sample can be done in under 2, 5, and 3 seconds for each dataset, respectively. Imputing the high-coverage 1KG dataset for the entire held-out set (12,551 SNPs across 1,002 haplotypes) takes about one second in a single conditional query; Impute5 with a real reference panel completes the same task in about four seconds. Inference is inexpensive by either route, and the cost that separates the methods is training rather than imputation. A systematic comparison of training time and memory against RBM and WGAN, and the resulting working range of GPC, is given in Section 4.5. A key advantage over other deep generative approaches is that convergence can be monitored via held-out log-likelihood, providing an objective stopping criterion rather than subjective visual inspection. Full training details for GPC and all baseline methods are provided in Section 4.4.

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[1] Url: https://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1012321

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