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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.

Time-series analysis is often affected by missing data, a common problem across several fields, including healthcare and environmental monitoring. Multiple Imputation by Chained Equations (MICE) has been prominent for imputing missing values through "fully conditional specification". We extend MICE using the Bayesian framework (tBayes-MICE), utilising Bayesian inference to impute missing values via Markov Chain Monte Carlo (MCMC) sampling to account for uncertainty in MICE model parameters and imputed values. We also include temporally informed initialisation and time-lagged features in the model to respect the sequential nature of time-series data. We evaluate the tBayes-MICE method using two real-world datasets (AirQuality and PhysioNet), and using both the Random Walk Metropolis (RWM) and the Metropolis-Adjusted Langevin Algorithm (MALA) samplers. Our results demonstrate that tBayes-MICE reduces imputation errors relative to the baseline methods over all variables and accounts for uncertainty in the imputation process, thereby providing a more accurate measure of imputation error. We also found that MALA mixed better than RWM across most variables, achieving comparable accuracy while providing more consistent posterior exploration. Overall, these findings suggest that the tBayes-MICE framework represents a practical and efficient approach to time-series imputation, balancing increased accuracy with meaningful quantification of uncertainty in various environmental and clinical settings.

Social conflict is a survival mechanism yielding both normal and pathological behaviors. Tounderstand its underlying principles, we collected behavioral and whole-brain neural data from mice advancing through stages of social conflict.

In this paper, we tackled just the first one in the list to show the effectiveness of9 ouralgorithm. Weagree that computations aresimple, i.e., elegant,once the18 aforementioned requirements have been elicited. Eliciting them, however,is definitely non-trivial and has not been19 explored in the literature so far for expectations. Our circuits are expressive enough to model larger datasets24 (see our answer to R#1.2) and learning them would scale: in manycases it is easier to learn aLC than aneural net25 (e.g., see [3]). 3. Approximate inference alternatives. Whenever we are able to compute expectations exactly for26 regression (Thm 1), we might want to consider approximations only to speed computations.

For our synthetic datasets, we conduct two additional rounds of instruction evolution to increase complexity. As shown in Figure 3, the scores of instructions across the three datasets consistently increase.

In this section, we first provide model parameters used for training the attack GANs. We then provide sample images from each cluster/class for each of the models, along with the generated noise using ourGAN models. In this section, we provide additional details for the defense approaches considered in this paper. B.1 RobustDeepClustering We provide hyperparameter values (Table 6) for training the GAN network for RUC, along with confusion matrices (Figures 37 - 39) and adversarial samples (Figures 40 - 42) obtained via our attack. Then, in Table 8 we provide the actual values used for generating the injection/detection bar plot figureinthemaintext.