Constrained Bayesian Methods for Testing Directional Hypotheses Restricted False Discovery Rates - Juniper Publishers

Biostatistics and Biometrics - Juniper Publishers

Abstract

Constrained Bayesian method (CBM) and the concept of false discovery rates (FDR) for testing directional hypotheses is considered in the paper. Here is shown that the direct application of CBM allows us to control FDR on the desired level. Theoretically it is proved that mixed directional false discovery rates (mdFDR) are restricted on the desired levels at the suitable choice of restriction levels at different statements of CBM. The correctness of the obtained theoretical results is confirmed by computation results of concrete examples.

Subject Classifications: 62F15; 62F03.

Keywords: Directional hypotheses; Constrained Bayesian method; False discovery rate; Mixed directional false discovery rates; False acceptance rate

Abbrevations: CBM: Constrained Bayesian method; DFDR: Directional False Discovery Rate; FDR: False Discovery Rates; MDFDR: Mixed Directional False Discovery Rates; FAR: False Acceptance Rate

Introduction

The traditional formulation of testing simple basic hypothesis versus composite alternative is a well studied problem in many scientific works [1-8]. The problem of making the sense about direction of difference between parameter values, defined by basic and alternative hypotheses, is important in many applications [9-17]. Here the decision whether the parameter outstrips or falls behind of the value defined by basic hypothesis is meaningful. For parametrical models, this problem can be stated as

Where θ is the parameter of the model, 0θ is known. These alternatives are called skewed or directional alternatives. The consideration of directional hypotheses found their applications in different realms. Among them are biology, medicine, technique and so on [17,18]. The appropriate tests “has just begun to stir up some interests in the educational and behavioral literature” [19-22]. Directional false discovery rate (DFDR) or mixed directional false discovery rate (mdFDR) are used when alternatives are skewed [17]. The optimal procedures controlling DFDR (or mdFDR) use two-tailed procedures assuming that directional alternatives are symmetrically distributed. Therefore, decision rule is symmetric in relation with the parameter’s value defined by basic hypothesis [14,23]. For the experiments where the distribution of the alternative hypotheses is skewed, the asymmetric decision rule, based on skew normal priors and used Bayesian methodology for testing when minimizing mdFDR, is offered in Bansal et al. [17]. There theoretically is proved “that a skewed prior permits a higher power in number of correct discoveries than if the prior is symmetric”. This result is confirmed by simulation study comparing the proposed rule with a frequentist’s rule and the rule offered in Benjamini, et al. [23]. Because CBM allows us to foresee the skewness by not only a prior probabilities but also by restriction levels in the constraints, it is expected that it will give more powerful decision rule in number of correct discoveries than existed symmetric or asymmetric in the prior decision rules. Therefore, different statements of CBM, for testing skewed hypotheses with restricted mdFDR, are considered below.

In Section 2 some possible statements of CBM for testing directional hypotheses are considered and the fact that FDR could be controlled on the desired level for each statement of CBM is proved. Concretization of the proposed theoretical results for the normally distributed directional hypotheses is given in Section 3. Computation results of concrete example for normal basic and truncated normal alternative hypotheses by simulation of the appropriate samples are given in section 4. Discussion of the obtained results and made conclusions are presented in sections 5 and 6, respectively.

CBM for testing directional hypotheses

Different statements of CBM for testing a set of hypotheses are given in Kachiashvili, et al. [24]; Kachiashvili, [25]; Kachiashvili et al. [26]. They differ from each other by the kind of restrictions put on the Type I or Type II errors made at testing. Let’s introduce the following denotations for statement of the problem of testing hypotheses [27]. Let the sample be generated from (px;θ) and the problem of interest is to test are disjoint subsets with iHpis the a priori probability of hypothesis is a prior density with support denotes the marginal density of x given is the set of solutions, where it being so that

associates each observation vector x with a certain decision

jΓ is the region of acceptance of hypothesis ,jH i.e., It is obvious that δ(x) is completely determined by the jΓ regions, i.e. and be the losses of incorrectly accepted and incorrectly rejected hypotheses. Then the total loss of incorrectly accepted and incorrectly rejected hypotheses is the following:

Adapting the made denotations to skewed hypotheses, let’s consider some kind of CBM, from all possible statements, for testing directional hypotheses (1). (Notation 1: numbering of the tasks, described below, is preserved from Kachiashvili, et al. [27] 2.1. Restrictions on the averaged probability of acceptance of true hypotheses (Task 1). In this case, CBM has the following form Kachiashvili, et al. [28]: to minimize the averaged loss of incorrectly accepted hypotheses

Where 1r is some real number determining the level of the averaged loss of incorrectly rejected hypotheses. For directional hypotheses (1) and loss functions

using concepts of posterior probabilities, the problem (3), (4) transforms in the following form Kachiashvili et al. [28]

subject to

The solution of the problem (6) and (7) by Lagrange undetermined multiplier method gives

where Lagrange multiplier λ₁ is determined so that in condition (7) equality takes place.

(Notation 2: for the statement (3), (4) as well as for other statements (see Tasks 2, 4 and 5, below), depending upon the choice of ,x there is a possibility that 1)(=xjδ for more than one j or 0)(=xjδ for all (),0,.

Let’s introduce denotations

and let’s call them individual average risks. Then for the average risk (6) we HAVE

At testing directional hypotheses, it is possible to make a false statement about choice among alternative hypotheses, i.e. to make a directional error, or a type III error [23]. For recognition of directional errors in the terms of false discovery rate (FDR) two variants of false discovery rate (FDR) were introduced in Benjamini et al. [29]: pure directional false discovery rate (pdFDR) and mixed directional false discovery rate (mdFDR), which are the following

It is shown that the FDR is an effective model selection criterion, as it can be translated into a penalty function. Therefore, FDR gives the opportunity to increase the power of the test in general case [30]. Both variants of FDR for directional hypotheses: pdFDR and mdFDR can be expressed by Type III error rates (ERRIII):

Here TIIIERR and KIIIERR are two different forms of Type III error rates, considered by different authors Mosteller, et al. [31]; Kaiser, [9]; Jones, et al. [13] and Shaffer, [14]) and IIISERR is the summary type III error rate ()IIISERR [25].

Here in after, if necessary, let’s ascribe the number of the task related to the considered CBM directly to this abbreviation.

Theorem 1. CBM 1 with restriction level of (7), at satisfying a condition

ensures a decision rule with less or equal to q i.e. with the condition

Proof. Because of the peculiarity of decision making rule of CBM, alongside of hypotheses acceptance regions there exist the regions of impossibility of making a decision [26,32]. Therefore, instead of condition

of the classical decision making procedures, the following condition is fulfilled in CBM

where imd is the abbreviation of the impossibility of making a decision

Taking into account (17), condition (7) can be rewritten as follows

From here follows that

Let’s denote Then from (18) we have

Taking into account (12), we write

This proves the theorem

Let’s call false acceptance rate (FAR) the following

Restrictions on the conditional probabilities of acceptance of each true hypothesis (Task 2)

To minimize (6) subject to

where Lagrange multipliers are determined so that in conditions (22) equalities takes place.

Theorem 2. CBM 2 with restriction level of (22), at satisfying a condition q, ensures a decision rule with mdFDR (i.e. with IIISERR) less or equal to q, i.e. with the condition

Proof. Taking into account (12), (17), condition

In our opinion these properties of and CBM are very interesting and useful. They bring the statistical hypotheses testing rule much close to the everyday decision-making rule when, at shortage of necessary information, acceptance of one of made suppositions is not compulsory.

The specific features of hypotheses testing regions of the Berger’s test and CBM, namely, the existence of the no-decision region in the test and the existence of regions of impossibility of making a unique or any decision in CBM give the opportunities to develop the sequential tests on their basis [2,36,26,28]. The sequential test was introduced by Wald in the middle of forty of last century [37,38]. Since Wald’s pioneer works, a lot of different investigations were dedicated to the sequential analysis problems (see, for example, Berger, et al. [39]; Ghosh, [40]; Ghosh, et al. [41]; Siegmund, [42]) and efforts to the development of this approach constantly increase as it has many important advantages in comparison with the parallel methods [43].

Application of CBM to different types of hypotheses (two and many simple, composite, directional and multiple hypotheses) with parallel and sequential experiments showed the advantage and uniqueness of the method in comparison with existing ones [24-29,44]. The advantage of the method is the optimality of made decisions with guaranteed reliability and minimality of necessary observations for given reliability. CBM uses not only loss functions and a priori probabilities for making decisions as the classical Bayesian rule does, but also a significance level as the frequentist method does. The combination of these opportunities improves the quality of made decisions in CBM in comparison with other methods. This fact is many times confirmed by application of CBM to the solution of different practical problems [45-47,32,44].

Finally, it must be noted that, the detailed investigation of different statements of CBM and the choice of optimal loss functions in the constrained statements of the Bayesian testing problem opens wide opportunities in statistical hypotheses testing with new, beforehand unknown and interesting properties. On the other hand, the statement of the Bayesian estimation problem as a constrained optimization problem gives new opportunities in finding optimal estimates with new, unknown beforehand properties, and it seems that these properties will advantageously differ from those of the approaches known today.

In our opinion, the proposed CBM are the ways for future, perspective investigations which will give researchers the opportunities for obtaining new perspective results in the theory and practice of statistical inferences and it completely corresponds to the thoughts of the well-known statistician B Efron [48]: “Broadly speaking, nineteenth century statistics was Bayesian, while the twentieth century was frequentist, at least from the point of view of most scientific practitioners. Here in the twenty-first century scientists are bringing statisticians much bigger problems to solve, often comprising millions of data points and thousands of parameters. Which statistical philosophy will dominate practice? My guess, backed up with some recent examples, is that a combination of Bayesian and frequentist ideas will be needed to deal with our increasingly intense scientific environment. This will be a challenging period for statisticians, both applied and theoretical, but it also opens the opportunity for a new golden age, rivaling that of Fisher, Neyman, and the other giants of the early 1900s.”

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Challenges to Solve Our Plastic Waste Problems - Juniper Publishers

Organic & Medicinal Chemistry - Juniper Publishers

Abstract

This year the campaign to stop plastics pollution has increase. On Social Media it is used the theme: #Beat Plastic Pollution. Such actions are crucial in the implementation of the 2030 Agenda for Sustainable Development and its SDGs, and the Paris Agreement on Climate Change, including through engaging in political advocacy, providing expertise and insights on potential policy choices, as well as being directly responsible for the implementation of many of these decisions [1]. At the moment, humanity makes about 300 million tons of plastic every year. And half of it is used only once (like bags and wrapping). Approximately 10% of it is recycled and the rest goes to garbage dumps/landfills and rivers. According to a recent study, 10 rivers (2 in Africa and 8 in Asia) transport 90% of all the plastic that ends up in oceans. The scale of pollution is awful and many of us don’t even realize how bad this situation is [2]. Countries in the world are mobilizing to ensure they become nations free of plastic bags and straws and polystyrenes takeaway boxes, among other undesirable plastics by the end of present year 2018. The ban on these plastics is an opportunity for us to help protect our planet and to strengthen our sustainable cultural practices [3]. 

Keywords: Plastics Pollution; Water Pollution; Land Pollution; Air Pollution; Sustainable Development; Climate Change

Discussion

Over the time, our oceans have increasingly become dumping grounds for different types of waste, including sewage, industrial waste, chemicals, plastics and litter. An estimated 80% of marine pollution originates from land-based sources including wastewater and nutrients loadings. Worldwide, pollution in coastal waters has increased exponentially during the last decades due to population growth and the increasing number of anthropogenic activities. Coral reefs are particularly vulnerable to wastewater and nutrient pollution, which consequently threatens the health and well‐being of hundreds of millions of people who depend on coral reef ecosystem services for nutrition, livelihoods and a safe living environment. With the influences of ocean warming and coral bleaching impacts, land-based pollution constitutes a significant additional threat that must be addressed with urgency.

you choose. There is the hawksbill turtle struggling to free itself from a plastic bag. The sea of polystyrene trash floating over a Caribbean nature reserve. Or the sperm whale washed ashore, its stomach filled with plastic waste. Pieces of plastic are sharp, brittle, toxic, and routinely found in the stomachs of dead fish, turtles, and marine mammals. Plastics can come with a range of hazardous additives and can act as a chemical sponge, soaking up and concentrating other pollutants. Marine species, including fish, seabirds and even marine mammals, can end up eating pieces of plastic, and at the same time get an additional dose of toxic chemicals.

Since the introduction of mass-produced plastics in the early 20th century, humanity has produced an estimated 8,300 million tons of the stuff. Around three-quarters has been thrown away, and 80% of that has drifted into the environment or gone into a landfill. Approximately, eight million tons a year end up in the ocean, it makes more than 5 trillion pieces and counting. It is an environmental catastrophe and a human one, too, as some people in parts of the developing world live ankle-deep in filthy, non-biodegrading plastic trash (Table 1). The long-term health implications for all of us remain uncertain, as ingested plastic works its way up to the food chain. Everyone agrees something must be done. From banning plastic straws to rebooting recycling systems to harnessing plastic-munching bacteria, there is no shortage of touted solutions. It is less clear what would work best. But fixing the plastic waste crisis is going to take some seriously joined-up thinking. If we make the wrong decisions now, we risk making the problem worse. If plastics didn’t exist, we would have to invent them. Generally made of oil-derived polymers, they can be hung with different chemical groups and spiced up with additives to give them wildly differing properties such as hardness, strength, density and heat-resistance. This makes them just the thing for everything from colorful, durable kids’ toys Figure 1.

We need to tackle the problem of marine debris head on. It’s not just an issue for environmentally conscious, it is an issue that ultimately affects human health. Man is a top predator that feeds on a variety of ocean fish, shellfish and other marine species. We face the same risks as the killer whale and polar bear. While any plastic or polystyrene pellets that may have been clogging the gut of the fish that is nicely presented on our dinner plate have been long removed, the toxic contaminants originating from that debris remain stored in the flesh we are about to eat. Food for thought indeed. Some scientists are suggesting that burning plastic could be the most environmentally-friendly way to get rid of it. They base it on the fact that plastics are composed of a mixture of hydrocarbons with varying chain length, mainly long chain hydrocarbons. They support the theory that there is not big difference with other fuels apart from some halogens and Sulphur that can be removed from any emissions using appropriate filters and sequesters [1-4].

Conclusion

When we deal with plastics pollution, we should be sober in adopting solutions. Climate Change and the Loss of Biodiversity remain our top environmental concerns, and we cannot afford to adopt plastic alternatives that increase, not decrease, our impact on the planet. A cotton tote bag or steel water bottle may generate higher carbon emissions over its lifetime. A rush to bioderived plastics may, as with bio-fuels, increase land cleared for crops. Plastics that degrade faster risk increasing the scourge of microplastic in the environment if not partnered with better waste management.

Some solutions to reduce plastic pollution are specific actions policy makers can take to improve waste management, promote eco-friendly alternatives, educate consumers, enable voluntary reduction strategies and successfully implement bans or levies on the use and sale of single-use plastics, etc. World Environment Day 2018 issued a call to action to individuals, governments, the public and the private sector to examine joint solutions to reduce the heavy burden of plastic pollution on our natural places, our wildlife and our own health. Now more than ever Zac Goldsmith quote is a proven statement: “Of all the waste we generate, plastics are perhaps the greatest symbol of our throwaway society. They are used, then forgotten, and they leave a terrible legacy”.

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Smiles and Tears in Prehistoric Art - Juniper Publishers

Archaeology & Anthropology - Juniper Publishers 

Summary

From a graphic point of view, smiles are prior to tears. While the smiles are documented in human graffiti from the Paleolithic period (Middle Magdalenian), tears are documented much later, in animals of post paleolithic periods. These data help us to make a few reflections about our way of seeing art, which may be influenced by our culture, too serious, for a few graphs that are not so much.

Keywords: Smile; Laughter; Paleolithic art; Evolution

The smile of Bourdois

The figure reproduced below is a cave engraving of the Bourdois shelter, located in the Vienne, France. They will agree with me that at first glance it does not seem to have anything of particular. It is a small human face endowed, of course, with a broad smile [1]. But, if we consider that the spelling in question is 15,000 years old, the thing changes, and the impression that the smile produces increases. Indeed, we are facing one of the first smiles in the History of Art. The smile of Bourdois is not, far from it, as famous as that of the Mona Lisa, and yet its transcendence is much greater. Even if only because a smile from 15,000 years ago contains many more puzzles than one of just four centuries. And if not, look closely at the face, is it possible to look at this smiling face for a long time without smiling? This reaction, almost instinctive, says a lot about the human species, about who we are and why we are here (Figure 1).

This enigmatic fascination, which all attentive observers share, is caused by the intimacy of a gesture of different and partly undefined nuances. And is that every smile always harbors a suspicion: the shadow of dissimulation. You can see in it the trace of deceit, submission or fear. But, although the shadow of a doubt looms over its inner light, the smile is, above all, an expression of pleasure and happiness. For a smile manages to stay true to itself and its own mystery. That is to say, what really hides is, neither more nor less, than the secret of happiness.

On the other hand, the fascinating attraction that a smile exerts can be understood as a power of seduction that, Freud did not hesitate to describe as erotic [2]. And although the genius of psychoanalysis was too often carried away by interpretations of a sexual nature, it is quite possible that this time it was not misguided. Almost all the specialists in gestural mimicry agree in affirming that the smile has an erotic function. Even scholars of human behavior have highlighted the erotic relationship of the smile in current primitive peoples [3]. This relationship may be very old, perhaps prehistoric. In paleolithic art we have four examples in which the smile is associated with anthropomorphic figures with upright sex [4]. These spellings seem to reflect a relationship between happiness and sexual satisfaction.

The female silhouettes of Angles-sur-l’Anglin

A few meters from the smiling face of Bourdois we have, in the same frieze, four sculpted female silhouettes, in a position that today we would not hesitate to describe as erotic. So much so, that Guthrie has compared these paleolithic profiles with the images of the Playboy. The comparison with current pornography deserves to be criticized at least in two essential points. The first is that porn is characterized by its seriousness, it is, in the words of Braudillard, deadly serious [5]. Which means that there has been a process of verification of the erotic in the pornographic, or, to put it another way, the vital and joyful component of the erotic has been eliminated, turning it into something mechanical and artificial, reduced to an act highly stereotyped And secondly, the pornographic in Western culture excludes the sacred, something that does not happen for example in the East.

However, as can be seen in three of the graphs reproduced below, the silhouettes of Angles-sur-l’Anglin, have an inescapable erotic tone. The pose of naked bodies has a disconcerting effect. It is almost impossible not to see in them the bodies of contemporary models or sex symbols. What does not lead us to ask the following question, what we see is conditioned by our pornographic aesthetics or does it have some biological basis? Look at the sinuous lines that frame the vulvas and delineate the hips and part of the legs. These parts of the female body are erotic in virtually all human cultures [6].

The reason why aesthetics uses these forms has a biological basis, that is, a “practical” sense. It is no coincidence that the beauty ideals of a tribe of Trobians and Westerners are so similar [7]. Nor that, apparently, these ideals, based on body proportions, have not changed over time. Well, they respond to a purpose, which is to stimulate reproduction, because according to certain studies, female hips and legs without visual indicators of women’s fertility. Furthermore, not only the silhouettes of Angles-sur-l’Anglin have that disconcertingly modern character; but there are other similar examples in the caves of Le Gabillou and La Magdeleine. The position of these female bodies, reminiscent of Goya’s Nude Maja, is eminently erotic. For example, the position of the arm behind the head in the bas-reliefs of La Magdeleine, is an expression of enjoyment that is also observed in sexual scenes from Roman times. In a painting of the House of the Restaurant of Pompeii (1st century), A woman bends her arm in this way while practicing sex with a man. The scene, of obvious interpretation in the Roman case, is not in the prehistoric. What is evoked in Paleolithic art is not the act itself, but something more complex whose final meaning escapes us. But what is interesting here is to point out that artists or paleolithic artists are the creators of a visual eroticism that is surely different from what we understand today. It was probably a happy and perhaps sacred eroticism, but the result of the complex world of seduction, the mystery of attraction, of creation and of life (Figure 2).

Prehistoric eroticism, and its Paleolithic graphic expression, have their roots in human evolution. Specifically, in the development of our particular mode of sexual reproduction. About two million years ago, our hominid relatives began a strategy of reproduction that we could describe as optimistic, since it consisted, mainly, of having sex a more or less constant pleasure. This fact was crucial in the evolution of our species. Our reproductive success (it is estimated that we are around 7,000 million people in the world) is unparalleled in the history of placental mammals. If we are a prolific species par excellence it is thanks to the intrinsic quality of our sex to provide us with pleasure at any time of the year. The other animal species either do not experience as much pleasure or are subject to short periods of heat. Human sexuality does not depend, like that of other mammals, on the olfactory stimuli and the hormonal chemistry of pheromones; but predominantly visual stimuli, based mainly on physical features and body proportions [7]. This explains, to a certain extent, the eroticism of artistic expressions. Today the pornographic market has reduced the erotic to its minimal expression. The most visited websites on the internet are, by far, pornographic. We are a species that we bet on the pleasure of reproduction. And natural selection has favored this optimistic strategy.

The Tears

From a graphic point of view, smiles are prior to tears. The tears that appear on the faces of the Tassili cows, studied by Le Quellec [8], seem to have a symbolic meaning. We have to go back to the Egyptian period (2000 BC) to identify the tears, not in a human, but in a cow again, which apparently cries because they are going to sacrifice their bull in a relief of a sarcophagus of Deir el- Bahari Are human feelings attributed or is it that these feelings are not exclusively human? Are historical cultures more pessimistic than prehistoric ones? About four million years ago, the Mesopotamian civilization left us a magnificent example of the existential pessimism that has developed in our culture in a way surreptitious as a principle of unquestionable reality. Since then, pessimism has slowly imposed itself, making us believe that the human species is sinful by nature. That’s what researchers think of the human imagination as Beltrán [9]. And it is that the real, in our world, is the serious thing, that is, the drama of life. In other words, more Freudian, our principle of reality is occupied by the drama of the serious. This is an automatism that operates mechanically in our culture without hardly questioning.

However, is optimism defining us as a species? The instinct of reproduction is the means by which the organic announces the joy of existence. Therefore, what we call optimism is the force that animates existence and drives reproduction. We can appreciate the laughter in the animals, the song of the birds, the tail of the dog, the purring of the cats. Is the joy of existence the engine of evolution? From this point of view, the upright position of the sapiens animals, allows the face to be seen, which is fundamental in the non-verbal communication of the smile. And besides, the bipedal position exposes the sexual organs, “shames” that all human cultures cover in some way. Shame is one of the fundamental axes of laughter. This feeling has played an important role in the development of human humor, in, for example, the phallic exhibitions.   

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