UNIVERSAL RISK BUDGETING

I juxtapose Cover’s vaunted universal portfolio selection algorithm ([Cover, TM (1991). Universal portfolios. Mathematical Finance, 1, 1–29]) with the modern representation of a portfolio as a certain allocation of risk among the available assets, rather than a mere allocation of capital. Thus, I define a Universal Risk Budgeting scheme that weights each risk budget, instead of each capital budget, by its historical performance record, á la Cover. I prove that my scheme is mathematically equivalent to a novel type of [Cover, TM and E Ordentlich (1996). Universal portfolios with side information. IEEE Transactions on Information Theory, 42, 348–363] universal portfolio that uses a new family of prior densities that have hitherto not appeared in the literature on universal portfolio theory. I argue that my universal risk budget, so-defined, is a potentially more perspicuous and flexible type of universal portfolio; it allows the algorithmic trader to incorporate, with advantage, his prior knowledge or beliefs about the particular covariance structure of instantaneous asset returns. Say, if there is some dispersion in the volatilities of the available assets, then the uniform or Dirichlet priors that are standard in the literature will generate a dangerously lopsided prior distribution over the possible risk budgets. In the author’s opinion, the proposed “Garivaltis prior” makes for a nice improvement on Cover’s timeless expert system, that is properly agnostic and open to different risk budgets from the very get-go. Inspired by [Jamshidian, F (1992). Asymptotically optimal portfolios. Mathematical Finance, 2, 131–150], the universal risk budget is formulated as a new kind of exotic option in the continuous time Black–Scholes market, with all the pleasure, elegance, and convenience that entails.

Alternative Dirichlet Priors for Estimating Entropy Via a Power Sum Functional

Entropy is a functional of probability and is a measurement of information contained in a system; however, the practical problem of estimating entropy in applied settings remains a challenging and relevant problem. The Dirichlet prior is a popular choice in the Bayesian framework for estimation of entropy when considering a multinomial likelihood. In this work, previously unconsidered Dirichlet type priors are introduced and studied. These priors include a class of Dirichlet generators as well as a noncentral Dirichlet construction, and in both cases includes the usual Dirichlet as a special case. These considerations allow for flexible behaviour and can account for negative and positive correlation. Resultant estimators for a particular functional, the power sum, under these priors and assuming squared error loss, are derived and represented in terms of the product moments of the posterior. This representation facilitates closed-form estimators for the Tsallis entropy, and thus expedite computations of this generalised Shannon form. Select cases of these proposed priors are considered to investigate the impact and effect on the estimation of Tsallis entropy subject to different parameter scenarios.

Co-Occurrence Estimation from Aggregated Data with Auxiliary Information

Complete co-occurrence data are unavailable in many applications, including purchase records and medical histories, because of their high cost or privacy protection. Even with such applications, aggregated data would be available, such as the number of purchasers for each item and the number of patients with each disease. We propose a method for estimating the co-occurrence of items from aggregated data with auxiliary information. For auxiliary information, we use item features that describe the characteristics of each item. Although many methods have been proposed for estimating the co-occurrence given aggregated data, no existing method can use auxiliary information. We also use records of a small number of users. With our proposed method, we introduce latent co-occurrence variables that represent the amount of co-occurrence for each pair of items. We model a probabilistic generative process of the latent co-occurrence variables by a multinomial distribution with Dirichlet priors. The parameters of the Dirichlet priors are parameterized with neural networks that take the auxiliary information as input, where neural networks are shared across different item pairs. The shared neural networks enable us to learn unknown relationships between auxiliary information and co-occurrence using the data of multiple items. The latent co-occurrence variables and the neural network parameters are estimated by maximizing the sum of the likelihood of the latent co-occurrence variables and the likelihood of the small records. We demonstrate the effectiveness of our proposed method using user-item rating datasets.

Exploring Symmetrical and Asymmetrical Dirichlet Priors for Latent Dirichlet Allocation

Latent Dirichlet Allocation (LDA) has gained much attention from researchers and is increasingly being applied to uncover underlying semantic structures from a variety of corpora. However, nearly all researchers use symmetrical Dirichlet priors, often unaware of the underlying practical implications that they bear. This research is the first to explore symmetrical and asymmetrical Dirichlet priors on topic coherence and human topic ranking when uncovering latent semantic structures from scientific research articles. More specifically, we examine the practical effects of several classes of Dirichlet priors on 2000 LDA models created from abstract and full-text research articles. Our results show that symmetrical or asymmetrical priors on the document–topic distribution or the topic–word distribution for full-text data have little effect on topic coherence scores and human topic ranking. In contrast, asymmetrical priors on the document–topic distribution for abstract data show a significant increase in topic coherence scores and improved human topic ranking compared to a symmetrical prior. Symmetrical or asymmetrical priors on the topic–word distribution show no real benefits for both abstract and full-text data.