Multiple instance learning (MIL) is often used in medical imaging to classify high-resolution 2D images by processing patches or classify 3D volumes by processing slices. However, conventional MIL approaches treat instances separately, ignoring contextual relationships such as the appearance of nearby patches or slices that can be essential in real applications. We design a synthetic classification task where accounting for adjacent instance features is crucial for accurate prediction. We demonstrate the limitations of off-the-shelf MIL approaches by quantifying their performance compared to the optimal Bayes estimator for this task, which is available in closed-form. We empirically show that newer correlated MIL methods still do not achieve the best possible performance when trained with ten thousand training samples, each containing many instances.

The reason to focus on DAE priors is explained in line 54-56 (also in line 14 and 35 of this rebuttal).

In this work, we offer a thorough analytical investigation into the role of shared hyperparameters in a hierarchical Bayesian model, examining their impact on information borrowing and posterior inference. Our approach is rooted in a non-asymptotic framework, where observations are drawn from a mixed-effects model, and a Gaussian distribution is assumed for the true effect generator. We consider a nested hierarchical prior distribution model to capture these effects and use the posterior means for Bayesian estimation. To quantify the effect of information borrowing, we propose an integrated risk measure relative to the true data-generating distribution. Our analysis reveals that the Bayes estimator for the model with a deeper hierarchy performs better, provided that the unknown random effects are correlated through a compound symmetric structure. Our work also identifies necessary and sufficient conditions for this model to outperform the one nested within it. We further obtain sufficient conditions when the correlation is perturbed. Our study suggests that the model with a deeper hierarchy tends to outperform the nested model unless the true data-generating distribution favors sufficiently independent groups. These findings have significant implications for Bayesian modeling, and we believe they will be of interest to researchers across a wide range of fields.

Our main technical contribution is the rigorous analysis of a Bayes estimator and of an approximate message passing (AMP) algorithm, both of which incorrectly assume a Gaussian setup.

We discuss the assumptions and implications of our results as well as related open problems. Other loss functions As mentioned in Section 1.1, standard arguments based on concentration Lemma 5 crucially relies on stationarity. Theorems 6.3 and 6.5], which, in turn, follow the arguments of [ In this section we prove the optimal lower bound in Theorem 7 for three states. Finally, we relate (39) formally to the minimax prediction risk under the KL divergence. In this section, we rigorously carry out the lower bound proof sketched in Section 3.2: In Section In Section 6.2.2, we make the steps in In Section 6.2.3, we choose a prior distribution on the transition probabilities and prove a lower bound on the resulting mutual information, thereby completing the proof of Theorem 1, with the added bonus that the construction is restricted to irreducible and M is shown in Figure 2. One can also verify that the spectral gap of M is Θ( We make use of the properties (P1)-(P3) in Section 6.2.1 to prove Lemma 9. Proof of Lemma 9.

Neural estimators are simulation-based estimators for the parameters of a family of statistical models, which build a direct mapping from the sample to the parameter vector. They benefit from the versatility of available network architectures and efficient training methods developed in the field of deep learning. Neural estimators are amortized in the sense that, once trained, they can be applied to any new data set with almost no computational cost. While many papers have shown very good performance of these methods in simulation studies and real-world applications, so far no statistical guarantees are available to support these observations theoretically. In this work, we study the risk of neural estimators by decomposing it into several terms that can be analyzed separately. We formulate easy-to-check assumptions ensuring that each term converges to zero, and we verify them for popular applications of neural estimators. Our results provide a general recipe to derive theoretical guarantees also for broader classes of architectures and estimation problems.

Likelihood-free approaches are appealing for performing inference on complex dependence models, either because it is not possible to formulate a likelihood function, or its evaluation is very computationally costly. This is the case for several models available in the multivariate extremes literature, particularly for the most flexible tail models, including those that interpolate between the two key dependence classes of `asymptotic dependence' and `asymptotic independence'. We focus on approaches that leverage neural networks to approximate Bayes estimators. In particular, we explore the properties of neural Bayes estimators for parameter inference for several flexible but computationally expensive models to fit, with a view to aiding their routine implementation. Owing to the absence of likelihood evaluation in the inference procedure, classical information criteria such as the Bayesian information criterion cannot be used to select the most appropriate model. Instead, we propose using neural networks as neural Bayes classifiers for model selection. Our goal is to provide a toolbox for simple, fast fitting and comparison of complex extreme-value dependence models, where the best model is selected for a given data set and its parameters subsequently estimated using neural Bayes estimation. We apply our classifiers and estimators to analyse the pairwise extremal behaviour of changes in horizontal geomagnetic field fluctuations at three different locations.

Regularized system identification has become a significant complement to more classical system identification. It has been numerically shown that kernel-based regularized estimators often perform better than the maximum likelihood estimator in terms of minimizing mean squared error (MSE). However, regularized estimators often require hyper-parameter estimation. This paper focuses on ridge regression and the regularized estimator by employing the empirical Bayes hyper-parameter estimator. We utilize the excess MSE to quantify the MSE difference between the empirical-Bayes-based regularized estimator and the maximum likelihood estimator for large sample sizes. We then exploit the excess MSE expressions to develop both a family of generalized Bayes estimators and a family of closed-form biased estimators. They have the same excess MSE as the empirical-Bayes-based regularized estimator but eliminate the need for hyper-parameter estimation. Moreover, we conduct numerical simulations to show that the performance of these new estimators is comparable to the empirical-Bayes-based regularized estimator, while computationally, they are more efficient.