A flexible class of dependence-aware multi-label loss functions

The idea to exploit label dependencies for better prediction is at the core of methods for multi-label classification (MLC), and performance improvements are normally explained in this way. Surprisingly, however, there is no established methodology that allows to analyze the dependence-awareness of MLC algorithms. With that goal in mind, we introduce a class of loss functions that are able to capture the important aspect of label dependence. To this end, we leverage the mathematical framework of non-additive measures and integrals. Roughly speaking, a non-additive measure allows for modeling the importance of correct predictions of label subsets (instead of single labels), and thereby their impact on the overall evaluation, in a flexible way. The well-known Hamming and subset 0/1 losses are rather extreme special cases of this function class, which give full importance to single label sets or the entire label set, respectively. We present concrete instantiations of this class, which appear to be especially appealing from a modeling perspective. The assessment of multi-label classifiers in terms of these losses is illustrated in an empirical study, clearly showing their aptness at capturing label dependencies. Finally, while not being the main goal of this study, we also show some preliminary results on the minimization of this parametrized family of losses.

Commutative L-algebras and measure theory

Measure and integration theory for finitely additive measures, including vector-valued measures, is shown to be essentially covered by a class of commutative L-algebras, called measurable algebras. The domain and range of any measure is a commutative L-algebra. Each measurable algebra embeds into its structure group, an abelian group with a compatible lattice order, and each (general) measure extends uniquely to a monotone group homomorphism between the structure groups. On the other hand, any measurable algebra X is shown to be the range of an essentially unique measure on a measurable space, which plays the role of a universal covering. Accordingly, we exhibit a fundamental group of X, with stably closed subgroups corresponding to a special class of measures with X as target. All structure groups of measurable algebras arising in a classical context are archimedean. Therefore, they admit a natural embedding into a group of extended real-valued continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra. Extending Loomis’ integration theory for finitely additive measures, it is proved that, modulo null functions, each integrable function can be represented by a unique continuous function on the Stone space.

Single motherhood, social independence and non-communicable disease (NCD) outcomes among young females (15-24 years old) in South Africa

Background: Non-communicable diseases (NCDs) acquired during youth follow into and affect adulthood. The association between young mother’s social independence and NCD status is of policy interest due to its effect on economic and social development. This study aimed to determine the causal relationship between social independence and NCD outcomes among young, single mothers in South Africa. Methods: Data from the South African National Income Dynamics Survey (NIDS) in 2008 and 2017 was used to determine if single mothers developed hypertension, diabetes or asthma by various indicators of social independence, including highest level of education and employment status. The sample was initially made-up of unmarried females (15-24 years old) without any children in 2008. Both fertility and social independence was followed-up to 2017. Results: In total, 66 young females developed an NCD by 2017 and 87% (n=57) of these women had a child in the interim period. Employment of young females increased from 4.78% in 2008 to 37.79% in 2017, but completion of secondary or tertiary education declined from 67.94% in 2008 to 56.01% in 2017. In addition, half (50.88%) of the young females were partially independent by 2017, with only 11.03% being fully independent at this time. Finally, logistic regression results showed that the likelihood of developing an NCD increased if young females with children were not socially independent. Conclusions: The relationship between social independence and NCDs suggest that policies and programmes in South Africa need to incorporate socioeconomic status as a determinant of disease and in particular, need to address socioeconomic indicators as additive measures and not autonomous indicators.

Single motherhood, social independence and non-communicable disease (NCD) outcomes among young females (15-24 years old) in South Africa

Background: Non-communicable diseases (NCDs) acquired during youth follow into and affect adulthood. The association between young mother’s social independence and NCD status is of policy interest due to its effect on economic and social development. This study aimed to determine the causal relationship between social independence and NCD outcomes among young, single mothers in South Africa. Methods: Data from the South African National Income Dynamics Survey (NIDS) in 2008 and 2017 was used to determine if single mothers developed hypertension, diabetes or asthma by various indicators of social independence, including highest level of education and employment status. The sample was initially made-up of unmarried females (15-24 years old) without any children in 2008. Both fertility and social independence was followed-up to 2017. Results: In total, 66 young females developed an NCD by 2017 and 87% (n=57) of these women had a child in the interim period. Employment of young females increased from 4.78% in 2008 to 37.79% in 2017, but completion of secondary or tertiary education declined from 67.94% in 2008 to 56.01% in 2017. In addition, half (50.88%) of the young females were partially independent by 2017, with only 11.03% being fully independent at this time. Finally, logistic regression results showed that the likelihood of developing an NCD increased if young females with children were not socially independent. Conclusions: The relationship between social independence and NCDs suggest that policies and programmes in South Africa need to incorporate socioeconomic status as a determinant of disease and in particular, need to address socioeconomic indicators as additive measures and not autonomous indicators.

Legitimate equilibrium

We present a general existence result for a type of equilibrium in normal-form games, which extends the concept of Nash equilibrium. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a legitimate equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a legitimate equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptually related equilibrium notion, and shows its existence under the conditions of finitely many players, separable metric action spaces and bounded Borel measurable payoff functions. Our proof borrows several ideas from (Vasquez (2017)), but is more direct as it does not make use of countably additive representations of finitely additive measures by (Yosida and Hewitt (1952)).

On Four Classical Measure Theorems

A subset B of an algebra A of subsets of a set Ω has property (N) if each B-pointwise bounded sequence of the Banach space ba(A) is bounded in ba(A), where ba(A) is the Banach space of real or complex bounded finitely additive measures defined on A endowed with the variation norm. B has property (G) [(VHS)] if for each bounded sequence [if for each sequence] in ba(A) the B-pointwise convergence implies its weak convergence. B has property (sN) [(sG) or (sVHS)] if every increasing covering {Bn:n∈N} of B contains a set Bp with property (N) [(G) or (VHS)], and B has property (wN) [(wG) or (wVHS)] if every increasing web {Bn1n2⋯nm:ni∈N,1≤i≤m,m∈N} of B contains a strand {Bp1p2⋯pm:m∈N} formed by elements Bp1p2⋯pm with property (N) [(G) or (VHS)] for every m∈N. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every σ-algebra has properties (N), (sN), (G) and (VHS). Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every σ-algebra has property (wN) and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property (wN) of a σ-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset B of an algebra A has property (wWHS) if and only if B has property (wN) and A has property (G).