Some Families of Jensen-like Inequalities with Application to Information Theory

It is well known that the traditional Jensen inequality is proved by lower bounding the given convex function, f(x), by the tangential affine function that passes through the point (E{X},f(E{X})), where E{X} is the expectation of the random variable X. While this tangential affine function yields the tightest lower bound among all lower bounds induced by affine functions that are tangential to f, it turns out that when the function f is just part of a more complicated expression whose expectation is to be bounded, the tightest lower bound might belong to a tangential affine function that passes through a point different than (E{X},f(E{X})). In this paper, we take advantage of this observation by optimizing the point of tangency with regard to the specific given expression in a variety of cases and thereby derive several families of inequalities, henceforth referred to as "Jensen-like" inequalities, which are new to the best knowledge of the author. The degree of tightness and the potential usefulness of these inequalities is demonstrated in several application examples related to information theory.

Net effect of environmental fluctuations in multiple global-change drivers across the tree of life

Jensen's inequality predicts that the response of any given system to average constant conditions is different from its average response to varying ones. Environmental fluctuations in abiotic conditions are pervasive on Earth; yet until recently, most ecological research has addressed the effects of multiple environmental drivers by assuming constant conditions. One could thus expect to find significant deviations in the magnitude of their effects on ecosystems when environmental fluctuations are considered. Drawing on experimental studies published during the last 30 years reporting more than 950 response ratios (n = 5,700), we present a comprehensive analysis of the role that environmental fluctuations play across the tree of life. In contrast to the predominance of interactive effects of global-change drivers reported in the literature, our results show that their cumulative effects were additive (58%), synergistic (26%), and antagonistic (16%) when environmental fluctuations were present. However, the dominant type of interaction varied by trophic level (autotrophs: interactive; heterotrophs: additive) and phylogenetic group (additive in Animalia; additive and positive antagonism in Chromista; negative antagonism and synergism in Plantae). In addition, we identify the need to tackle how complex communities respond to fluctuating environments, widening the phylogenetic and biogeographic ranges considered, and to consider other drivers beyond warming and acidification as well as longer timescales. Environmental fluctuations must be taken into account in experimental and modeling studies as well as conservation plans to better predict the nature, magnitude, and direction of the impacts of global change on organisms and ecosystems.

Why are "suboptimal" temperatures preferred in a tropical intertidal ectotherm?

In thermally extreme environments it is challenging for organisms to maximize performance due to risks associated with stochastic variation in temperature and, subsequently, over evolutionary time minimizing the exposure to risk can serve as one of the mechanisms that result in organisms preferring suboptimal temperatures. We tested this hypothesis in a slow-moving intertidal snail on tropical rocky shores, where temperature variability increases with time from 30 min to 20 h when recorded at 30 min intervals (due to short-term environmental autocorrelation where temperatures closer in time are more similar as compared to temperatures over a long period of time). Failure to accommodate temporal variation in thermal stress by selecting cool habitats can result in mortality. Thermal performance curves for different traits (heart rate and locomotion) were measured and compared to the snail's thermal preferences in both the field and laboratory. Predicted performances of the snails were simulated based on thermal performance curves for different traits over multiple time scales and simulated carryover effects. A strong mismatch was found between physiological and behavioural thermal maxima of the snails (physiological thermal maximum being higher by ~ 7 °C), but the snails avoided these maxima and sought temperatures 7 - 14 °C cooler. Such a risk-averse strategy can be explained by their predicted performances where the snails should make decisions about preferred temperatures based on time periods ≥ 5 h to avoid underestimating the temporal variation in body temperature. In extreme and stochastic environments, where the temporal variation in environmental conditions can lead to substantial divergence between instantaneous and time-averaged thermal performances, "cooler is better" and "suboptimal" body temperatures are preferred as they provide sufficient buffer to reduce mortality risk from heat stress.

Responses to Developmental Temperature Fluctuation in Life History Traits of Five Drosophila Species (Diptera: Drosophilidae) from Different Thermal Niches

Temperature has profound effects on biochemical processes as suggested by the extensive variation in performance of organisms across temperatures. Nonetheless, the use of fluctuating temperature (FT) regimes in laboratory experiments compared to constant temperature (CT) regimes is still mainly applied in studies of model organisms. We investigated how two amplitudes of developmental temperature fluctuation (22.5/27.5 °C and 20/30 °C, 12/12 h) affected several fitness-related traits in five Drosophila species with markedly different thermal resistance. Egg-to-adult viability did not change much with temperature except in the cold-adapted D. immigrans. Developmental time increased with FT among all species compared to the same mean CT. The impact of FT on wing size was quite diverse among species. Whereas wing size decreased quasi-linearly with CT in all species, there were large qualitative differences with FT. Changes in wing aspect ratio due to FT were large compared to the other traits and presumably a consequence of thermal stress. These results demonstrate that species of the same genus but with different thermal resistance can show substantial differences in responses to fluctuating developmental temperatures not predictable by constant developmental temperatures. Testing multiple traits facilitated the interpretation of responses to FT in a broader context.

Individual and Population Differences Shape Species Interactions and Natural Selection

Trait variation is central to our understanding of species interactions, and trait variation arising within species is increasingly recognized as an important component of community ecology. Ecologists generally consider intraspecific variation either among or within populations, yet these differences can interact to create patterns of species interactions. These differences can also affect species interactions by altering processes occurring at distinct scales. Specifically, intraspecific variation may shape species interactions simply by shifting a population's position along a trait-function map or by shifting the relationship between traits and their ecological function. I test these ideas by manipulating within- and among-population intraspecific variation in wild populations of a gall-forming insect before quantifying species interactions and phenotypic selection. Within- and among-population differences in gall size interact to affect attack rates by an enemy community, but among-population differences were far more consequential. Intraspecific differences shaped species interactions by both shifting the position of populations along the trait-function map and altering the relationship between traits and their function, with ultimate consequences for patterns of natural selection. I suggest that intraspecific variation can affect communities and natural selection by acting through individual- and population-level mechanisms.

About the sharpness of the Jensen inequality

The main aim of this paper is to give an improvement of the recent result on the sharpness of the Jensen inequality. The results given here are obtained using different Green functions and considering the case of the real Stieltjes measure, not necessarily positive. Finally, some applications involving various types of f-divergences and Zipf-Mandelbrot law are presented.

Refinement of Jensen's inequality and estimation of f- and Rényi divergence via Montgomery identity

Jensen's inequality is important for obtaining inequalities for divergence between probability distribution. By applying a refinement of Jensen's inequality (Horváth et al. in Math. Inequal. Appl. 14:777-791, 2011) and introducing a new functional based on an f-divergence functional, we obtain some estimates for the new functionals, the f-divergence, and Rényi divergence. Some inequalities for Rényi and Shannon estimates are constructed. The Zipf-Mandelbrot law is used to illustrate the result. In addition, we generalize the refinement of Jensen's inequality and new inequalities of Rényi Shannon entropies for an m-convex function using the Montgomery identity. It is also given that the maximization of Shannon entropy is a transition from the Zipf-Mandelbrot law to a hybrid Zipf-Mandelbrot law.

On the systematic underestimation of methane conversion factors in IPCC guidance

This paper documents a systematic underestimation in how the Tier 2 methane conversion factors (MCF) are calculated in IPCC (2006) guidelines for liquid manure management. The first issue is the use of annual average temperature as an input to a non-linear function describing methane production. As expected based on Jensen's inequality, the MCF calculated based on annual average temperature is always an underestimate. In regions with large intra-annual temperature ranges, such as temperate climates, the underestimation can exceed 30%. A second issue is the lack of consideration for volatile solids retention time. Future updates to the IPCC methodology should therefore account for intra-annual temperature regime and retention time-not simply annual average temperature.

Age-Dependent Developmental Response to Temperature: An Examination of the Rarely Tested Phenomenon in Two Species (Gypsy Moth (Lymantria dispar) and Winter Moth (Operophtera brumata))

The pervading paradigm in insect phenology models is that the response to a given temperature does not vary within a life stage. The developmental rate functions that have been developed for general use, or for specific insects, have for the most part been temperature-dependent but not age-dependent, except where age is an ordinal variable designating the larval instar. Age dependence, where age is a continuous variable, is not often reported (or investigated), and is rarely included in phenology models. I provide a short review of the seldom-investigated phenomenon of age dependence in developmental response to temperature, and compare the derivation of the winter moth egg phenology model by Salis et al. to the derivation of another egg phenology model with age-dependent responses to temperature I discuss some probable reasons for the discrepancies (acknowledged by Salis et al.) between modelled and observed developmental rates of the winter moth, and discuss the contribution that geographically robust phenology models can make to estimates of species distributions.

On generalization of refinement of Jensen's inequality using Fink's identity and Abel-Gontscharoff Green function

In this paper, we formulate new Abel-Gontscharoff type identities involving new Green functions for the 'two-point right focal' problem. We use Fink's identity and a new Abel-Gontscharoff-type Green's function for a 'two-point right focal' to generalize the refinement of Jensen's inequality given in (Horváth and Pečarić in Math. Inequal. Appl. 14: 777-791, 2011) from convex function to higher order convex function. Also we formulate the monotonicity of the linear functional obtained from these identities using the recent theory of inequalities for n-convex function at a point. Further we give the bounds for the identities related to the generalization of the refinement of Jensen's inequality using inequalities for the Cebyšev functional. Some results relating to the Grüss and Ostrowski-type inequalities are constructed.

Levinson's type generalization of the Jensen inequality and its converse for real Stieltjes measure

We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi's inequality. The obtained results are then applied for establishing new mean-value theorems. The results from this paper represent a generalization of several recent results.

A Paradoxical Result in Estimating Regression Coefficients

This paper presents a counterintuitive result regarding the estimation of a regression slope co-efficient. Paradoxically, the precision of the slope estimator can deteriorate when additional information is used to estimate its value. In a randomized experiment, the distribution of baseline variables should be identical across treatments due to randomization. The motivation for this paper came from noting that the precision of slope estimators deteriorated when pooling baseline predictors across treatment groups.

A Univariate Extension of Jensen's Inequality

The result in this paper explains some of the qualitative nature of Jensen's inequality. It is shown that the more disperse the distribution of a random variable is, the smaller is the expectation of any concave function of it. This result can be used to show the inadequacy of some current methods of reporting environmental data by using geometric means, and it extends the result of I. Billick, D. Shier, and C. H. Spiegelman, where symmetry of the error in environmental measurements is assumed.