Properties of the bridge sampler with a focus on splitting the MCMC sample

Computation of normalizing constants is a fundamental mathematical problem in various disciplines, particularly in Bayesian model selection problems. A sampling-based technique known as bridge sampling (Meng and Wong in Stat Sin 6(4):831–860, 1996) has been found to produce accurate estimates of normalizing constants and is shown to possess good asymptotic properties. For small to moderate sample sizes (as in situations with limited computational resources), we demonstrate that the (optimal) bridge sampler produces biased estimates. Specifically, when one density (we denote as $$p_2$$) is constructed to be close to the target density (we denote as $$p_1$$) using method of moments, our simulation-based results indicate that the correlation-induced bias through the moment-matching procedure is non-negligible. More crucially, the bias amplifies as the dimensionality of the problem increases. Thus, a series of theoretical as well as empirical investigations is carried out to identify the nature and origin of the bias. We then examine the effect of sample size allocation on the accuracy of bridge sampling estimates and discovered that one possibility of reducing both the bias and standard error with a small increase in computational effort is by drawing extra samples from the moment-matched density $$p_2$$ (which we assume easy to sample from), provided that the evaluation of $$p_1$$ is not too expensive. We proceed to show how the simple adaptive approach we termed “splitting” manages to alleviate the correlation-induced bias at the expense of a higher standard error, irrespective of the dimensionality involved. We also slightly modified the strategy suggested by Wang et al. (Warp bridge sampling: the next generation, Preprint, 2019. arXiv:1609.07690) to address the issue of the increase in standard error due to splitting, which is later generalized to further improve the efficiency. We conclude the paper by offering our insights of the application of a combination of these adaptive methods to improve the accuracy of bridge sampling estimates in Bayesian applications (where posterior samples are typically expensive to generate) based on the preceding investigations, with an application to a practical example.     

Estimating the impact of the Global Financial Crisis on poverty and deprivation

Although the Australian economy has avoided going into recession since the onset of the global financial crisis, relatively little is known about the social impact of the crisis. Survey and anecdotal evidence presented by a number of community sector NGOs suggest that the demand for emergency relief and other services rose in the aftermath of the crisis. This article reports results derived from two national surveys on poverty and social disadvantage that were conducted in 2006 and 2010 – prior to and after the crisis struck. The survey data allow income‐based poverty rates to be estimated as well as the extent and nature of deprivation, where deprivation is defined as being unable to afford items that a majority regards as essential: things that no‐one in Australia should have to go without. Poverty and deprivation are not alternative indicators but can be combined into a multi‐dimensional measure of consistent poverty. The paper examines how conventional (income) poverty, deprivation and consistent poverty changed between 2006 and 2010, and analyses the sensitivity of the results to alternative definitions. The findings provide the first comprehensive evidence on how the profile of social disadvantage in Australia changed in the period that spans the financial crisis.     

Axiomatizations of public good indices with a priori unions

We provide axiomatizations for six variants of the Public Good Index (PGI) for games with a priori unions. Two such coalitional PGIs have been introduced and alternatively axiomatized in Alonso-Meijide et al. (Working paper 18, 2008b). They assign power in two steps. In the first step, power is distributed between unions according to the PGI of the quotient game. In a second step, the Solidarity PGI splits power equally among union members while the Owen Extended PGI takes into account so-called essential parts. The other four coalitional PGIs have been introduced in Holler and Nohn (Homo Oeconomicus 26, 2009). The first variant elaborates the original idea of Holler (Political Studies 30:262–271, 1982) that the coalitional value is a public good and only minimal winning coalitions of the quotient game are relevant. The remaining three variants also use the two-step distribution where, however, on the member stage they take into account the possibilities of players to threaten their partners through leaving their union.     

Sex Differences in Perceived Family Functioning and Family Resources in Hong Kong Families: Implications for Social Work Practice

Although enhancing family functioning has become a rising concern of social workers in Chinese contexts such as Hong Kong, little has been known on perceived family functioning and family resources. To fill in this knowledge gap, this article reports part of the results of a telephone survey conducted in Hong Kong, aiming to identify the sex differences in these two areas. The results of the study have shown that Chinese women perceived better affective involvement, one of the crucial indicators of family functioning, than Chinese men. However, among the different family resources, men rated better physical and mental well-being whereas women's social connection was stronger than men's. Further analysis have shown that the linkage of three family resources (namely stress coping efficacy, time spent with family and income) to perceived family functioning was statistically significant irrespective of genders. Implications of this study for social work practice are discussed at the end of the article.     

Welfare attitudes and social class: the case of Hong Kong in comparative perspective

Based on a survey of Hong Kong residents, this article explores the attitudes towards the welfare state and whether or not there are significant differences between different social classes with regard to their approval of the welfare state. The findings were then compared with those for Sweden and the USA. The study shows that Hong Kong residents strongly approve of the welfare state. The strength of their support is similar to that in Sweden and is far stronger than in the USA. In Hong Kong, the influence of social class on attitudes towards the welfare state is negligible. In some cases, the privileged classes expressed greater approval for the welfare state than the underprivileged classes. This is in striking contrast to the experiences in Sweden and the USA where the underprivileged classes are more supportive of the welfare state than are the privileged classes.     

A note on inference for P(X < Y) for right truncated exponentially distributed data

In this paper, a likelihood based analysis is developed and applied to obtain confidence intervals and p values for the stress-strength reliability R  =  P(X  <  Y) with right truncated exponentially distributed data. The proposed method is based on theory given in Fraser et al. (Biometrika 86:249–264, 1999) which involves implicit but appropriate conditioning and marginalization. Monte Carlo simulations are used to illustrate the accuracy of the proposed method.