**Law of Total Expectation**

The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, Adam’s law, and the smoothing theorem, among other names, states that if X is a random variable whose expected value E( X ) is defined, and Y is any random variable on the same probability space, then E(X) = E(E(X|Y)) , i.e., the expected value of the conditional expected value of X given Y is the same as the expected value of X. … **Latent Topic Conversational Model (LTCM)**

Latent variable models have been a preferred choice in conversational modeling compared to sequence-to-sequence (seq2seq) models which tend to generate generic and repetitive responses. Despite so, training latent variable models remains to be difficult. In this paper, we propose Latent Topic Conversational Model (LTCM) which augments seq2seq with a neural latent topic component to better guide response generation and make training easier. The neural topic component encodes information from the source sentence to build a global ‘topic’ distribution over words, which is then consulted by the seq2seq model at each generation step. We study in details how the latent representation is learnt in both the vanilla model and LTCM. Our extensive experiments contribute to better understanding and training of conditional latent models for languages. Our results show that by sampling from the learnt latent representations, LTCM can generate diverse and interesting responses. In a subjective human evaluation, the judges also confirm that LTCM is the overall preferred option. … **ChIMP**

Information fusion is an essential part of numerous engineering systems and biological functions, e.g., human cognition. Fusion occurs at many levels, ranging from the low-level combination of signals to the high-level aggregation of heterogeneous decision-making processes. While the last decade has witnessed an explosion of research in deep learning, fusion in neural networks has not observed the same revolution. Specifically, most neural fusion approaches are ad hoc, are not understood, are distributed versus localized, and/or explainability is low (if present at all). Herein, we prove that the fuzzy Choquet integral (ChI), a powerful nonlinear aggregation function, can be represented as a multi-layer network, referred to hereafter as ChIMP. We also put forth an improved ChIMP (iChIMP) that leads to a stochastic gradient descent-based optimization in light of the exponential number of ChI inequality constraints. An additional benefit of ChIMP/iChIMP is that it enables eXplainable AI (XAI). Synthetic validation experiments are provided and iChIMP is applied to the fusion of a set of heterogeneous architecture deep models in remote sensing. We show an improvement in model accuracy and our previously established XAI indices shed light on the quality of our data, model, and its decisions. … **Magnitude Bounded Matrix Factorisation (MBMF)**

Low rank matrix factorisation is often used in recommender systems as a way of extracting latent features. When dealing with large and sparse datasets, traditional recommendation algorithms face the problem of acquiring large, unrestrained, fluctuating values over predictions especially for users/items with very few corresponding observations. Although the problem has been somewhat solved by imposing bounding constraints over its objectives, and/or over all entries to be within a fixed range, in terms of gaining better recommendations, these approaches have two major shortcomings that we aim to mitigate in this work: one is they can only deal with one pair of fixed bounds for all entries, and the other one is they are very time-consuming when applied on large scale recommender systems. In this paper, we propose a novel algorithm named Magnitude Bounded Matrix Factorisation (MBMF), which allows different bounds for individual users/items and performs very fast on large scale datasets. The key idea of our algorithm is to construct a model by constraining the magnitudes of each individual user/item feature vector. We achieve this by converting from the Cartesian to Spherical coordinate system with radii set as the corresponding magnitudes, which allows the above constrained optimisation problem to become an unconstrained one. The Stochastic Gradient Descent (SGD) method is then applied to solve the unconstrained task efficiently. Experiments on synthetic and real datasets demonstrate that in most cases the proposed MBMF is superior over all existing algorithms in terms of accuracy and time complexity. …

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**24**
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Mar 2020

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