Achieving a balance between lightweight design and high performance remains a significant challenge for speech enhancement (SE) tasks on resource-constrained devices. Existing state-of-the-art methods, such as MUSE, have established a strong baseline with only 0.51M parameters by introducing a Multi-path Enhanced Taylor (MET) transformer and Deformable Embedding (DE). However, an in-depth analysis reveals that MUSE still suffers from efficiency bottlenecks: the MET module relies on a complex "approximate-compensate" mechanism to mitigate the limitations of Taylor-expansion-based attention, while the offset calculation for deformable embedding introduces additional computational burden. This paper proposes IMSE, a systematically optimized and ultra-lightweight network. We introduce two core innovations: 1) Replacing the MET module with Amplitude-Aware Linear Attention (MALA). MALA fundamentally rectifies the "amplitude-ignoring" problem in linear attention by explicitly preserving the norm information of query vectors in the attention calculation, achieving efficient global modeling without an auxiliary compensation branch. 2) Replacing the DE module with Inception Depthwise Convolution (IDConv). IDConv borrows the Inception concept, decomposing large-kernel operations into efficient parallel branches (square, horizontal, and vertical strips), thereby capturing spectrogram features with extremely low parameter redundancy. Extensive experiments on the VoiceBank+DEMAND dataset demonstrate that, compared to the MUSE baseline, IMSE significantly reduces the parameter count by 16.8\% (from 0.513M to 0.427M) while achieving competitive performance comparable to the state-of-the-art on the PESQ metric (3.373). This study sets a new benchmark for the trade-off between model size and speech quality in ultra-lightweight speech enhancement.

This paper is concerned with forecast error, particularly in relation to loss reserving. This is generally regarded as consisting of three components, namely parameter, process and model errors. The first two of these components, and their estimation, are well understood, but less so model error. Model error itself is considered in two parts: one part that is capable of estimation from past data (internal model error), and another part that is not (external model error). Attention is focused here on internal model error. Estimation of this error component is approached by means of Bayesian model averaging, using the Bayesian interpretation of the LASSO. This is used to generate a set of admissible models, each with its prior probability and the likelihood of observed data. A posterior on the model set, conditional on the data, results, and an estimate of model error (contained in a loss reserve) is obtained as the variance of the loss reserve according to this posterior. The population of models entering materially into the support of the posterior may turn out to be thinner than desired, and bootstrapping of the LASSO is used to gain bulk. This provides the bonus of an estimate of parameter error also. It turns out that the estimates of parameter and model errors are entangled, and dissociation of them is at least difficult, and possibly not even meaningful. These matters are discussed. The majority of the discussion applies to forecasting generally, but numerical illustration of the concepts is given in relation to insurance data and the problem of insurance loss reserving.

We introduce Density sketches (DS): a succinct online summary of the data distribution. DS can accurately estimate point wise probability density. Interestingly, DS also provides a capability to sample unseen novel data from the underlying data distribution. Thus, analogous to popular generative models, DS allows us to succinctly replace the real-data in almost all machine learning pipelines with synthetic examples drawn from the same distribution as the original data. However, unlike generative models, which do not have any statistical guarantees, DS leads to theoretically sound asymptotically converging consistent estimators of the underlying density function. Density sketches also have many appealing properties making them ideal for large-scale distributed applications. DS construction is an online algorithm. The sketches are additive, i.e., the sum of two sketches is the sketch of the combined data. These properties allow data to be collected from distributed sources, compressed into a density sketch, efficiently transmitted in the sketch form to a central server, merged, and re-sampled into a synthetic database for modeling applications. Thus, density sketches can potentially revolutionize how we store, communicate, and distribute data.

Deep Gaussian processes (DGPs) are increasingly popular as predictive models in machine learning (ML) for their non-stationary flexibility and ability to cope with abrupt regime changes in training data. Here we explore DGPs as surrogates for computer simulation experiments whose response surfaces exhibit similar characteristics. In particular, we transport a DGP's automatic warping of the input space and full uncertainty quantification (UQ), via a novel elliptical slice sampling (ESS) Bayesian posterior inferential scheme, through to active learning (AL) strategies that distribute runs non-uniformly in the input space -- something an ordinary (stationary) GP could not do. Building up the design sequentially in this way allows smaller training sets, limiting both expensive evaluation of the simulator code and mitigating cubic costs of DGP inference. When training data sizes are kept small through careful acquisition, and with parsimonious layout of latent layers, the framework can be both effective and computationally tractable. Our methods are illustrated on simulation data and two real computer experiments of varying input dimensionality. We provide an open source implementation in the "deepgp" package on CRAN.

Integrated Mean Squared Error (IMSE) is a version of the usual mean squared error criterion, averaged over all possible training sets of a given size. If it could be observed, it could be used to determine optimal network complexity or optimal data subsets for efficient training. We show that two common methods of cross-validating average squared error deliver unbiased estimates of IMSE, converging to IMSE with probability one. These estimates thus make possible approximate IMSE-based choice of network complexity. We also show that two variants of cross validation measure provide unbiased IMSE-based estimates potentially useful for selecting optimal data subsets. 1 Summary To begin, assume we are given a fixed network architecture.

Integrated Mean Squared Error (IMSE) is a version of the usual mean squared error criterion, averaged over all possible training sets of a given size. If it could be observed, it could be used to determine optimal network complexity or optimal data subsets for efficient training. We show that two common methods of cross-validating average squared error deliver unbiased estimates of IMSE, converging to IMSE with probability one. These estimates thus make possible approximate IMSE-based choice of network complexity. We also show that two variants of cross validation measure provide unbiased IMSE-based estimates potentially useful for selecting optimal data subsets. 1 Summary To begin, assume we are given a fixed network architecture.

Integrated Mean Squared Error (IMSE) is a version of the usual mean squared error criterion, averaged over all possible training sets of a given size. If it could be observed, it could be used to determine optimal network complexity or optimal data subsets forefficient training. We show that two common methods of cross-validating average squared error deliver unbiased estimates of IMSE, converging to IMSE with probability one. We also show that two variants of cross validation measure provide unbiased IMSE-based estimates potentially useful for selecting optimal data subsets. 1 Summary To begin, assume we are given a fixed network architecture. Let zN denote a given set of N training examples.