Minimizing the Mean Squared Error (MSE) is a key objective in machine learning and is commonly used for imputing missing values. While this approach provides accurate point estimates, it introduces systematic biases in downstream analyses. These biases affect key parameters such as variance, prevalence, correlation, slope, and explained variance. The root cause is that imputed values optimized for MSE are averages, which reduce the natural variability in the data. This paper demonstrates that adding noise to imputed values can effectively eliminate these biases. The required noise level is proportional to the MSE. Using a toy example in a multivariate normal setting, we compare two methods: predictive imputation, which minimizes MSE, and stochastic imputation, which incorporates random noise. Simulation results show that predictive methods systematically introduce bias, while stochastic methods preserve the data's natural variability and produce unbiased estimates. We also evaluate three popular imputation tools -- missForest, softImpute, and mice -- and observe consistent biases in predictive methods. These findings highlight that MSE is an inadequate measure of imputation quality, as it prioritizes accuracy over variability. Incorporating noise into imputation methods is essential to prevent biases and ensure valid downstream analyses, underscoring the importance of stochastic approaches for handling incomplete data.

AQuAcurrently includes a variety of datasets for different classification problems, varying in the number of classes, sources of annotations, and data modalities. All datasets except those marked with are multi-class.

In the supplementary material, we provide additional information and details in A.1. This section covers the introduction of data, key parameter settings, comparisons with baselines, optimization methods, and the algorithm process of our method. Furthermore, A.2 presents supplementary experiments for our model, including visualization experiments and replication studies. Additionally, we discuss the reasons behind utilizing hypergraphs as the temporal encoder in A.3. Finally, the limitations and broader impacts of our work are discussed in A.4. A.1 Data and Implementation Details Data. The statistical information of the aforementioned four real-world datasets is presented in Table 4.

In this section we motivate the design choices and inductive biases that we encode into our neural encoder network e, which is the network that is used to model the relative accuracies of the weak supervision sources λ. Recall that we model the probability of a particular sample x X having the class label y Y = {1,...,C}as Pθ(y|λ) = softmax(s)yP(y), (4) s = θ(λ,x)Tλ RC . Connection to prior PGM models We now motivate this choice by deriving a less expressive variant of it from the standard Markov Random Field (MRF) used in the related work. If we view the attention scores θ(λ,x) Rm, that assign sample-dependent accuracies to each labeling function, as sample-independent parameters θ1 and, by that, drop the features from the equation - as is done in the related work [30, 32, 19, 11] - we can rewrite Eq. 4 as exp θT1 1 {λ = y} P We can recognize Pθ as a distribution from the exponential familiy, and more specifically as a pairwise MRF, or factor graph, with canonical parameters θ = (θ1,θ2) and corresponding sufficient statistics, or factors, φ(λ,y) = (φ1(λ,y),φ2(λ)), as well as the log partition function Zθ. The accuracy factors and parameters φ1,θ1 are the core component of this model and sometimes take the form φ1(λy) = λy in binary models as in [30, 19, 11]. The label-independent factors φ2(λ) have, as can be seen from the derivation above, no direct influence on the latent label posterior, but are often used to model labeling propensities 1 {λ 6= 0}and correlation dependencies 1 {λi = λj}, which can be important for PGM parameter learning, but are susceptible to misspecifications [39, 11, 8].

The classes are airplanes, cars, birds, cats, deer, dogs, frogs, horses, ships, and trucks, and they are all mutually exclusive.

In the supplementary material, we provide additional information and details in A.1. This section covers the introduction of data, key parameter settings, comparisons with baselines, optimization methods, and the algorithm process of our method. The statistical information of the aforementioned four real-world datasets is presented in Table 4. These datasets primarily consist of daily spatio-temporal statistics in the United States. We perform 2 dynamic routing iterations.