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.

Many machine learning systems make constrained decisions by optimizing factorized objectives, but the context-specific objective is often treated as fixed. We study contextual decision-weight learning: from logged decisions and proxy outputs, learn an optimizer-facing weight vector w(x) over interpretable decision factors z(x,d), rather than a direct policy or generic predictive score. We propose OTSS, an output-targeted soft-segmentation model that deploys the personalized decision-ready weight vector. At the function-class level, the theory highlights a hard-versus-soft distinction. Hard partitions incur an approximation-estimation tradeoff under overlap, while a realizable fixed-K soft class removes the hard-partition approximation floor and attains a parametric rate. We evaluate OTSS in controlled benchmarks with finite evaluation libraries, where the true weight vector and downstream regret can be computed exactly. In the representative overlap setting, OTSS attains the lowest mean regret among the comparators, including EM mixture regression, the strongest soft-mixture baseline in our comparison; it matches EM on coefficient recovery while running about two orders of magnitude faster. In a matched K=5 benchmark, OTSS remains competitive under hard-routed truth and improves as heterogeneity becomes softer and sample size grows. On a fixed Complete Journey retail anchor with real household covariates and action geometry, OTSS again achieves the lowest mean-regret point estimate.

Conditional Average Treatment Effect (CATE) estimation in practice demands three properties simultaneously: heterogeneous effects τ(x), calibrated uncertainty over them, and robustness to the heavy tails that contaminate real outcome data. Meta-learners (Künzel et al., 2019) give (i); causal forests and BART give (i)-(ii) with Gaussian-tail assumptions; no widely used tool gives all three. We present Bayesian X-Learner, an X-Learner built on cross-fitted doubly robust pseudo-outcomes (Kennedy, 2020) with a full MCMC posterior over τ(x) via a Welsch redescending pseudo-likelihood. On Hill's IHDP benchmark the default configuration attains mean εPEHE = 0.56 on 5 replications (lowest mean; differences from S-/T-/X-learners, full-config Causal BART, and a causal forest baseline are not significant at α = 0.05, and rank ordering is unstable at 10 replications -- IHDP comparisons are competitive rather than dominant). On contaminated "whale" DGPs with up to 20-25% tail density, a one-flag extension (contamination_severity) that selects a Huberδ nuisance loss per Huber's minimax-δ relation recovers RMSE 0.13 with tight credible intervals (single-cross-fit 30-seed coverage 83% [Wilson 66%, 93%] at 20% density; modularBayes pooling with Bayesian-bootstrap nuisance draws restores nominal 95% coverage). We validate on the Hillstrom email-marketing RCT (N = 42,613), demonstrating consistent behaviour on real heavy-tailed outcome data, and report covariate-stratified τ(x) coverage across covariate quintiles to substantiate calibration for heterogeneous effects beyond scalar summaries. We draw a clean distinction between tails-as-contamination (handled by Welsch + Huber nuisance) and tails-as-signal (handled by a tail-aware CATE basis); an empirical probe confirms a tail-aware basis recovers τtail with full subgroup coverage, while the library's Hill-estimator path is contamination-directed and should not be used for heterogeneous τ. We map six empirical boundaries (contamination ceiling, clean-data efficiency cost, basis sensitivity, sample size, treatment type, compute) and show where other tools are preferable. Code and reproducible benchmarks are released.

The source code of Minigrid and Miniworld can be found at https://github.com/ To run the experiments, we have implemented the following functionalities: 1. implemented the human trajectory saving for MiniGrid-FourRooms-v0 (copied the ManualControlclass from Minigrid and added 38 lines of code, which are mostly calling data saving functions); 2. implemented the human trajectory saving for MiniWorld-FourRooms-v0 (copied the ManualControlclass from Miniworld and added 45 lines of code, which are mostly calling data saving functions); 3. implemented data saving and plotting for MiniGrid-FourRooms-v0 (33 lines of code, mostly for Matplotlib); 4. implemented data saving and plotting for MiniWorld-FourRooms-v0 (33 lines of code, mostly for Matplotlib). In total, the implementation of this new functionality required 149 lines of code. The source code is hosted on GitHub. We bear all the responsibility in case of violation of rights.

A.1 Additional experimental results We further introduce our additional experiments in this section. In our main article, we compared our model FREED with baseline models REINVENT and MORLD. For fairer comparison of quality scores, we also performed multi-objective optimization of REINVENT and MORLD on both quality score (pharmacochemical filter score) and docking score as follows. Table 1 in the main text shows that such an implicit method is not enough to achieve nearly perfect filter scores as our model did. Also, as shown in Table 1 REINVENT showed deteriorated performance when jointly trained with filter scores, in terms of hit ratio and top 5% scores, implying that multiobjective optimization is more difficult than explicitly constrained optimization. Such a result was consistent for all three targets. The two baseline models REINVENT and MORLD that are jointly trained to maximize filter scores are noted as REINVENT w/ filter and MORLD w/ filter.