Introduction: The Missing Data Problem in Scientific Simulators

From modeling the spread of diseases in epidemiology to simulating gravitational waves in astrophysics, computer simulations have become an indispensable tool for understanding complex phenomena. These mechanistic models—often called simulators—allow scientists to explore how theoretical models behave under various conditions, testing hypotheses and estimating parameters that are otherwise inaccessible through experiments.

However, fitting these simulators to real-world data poses a major challenge. The likelihood function—the probability of observing the data given the model parameters, \( p(\mathbf{x} \mid \theta) \)—is often intractable or computationally expensive to evaluate. Simulation-Based Inference (SBI) methods overcome this obstacle by relying on forward simulations instead of explicit likelihood evaluations. By repeatedly simulating data under different parameters, SBI learns to infer posteriors \( p(\theta \mid \mathbf{x}) \) even when likelihoods are unavailable.

But there’s a catch: SBI assumes the observational data is complete and clean. In reality, datasets are often riddled with missing values due to sensor failures, instrument limits, or incomplete sampling. Naively filling in missing entries with zeros or mean values may seem practical—but it can distort relationships between variables and bias the inferred parameters.

The effect of missing data on SBI. As the percentage of missing data (ε) increases, posteriors estimated with a naive imputation method (zero-filling) become biased and move away from the true parameter value (black cross).

Figure 1. As the percentage of missing data (\( \varepsilon \)) increases, naive zero-filling imputation leads to biased posteriors that drift away from the true parameter (black cross).

This is the central challenge addressed by Verma et al. in “Robust Simulation-Based Inference under Missing Data via Neural Processes.” The authors introduce RISE—short for Robust Inference under imputed SimulatEd data—an approach that performs parameter inference and data imputation jointly. By coupling these tasks in a single amortized framework, RISE mitigates bias and produces reliable estimates even under extensive missingness.

In this post, we’ll unpack the problem of missing data in SBI, explore the RISE methodology, and highlight its performance across benchmark and real-world tasks.

Background: SBI and the Challenge of Incomplete Data

Before diving into RISE, it’s important to understand the tools it builds upon—particularly Neural Posterior Estimation (NPE) and the different mechanisms of missing data.

Neural Posterior Estimation (NPE) in a Nutshell

In Bayesian inference, we are interested in the posterior distribution \( p(\theta \mid \mathbf{x}) \)—the probability of parameters \( \theta \) given observed data \( \mathbf{x} \). Instead of directly computing this (which is infeasible when likelihoods are intractable), NPE learns to approximate it using a neural network \( q_{\phi}(\theta \mid \mathbf{x}) \) with parameters \( \phi \).

The training pipeline is simple yet effective:

  1. Sample a parameter vector \( \theta_i \) from its prior \( p(\theta) \).
  2. Simulate data \( \mathbf{x}_i \sim p(\cdot \mid \theta_i) \) using the simulator.
  3. Train the network to maximize the posterior probability of \( \theta_i \) given \( \mathbf{x}_i \).

The training objective minimizes a negative log-likelihood loss:

\[ \ell_{\text{NPE}}(\phi) = -\frac{1}{n} \sum_{i=1}^{n} \log q_{\phi}(\theta_i \mid \mathbf{x}_i) \]

Once trained, NPE allows instant posterior predictions for new observations. This “train once, infer many times” property is known as amortization—one of SBI’s key advantages.

The Three Types of Missing Data

Real-world data rarely comes fully observed. Each sample can be split into observed and missing parts:

\[ \mathbf{x} = (\mathbf{x}_{\text{obs}}, \mathbf{x}_{\text{mis}}) \]

Understanding how the missingness occurs is crucial. Rubin’s classic taxonomy (1976) defines three mechanisms:

  1. Missing Completely at Random (MCAR): Missingness occurs independently of both observed and true values (e.g., random sensor drops).
  2. Missing at Random (MAR): Missingness depends on observed data but not on missing values (e.g., respondents of certain demographics skipping specific questions).
  3. Missing Not at Random (MNAR): Missingness depends on the missing values themselves (e.g., high-income earners being less likely to report income).

Most naive imputation techniques assume MCAR and fail when data are MAR or MNAR, introducing biases that compound inference errors. A robust SBI method must handle all three.

The Core Method: How RISE Works

The authors formalized the missing data problem within SBI, showing that inference under incomplete data inherently involves integrating over possible missing value configurations.

Two-Step Imputation Isn’t Enough

Ideally, the SBI posterior given observed data is expressed as:

\[ p_{\text{SBI}}(\theta \mid \mathbf{x}_{\text{obs}}) = \int p_{\text{SBI}}(\theta \mid \mathbf{x}_{\text{obs}}, \mathbf{x}_{\text{mis}}) \, p_{\text{true}}(\mathbf{x}_{\text{mis}} \mid \mathbf{x}_{\text{obs}}) \, d\mathbf{x}_{\text{mis}} \]

Here, the inference term depends on complete data, while the imputation term depends on the distribution of missing values. However, in practice \( p_{\text{true}}(\mathbf{x}_{\text{mis}} \mid \mathbf{x}_{\text{obs}}) \) is unknown, forcing us to estimate it:

\[ \hat{p}_{\text{SBI}}(\theta \mid \mathbf{x}_{\text{obs}}) = \int p_{\text{SBI}}(\theta \mid \mathbf{x}_{\text{obs}}, \mathbf{x}_{\text{mis}}) \, \hat{p}(\mathbf{x}_{\text{mis}} \mid \mathbf{x}_{\text{obs}}) \, d\mathbf{x}_{\text{mis}} \]

If the estimated imputation distribution \(\hat{p}\) diverges from the true one, the resulting posterior becomes biased—demonstrating that accurate imputation is inseparable from valid inference.

The RISE Framework: Joint Imputation and Inference

Rather than performing imputation first and inference second, RISE couples them together. It learns both the imputation distribution and inference posterior simultaneously, allowing shared representations that naturally correct each other during training.

RISE’s objective function combines two log-likelihood terms: [ \hat{\phi}, \hat{\varphi} = \arg\min_{\phi, \varphi}

  • \mathbb{E}{(\mathbf{x}{\text{obs}},\theta)} \mathbb{E}{\mathbf{x}{\text{mis}}} \left[ \log \hat{p}{\varphi}(\mathbf{x}{\text{mis}} \mid \mathbf{x}_{\text{obs}})
  • \log q_{\phi}(\theta \mid \mathbf{x}{\text{obs}}, \mathbf{x}{\text{mis}}) \right] ] This RISE loss jointly optimizes:
  • Imputation Network (\( \hat{p}_{\varphi} \)): learns missing data distributions.
  • Inference Network (\( q_{\phi} \)): approximates posteriors using imputed data.

As a unified amortized model, RISE reduces bias and adapts across missingness types (MCAR, MAR, MNAR) without separate retraining.

Building the Imputation Model: Neural Processes

RISE employs Neural Processes (NPs) to parameterize its imputation model \( \hat{p}_{\varphi}(\mathbf{x}_{\text{mis}} \mid \mathbf{x}_{\text{obs}}) \).

NPs fuse the strengths of neural networks and probabilistic processes: they learn distributions over functions that map input locations to outputs while quantifying predictive uncertainty. This makes them ideal for imputing missing values where uncertainty is key.

Each NP comprises:

  • Encoder \( \hat{p}_{\beta}(\tilde{z} \mid C) \): produces a latent representation \(\tilde{z}\) of the context (observed data and their locations).
  • Decoder \( \hat{p}_{\alpha}(x_{\text{mis},i} \mid c_{\text{mis},i}, \tilde{z}) \): predicts Gaussian distributions for each missing value at location \(c_{\text{mis},i}\).

A plate diagram illustrating the generative model for the Neural Process-based imputation in RISE. The observed data (X_obs, C_obs) informs a latent representation (Z̃), which is then used to generate the missing data (X_mis).

Figure 2. Plate diagram of the NP-based imputation model. Observed data (\( X_{\text{obs}}, C_{\text{obs}} \)) generate a latent representation \( \tilde{Z} \), used to infer missing values \( X_{\text{mis}} \).

The NP predictive distribution is formulated as:

\[ \hat{p}_{\varphi}(\mathbf{x}_{\text{mis}} \mid \mathbf{c}_{\text{mis}}, C) = \int \hat{p}_{\beta}(\tilde{z} \mid C) \prod_{i=1}^{k} \hat{p}_{\alpha}(x_{\text{mis},i} \mid c_{\text{mis},i}, \tilde{z}) \, d\tilde{z} \]

This design yields uncertainty-aware imputations, ensuring robust inference downstream.

Meta-Learning Missingness: RISE-Meta

Beyond accuracy, RISE also aims for adaptability. Its meta-learning variant—RISE-Meta—trains over varying missingness levels (\( \varepsilon = 10\%, 25\%, 60\% \)), effectively amortizing across tasks. As a result, RISE-Meta generalizes to new datasets with unknown amounts of missing data, avoiding costly retraining.

Experiments: Putting RISE to the Test

RISE was rigorously evaluated across synthetic SBI benchmarks and real-world settings, assessing both inference accuracy and imputation effectiveness.

Benchmark Performance on SBI Tasks

RISE was compared to baselines including NPE-NN (joint NPE plus neural network imputation), Wang et al. (2024), and Simformer (Gloeckler et al., 2024). Tests were run on four standard models: Ricker, Ornstein–Uhlenbeck (OUP), Generalized Linear Model (GLM), and Gaussian Linear Uniform (GLU).

RISE consistently outperforms baseline methods across four benchmark datasets, for both MCAR and MNAR missingness, and at varying levels of missing data (ε). It achieves higher NLPP and lower C2ST scores, indicating more accurate posterior estimates.

Figure 3. RISE achieves more accurate posterior estimates across varying missingness levels and mechanisms compared to all baselines.

RISE exhibits superior results in NLPP, C2ST, and MMD metrics. The advantage grows as missingness rises—demonstrating its robustness under extreme data loss scenarios.

Real-World Case Study: Hodgkin–Huxley Neuron Model

To showcase practical impact, the authors applied RISE to the Hodgkin–Huxley model—a canonical neuroscience simulator modeling neuronal ion channel dynamics.

Posterior estimates for the Hodgkin-Huxley model. RISE’s posteriors (purple) remain centered on the true parameter value (black cross) even with 60% missing data. In contrast, the baseline NPE-NN’s posteriors (orange) drift significantly away as missingness increases.

Figure 4. Posterior distributions for the Hodgkin–Huxley model under increasing missingness. RISE (purple) remains consistent with the true parameters, while NPE-NN (orange) deviates as data gaps widen.

Even with 60% of data missing, RISE maintained accurate posterior localization. Comparative coverage analyses also revealed better calibration.

Generalization Across Unknown Missingness Levels

Next, RISE-Meta was tested on data with randomly sampled missingness rates \( \varepsilon \sim \mathcal{U}(0,1) \). Despite not knowing precise sparsity at test time, RISE-Meta generalized well.

Generalization results. When tested on datasets with unknown levels of missingness, RISE-Meta achieves a lower Maximum Mean Discrepancy (MMD) than the baseline, showing its ability to generalize effectively.

Figure 5. When evaluated on varied missingness, RISE-Meta achieves lower MMD scores, confirming strong generalization.

Ablation Studies: Validating Design Choices

Two crucial ablation tests examined RISE’s architecture:

  • Imputation Performance: The NP-based imputation network was benchmarked on bioactivity datasets (Adrenergic and Kinase assays), achieving state-of-the-art \( R^2 \) scores against QSAR, Collective Matrix Factorization, and Deep Neural Net baselines.
  • Joint vs. Separate Training: Comparing RISE’s joint training to separate imputation–inference strategies confirmed that the joint approach yields lower RMSE—proving synergy between its components.

Ablation studies confirm the effectiveness of RISE’s design. The NP-based imputation model is state-of-the-art (left), and the joint learning strategy outperforms a separate, two-step approach (right).

Figure 6. RISE’s imputation performs best (left), and joint learning beats two-step baselines (right).

Why RISE Matters: Implications and Outlook

Missing data is ubiquitous in science, and handling it poorly can bias discoveries. RISE marks a milestone by directly integrating imputation and inference in SBI workflows, making likelihood-free inference resilient to incomplete data.

Key takeaways:

  • Robustness: RISE offers unbiased posteriors even under high missingness.
  • Amortization: Once trained, it generalizes efficiently to new samples.
  • Generalization: The RISE-Meta variant handles unknown missingness without retraining.
  • Uncertainty Awareness: Neural Processes quantify imputation uncertainty, improving model confidence.

While RISE inherits calibration challenges from NPE and may face limits under highly nonlinear data patterns, it fundamentally extends SBI’s applicability to the messy realities of scientific data.

By bridging the gap between imperfect observations and rich simulators, RISE ensures that missing values no longer mean missing insights.