Likelihood, entropy and species diversity; some comparisons in a Sumatran forest

In attempts to conserve the species diversity of trees in tropical forests, monitoring of diversity in inventories is essential. For effective monitoring it is crucial to be able to make meaningful comparisons between different regions, or comparisons of the diversity of a region at different times. Many species diversity measures have been defined, including the well-known abundance and entropy measures. All such measures share a number of problems in their effective practical use. However, probably the most problematic is that they cannot be used to meaningfully assess changes, since thay are only concerned with the number of species or the proportions of the population/sample which they constitute. A natural (though simplistic) model of a species frequency distribution is the multinomial distribution. It is shown that the likelihood analysis of samples from such a distribution are closely related to a number of entropy-type measures of diversity. Hence a comparison of the species distribution on two plots, using the multinomial model and likelihood methods, leads to generalised cross-entropy as the LRT test statistic of the null that the species distributions are the same. Data from 30 contiguous plots in a forest in Sumatra are analysed using these methods. Significance tests between all pairs of plots yield extremely low p-values, indicating strongly that it ought to been "Obvious" that the observed species distributions are different on different plots. In terms of how different the plots are, and how these differences vary over the whole study site, a display of the degrees of freedom of the test, (equivalent to the number of shared species) seems to be the most revealing indicator, as well as the simplest.

Maximum-likelihood estimator for coarse carrier frequency offset estimation in OFDM systems

Orthogonal frequency division multiplexing (OFDM) systems are more sensitive to carrier frequency offset (CFO) compared to the conventional single carrier systems. CFO destroys the orthogonality among subcarriers, resulting in inter-carrier interference (ICI) and degrading system performance. To mitigate the effect of the CFO, it has to be estimated and compensated before the demodulation. The CFO can be divided into an integer part and a fractional part. In this paper, we investigate a maximum-likelihood estimator (MLE) for estimating the integer part of the CFO in OFDM systems, which requires only one OFDM block as the pilot symbols. To reduce the computational complexity of the MLE and improve the bandwidth efficiency, a suboptimum estimator (Sub MLE) is studied. Based on the hypothesis testing method, a threshold Sub MLE (T-Sub MLE) is proposed to further reduce the computational complexity. The performance analysis of the proposed T-Sub MLE is obtained and the analytical results match the simulation results well. Numerical results show that the proposed estimators are effective and reliable in both additive white Gaussian noise (AWGN) and frequency-selective fading channels in OFDM systems.

What determines the likelihood of species discovery in marine holozooplankton: is size, range or depth important?

The relationship between date of first description and size, geographic range and depth of occurrence is investigated for 18 orders of marine holozooplankton (comprising over 4000 species). Results of multiple regression analyses suggest that all attributes are linked, which reflects the complex interplay between them. Partial correlation coefficients suggest that geographic range is the most important predictor of description date, and shows an inverse relationship. By contrast, size is generally a poor indicator of description date, which probably mirrors the size-independent way in which specimens are collected, though there is clearly a positive relationship between both size and depth (for metabolic/trophic reasons), and size and geographic range. There is also a positive relationship between geographic range and depth that probably reflects the near constant nature of the deep-water environment and the wide-ranging currents to be found there. Although we did not explicitly incorporate either abundance or location into models predicting the date of first description, neither should be ignored.

Maximum likelihood estimation of higher-order integer-valued autoregressive processes

In this article, we extend the earlier work of Freeland and McCabe [Journal of time Series Analysis (2004) Vol. 25, pp. 701–722] and develop a general framework for maximum likelihood (ML) analysis of higher-order integer-valued autoregressive processes. Our exposition includes the case where the innovation sequence has a Poisson distribution and the thinning is binomial. A recursive representation of the transition probability of the model is proposed. Based on this transition probability, we derive expressions for the score function and the Fisher information matrix, which form the basis for ML estimation and inference. Similar to the results in Freeland and McCabe (2004), we show that the score function and the Fisher information matrix can be neatly represented as conditional expectations. Using the INAR(2) speci?cation with binomial thinning and Poisson innovations, we examine both the asymptotic e?ciency and ?nite sample properties of the ML estimator in relation to the widely used conditional least
squares (CLS) and Yule–Walker (YW) estimators. We conclude that, if the Poisson assumption can be justi?ed, there are substantial gains to be had from using ML especially when the thinning parameters are large.

Maximum likelihood estimation of higher-order integer-valued autoregressive processes

Corrigendum Vol. 30, Issue 2, 259, Article first published online: 15 MAR 2009 to correct the order of authors names: Bu R., K. Hadri, and B. McCabe.

Predicting the likelihood of carrying a BRCA1 or BRCA2 mutation: validation of BOADICEA, BRCAPRO, IBIS, Myriad and the Manchester scoring system using data from UK genetics clinics

Objectives: Genetic testing for the breast and ovarian cancer susceptibility genes BRCA1 and BRCA2 has important implications for the clinical management of people found to carry a mutation. However, genetic testing is expensive and may be associated with adverse psychosocial effects. To provide a cost-efficient and clinically appropriate genetic counselling service, genetic testing should be targeted at those individuals most likely to carry pathogenic mutations. Several algorithms that predict the likelihood of carrying a BRCA1 or a BRCA2 mutation are currently used in clinical practice to identify such individuals.

Predator diversity enhances secondary production and decreases the likelihood of trophic cascades

We manipulated the diversity of top predators in a three trophic level marine food web. The food web included four top benthic marine fish predators (black goby, rock goby, sea scorpion and shore rockling), an intermediate trophic level of small fish, and a lower trophic level of benthic invertebrates. We kept predator density constant and monitored the response of the lower trophic levels. As top predator diversity increased, secondary production increased. We also observed that in the presence of the manipulated fish predators, the density of small gobiid fish (intermediate consumers) was suppressed, releasing certain groups of benthic invertebrates (caprellid amphipods, copepods, nematodes and spirorbid worms) from heavy intermediate predation pressure. We attribute the mechanism responsible for this trophic cascade to a trait-mediated indirect interaction, with the small gobiid fish changing their use of space in response to altered predator diversity. In the absence of top fish predators, a full-blown trophic cascade occurs. Therefore the diversity of predators reduces the likelihood of trophic cascades occurring and hence provides insurance against the loss of an important ecosystem function (i.e. secondary production).

Social support and the likelihood of maintaining and improving levels of physical activity:the Whitehall II Study

Background: Evidence on the association between social support and leisure time physical activity (LTPA) is scarce and mostly based on cross-sectional data with different types of social support collapsed into a single index. The aim of this study was to investigate whether social support from the closest person was associated with LTPA.

Nonresident Father Involvement with Children and Divorced Women's Likelihood of Remarriage

Although remarriage is a relatively common transition, little is known about how nonresident fathers affect divorced mothers’ entry into remarriage. Using the 1979–2010 rounds of the National Longitudinal Study of Youth 1979, the authors examined the likelihood of remarriage for divorced mothers (N = 882) by nonresident father contact with children and payment of child support. The findings suggest that maternal remarriage is positively associated with nonresident father contact but not related to receiving child support.

Nonresident Father Involvement with Children and Divorced Women's Likelihood of Remarriage

Although remarriage is a relatively common transition, we know little about how nonresident fathers affect divorced mothers’ entry into remarriage. Using the 1979-2010 rounds of the National Longitudinal Study of Youth 1979, we examined the likelihood of remarriage for divorced mothers (n=882) by nonresident father contact with children and payment of child support. The findings suggest that maternal remarriage is positively associated with nonresident father contact but not related to receiving child support.

Efficient Marginal Likelihood Computation for Gaussian Process Regression

In a Bayesian learning setting, the posterior distribution of a predictive model arises from a trade-off between its prior distribution and the conditional likelihood of observed data. Such distribution functions usually rely on additional hyperparameters which need to be tuned in order to achieve optimum predictive performance; this operation can be efficiently performed in an Empirical Bayes fashion by maximizing the posterior marginal likelihood of the observed data. Since the score function of this optimization problem is in general characterized by the presence of local optima, it is necessary to resort to global optimization strategies, which require a large number of function evaluations. Given that the evaluation is usually computationally intensive and badly scaled with respect to the dataset size, the maximum number of observations that can be treated simultaneously is quite limited. In this paper, we consider the case of hyperparameter tuning in Gaussian process regression. A straightforward implementation of the posterior log-likelihood for this model requires O(N^3) operations for every iteration of the optimization procedure, where N is the number of examples in the input dataset. We derive a novel set of identities that allow, after an initial overhead of O(N^3), the evaluation of the score function, as well as the Jacobian and Hessian matrices, in O(N) operations. We prove how the proposed identities, that follow from the eigendecomposition of the kernel matrix, yield a reduction of several orders of magnitude in the computation time for the hyperparameter optimization problem. Notably, the proposed solution provides computational advantages even with respect to state of the art approximations that rely on sparse kernel matrices.

Improving parameter learning of Bayesian nets from incomplete data

This paper addresses the estimation of parameters of a Bayesian network from incomplete data. The task is usually tackled by running the Expectation-Maximization (EM) algorithm several times in order to obtain a high log-likelihood estimate. We argue that choosing the maximum log-likelihood estimate (as well as the maximum penalized log-likelihood and the maximum a posteriori estimate) has severe drawbacks, being affected both by overfitting and model uncertainty. Two ideas are discussed to overcome these issues: a maximum entropy approach and a Bayesian model averaging approach. Both ideas can be easily applied on top of EM, while the entropy idea can be also implemented in a more sophisticated way, through a dedicated non-linear solver. A vast set of experiments shows that these ideas produce significantly better estimates and inferences than the traditional and widely used maximum (penalized) log-likelihood and maximum a posteriori estimates. In particular, if EM is adopted as optimization engine, the model averaging approach is the best performing one; its performance is matched by the entropy approach when implemented using the non-linear solver. The results suggest that the applicability of these ideas is immediate (they are easy to implement and to integrate in currently available inference engines) and that they constitute a better way to learn Bayesian network parameters.