New Results for the MAP Problem in Bayesian Networks

This paper presents new results for the (partial) maximum a posteriori (MAP) problem in Bayesian networks, which is the problem of querying the most probable state configuration of some of the network variables given evidence. First, it is demonstrated that the problem remains hard even in networks with very simple topology, such as binary polytrees and simple trees (including the Naive Bayes structure). Such proofs extend previous complexity results for the problem. Inapproximability results are also derived in the case of trees if the number of states per variable is not bounded. Although the problem is shown to be hard and inapproximable even in very simple scenarios, a new exact algorithm is described that is empirically fast in networks of bounded treewidth and bounded number of states per variable. The same algorithm is used as basis of a Fully Polynomial Time Approximation Scheme for MAP under such assumptions. Approximation schemes were generally thought to be impossible for this problem, but we show otherwise for classes of networks that are important in practice. The algorithms are extensively tested using some well-known networks as well as random generated cases to show their effectiveness.Comment: A couple of typos were fixed, as well as the notation in part of section 4, which was misleading. Theoretical and empirical results have not change

A Family of Maximum Entropy Densities Matching Call Option Prices

We investigate the position of the Buchen-Kelly density in a family of entropy maximising densities which all match European call option prices for a given maturity observed in the market. Using the Legendre transform which links the entropy function and the cumulant generating function, we show that it is both the unique continuous density in this family and the one with the greatest entropy. We present a fast root-finding algorithm that can be used to calculate the Buchen-Kelly density, and give upper boundaries for three different discrepancies that can be used as convergence criteria. Given the call prices, arbitrage-free digital prices at the same strikes can only move within upper and lower boundaries given by left and right call spreads. As the number of call prices increases, these bounds become tighter, and we give two examples where the densities converge to the Buchen-Kelly density in the sense of relative entropy when we use centered call spreads as proxies for digital prices. As pointed out by Breeden and Litzenberger, in the limit a continuous set of call prices completely determines the density.Comment: 22 pages, 6 figure

The Brazilian Trade Liberalisation: the regionalized impacts on the productive chain of cotton, textile and apparel industries

During the last 8 years Brazil has experiencing a large program of commercial overture. A lot of administrative and tax barriers had suppressed and the textile sector had been one of the most affected by this process. The paper analyses the regional impact of this process over the so-called textile chain (cotton-textile-confections) utilising dates of production and sales from 1980 until 1994 for the most important production regions in Brazil. These informations are analyzed by nonparametrical statistics. In a synthesis, the commercial overture that happened in Brazil during the last 8 years didn't change the structural tendencies in the spatial division of work of the production links of Cotton, Textile and Confections. It incremented the Cotton March to West, the concurrence among decadent textiles regions as Pernambuco, Minas Gerais and S?o Paulo and those, more dynamics, as Santa Catarina and Ceara. The commercial overture has been a double, because are growing the exports and imports figures of Textile. On the other hand those that are still using old technologies, are closing the plants. But the price of this adjusting process was paid by the cotton's farmers and, maybe, by the small and medium firms of Confections. The new regions are growing due to different factors. Ceara, in the northwest of Brazil, due to fiscal incentives by SUDENE and advantages from direct importation of cotton from Africa. Santa Catarina, in the south, due to a special characteristic of its industrial organisation and a new territorial organisation of production buying cotton from Argentina, Paraguay and Parana (Brazil) and inputs textile from EEC.

The Impact of Health Status and Out-of-Pocket Medical Expenditures on Annuity Valuation

This paper describes how differences in health status at retirement can influence the decision to purchase a life annuity. We extend previous research on annuitization decisions by incorporating the effect of health differentials via differences in survival throughout the latter portion of life. Next, we consider how precautionary savings motivated by uncertain out-of-pocket medical expenses influence annuitization decisions. Our results show that annuities become less attractive to people facing uncertain medical expenses. While full annuitization would still be optimal if annuity markets were truly complete and both life- and health-contingent, lacking this, annuity equivalent wealth values are much lower for those in poor health, as compared to persons in good health.

Approximation Complexity of Maximum A Posteriori Inference in Sum-Product Networks

We discuss the computational complexity of approximating maximum a posteriori inference in sum-product networks. We first show NP-hardness in trees of height two by a reduction from maximum independent set; this implies non-approximability within a sublinear factor. We show that this is a tight bound, as we can find an approximation within a linear factor in networks of height two. We then show that, in trees of height three, it is NP-hard to approximate the problem within a factor $2^{f(n)}$ for any sublinear function $f$ of the size of the input $n$. Again, this bound is tight, as we prove that the usual max-product algorithm finds (in any network) approximations within factor $2^{c \cdot n}$ for some constant $c < 1$. Last, we present a simple algorithm, and show that it provably produces solutions at least as good as, and potentially much better than, the max-product algorithm. We empirically analyze the proposed algorithm against max-product using synthetic and realistic networks.Comment: 18 page