Evidence and Phylogenetic Inference

-keeping the Popperian and Bayesian spirits alive

Most of problems in phylogeny are not solved by simple hypotheses (i.e., Arthropoda and Onycophora as sister clades), they are rather assessed by composed hypotheses that should not be addressed as it was just one (let's make a reliable phylogenetic reconstruction of the group you are most interested in). Therefore, sorting this out is a matter of two components: Popperian vision of confirmation and Bayesian analysis. It is possible that the aforesaid will always sound odd, since Popper always expressed his disagreement about the Bayesian test. Nonetheless, the purpose of this article is to give some attention on the points that actually join these two approaches.

Evidence and Popper

Searching for evidence is an everyday task for the quizzical spirits who make science as what it really is: a constant pursuit of the truth about the processes that underlie the phenomena of life. However, there is no absolute certain of anything, and what we call truth would rather require a level of belief. This level of belief comes with the evidence of the hypothesis, which is what we think is the possible explanation of a particular event, or a statement that can not be defeated by evidence per se.

Sober (2009), states that the evidence we have do not render our theories true, but if we follow an argument of deductive logic, “the conclusion must be true if the premises are”. Hence, if the premises are true, there is nothing wrong in believing the conclusion. This concept is the antithesis of what Popper has proposed in the past about the power of a hypothesis, which relies on how improbable it is. The aforementioned, because in contrast to a scientist who accepts the most highly probable hypotheses, scientists according to the philosophy of Popper (1963) seek for explanations that are not subjected to a limited number of observations.

At this point, it is very necessary to bring up some of the important concepts of Sir. Karl Popper, which have contributed to this crucial step of the scientific method. First, falsifiability is a necessary criterion for scientific ideas, for it is the logical possibility that a statement could be false because of a particular observation or an experiment (Helfenbein and DeSalle R. 2005). Thereupon, a common feature of every scientific hypothesis and theory is to be “falsiable”, which not necessarily implies that they are false. If there is no degree of falsifiability in the theory or hypothesis you are formulating, it could possibly mean that it behaves as a universal law or you are facing an artifact, which is undesirable in scientific framework. Up to this point, I totally agree with Popper's statement, because one good conclusion is derived from a good hypothesis that could be contrasted with the evidence and other researches could arrive at the very same conclusion using different experiments.

The only issue I take on Popper's proposal is because of the formulated concept of corroboration, which implies that hypotheses should stand up to the most severe tests. In the cases where hypotheses were not falsified under these tests, we call the concept of degree of corroboration, which is the “appraisal of the worth of the hypothesis” (Helfenbein and DeSalle R. 2005). So, if a hypothesis has not been falsified, it has been corroborated.

To put this into context, I am enormously fond on molluscs, whose morphological and behavioral traits have defined them as a monophyletic group. Among several features, the veliger larvae constitutes the synapomorphy of this clade. Therefore the veliger larvae is the evidence of this hypothesis of monophyly. Through several analyses, we could be positive sure that it is a powerful conclusion that suits the given evidence just fine. Thus, the Popperian concept of a good hypothesis could really be in trouble when testing, because there is not such unlimited number of available observations.

Bayes and Popper

“There is not even anything irrational in relying for practical purposes upon well-tested theories, for no more rational course of action is open to us. (Popper, 1963)

First, I agree with the point of Bernardo (1999): "we are able to embed a Popperian take on the goal and methods of science into a genuine Bayesian model of hypothesis testing(...)". This idea is supported mainly because Popper’s judgement that an idea must be falsifiable could interpreted as a manifestation of the Bayesian conservation-of-probability rule (Yudowsky, 2014).

The perks of Bayesian analysis

Then, why is it preferable to use a Bayesian framework in phylogeny? Because it is flexible and allows you to use an evolution model. The prior probability that is calculated is nothing else but the probability of the model when you have not take a look at the data. Thus, the posterior probability is the probability of your model (hypothesis) given the data. These two components are explicit, which allows further assessment. Moreover, the Bayesian approach also takes account on the likelihood, which is included on its theorem. The likelihood is the probability of the data given the model, which gives us useful information, but not all the information we need.

On the other hand, due to the flexibility of the method, it allows using complex models and large sets of data. This property always provides the possibility of enlarge or enhance the method (Heaps et al., 2014). And finally, you can get a clearer grasp on the solution of your problem with the posterior probability.

Thence, the tests that are based on the calculus of probabilities, following an evolution model seem to be a powerful way to evaluate a hypothesis and determine how close we are to get a suitable likely answer. In fact, contemporary science tends to accept robust hypotheses that have been supported, rather than hypotheses of what can not possibly be refuted for its highly degree of improbability. Therefore, using a Bayesian approach would give very informative results and could allow us to perform further analyses. As a matter of being judgemental about an statement, Popper's philosophy is strong and it is an important startpoint for the scientific thinking: hypotheses must stand for severe tests. Nonetheless, in any case, choosing the approach to work with is subjected to personal criteria and to the aims of the study.

References

Bernardo J M. (1999) “Nested Hypothesis Testing: The Bayesian Reference Criterion”, in J. Bernardo et al. (eds.): Bayesian Statistics 6: Proceedings of the Sixth Valencia Meeting, 101–130 (with discussion), Oxford University Press, Oxford.

Helfenbein KG, DeSalle R. (2005) Falsifications and corroborations: Karl Popper’s influence on systematics. Molecular phylogenetics and evolution, 35(1), 271-280.

Heaps SE, Nye TM, Boys RJ, Williams TA, Embley TM. (2014) Bayesian modelling of compositional heterogeneity in molecular phylogenetics. Stat Appl Genet Mol Biol. 13, 589-609.

Popper KR. (1963) Conjectures and Refutations: The Growth of Scientific Knowledge. New York: Harper.

Sober E. (2008) Evidence and Evolution: The Logic Behind the Science. Cambridge University Press.

Addittional references

How Do Hypothesis Tests Provide Scientific Evidence? Reconciling Karl Popper and Thomas Bayes Departmental Seminar, Philosophy Department of Uppsala University, Uppsala.

Yudkowsky E. An Intuitive Explanation of Bayes'Theorem.http://yudkowsky.net/rational/bayes

Bayesianism and Popperian spirit... sounds like contradiction.

Hypotheses and theories are part of the science in a wide sense; both are the basis for its development and these are the essence to understand the Popperian ideas.

According to different dictionaries science is

“A systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe”

It is important to know the meaning of this word to understand the subsequent discussion, but also the meaning of the words theory and hypothesis has to be clear in our minds. So, a theory

“Is a contemplative and rational type of abstract or generalizing thinking or the results of such thinking.”

“In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with scientific method.”

And hypothesis is,

            “A proposed explanation for a phenomenon… and one can test it.”

I agree with some ideas of Sr. Karl R. Popper, which can be applied in the scientific life. However, I think that is not necessary to have them as the only thinking idea, because, even when there is a different stream of thought considered opposite to Popper ideas, it can works well with both in the evaluation of hypotheses and the evidence.

First, the principal idea that many people know about Popper is ‘Falsifiability’ or refutability, the logical possibility that a statement could be false by a particular observation or an experiment, but something “falsifiable” does not mean it is false. This idea is a little easy to understand because any statement that is formulated could be tested and will be falsifiable. It occurs in the way of singular and universal statements. So, if you have formulated a theory or a hypothesis is necessary that both have a degree of falsifiability, otherwise, you are in front of something totally true or an artifact.

Based on that, Popper concluded that a hypothesis, proposition, or theory is "scientific" if it is, among other things, falsifiable. That is, falsifiability is a necessary criterion for scientific ideas, but is not sufficient. Things that cannot be tested are strange to understand and would need to include a term as is ‘faith’. In addition, which contributions to science raise whether someone would answer a problem whose solution is already known or propose a theory adorned with hypothesis that prevent their falsifiability (ad hoc hypothesis).

Popper's view is not equivalent with confirmation and does not guarantee that the theory is true or even partially true. I think that if something does not falsify a statement, you should not conclude that is true, maybe it was the wrong way to apply the falsifiability, but neither is evidence of a statement confirmed.

People used to practice inductive thinking, arriving to general ideas from the particular ones. This class of thinking is appropriated to educate the scientific mind of children or people who wants to stay in science because, you can generate a global idea from many singular statement and this capacity of thinking is recognize in many Scientifics. However, it has a problem and Popper proposed falsification as a solution to the induction. The issue is that although a singular existential statement cannot be used to affirm a universal statement, it can be used to show that one is false. It is known like modus tollens, a rule of inference.

The famous example of swans is bringing here,

The singular observation of a ‘white swan’ cannot be used to affirm the universal statement ‘all swans are white’.

    The singular observation of a black swan show that 'all swans are white' is false.

Karl Popper's philosophy of science uses modus tollens as the central method of disconfirming, or falsifying, scientific hypotheses, is an useful tool that assist in discerning what hypothesis are really remarkable in science.

In addition, thanks to the inverse relationship between falsifiability and probability, proposed by Popper, is necessary formulated improbable theories in science; it has more sense than search for those in which there is some degree of confirmation.

It is relevant to cite Helfenbein & DeSalle (2005) who says, “The popperian spirit or critical attitude toward hypotheses is fundamental to all science”.

But as I said before there is another way of thinking and in many cases contradict the Popperian ideas, it is because Bayesianism assigns ‘degrees of belief’ that is like confirmation. Bayesian inference is an evidence-relationship, or confirmationist approach, and Popper’s corroboration is a non-bayesian test to the evaluation of hypotheses (Mayo, 1996). Also, Bayesianism allows informative priors and the prior knowledge or results of a previous model can be used to inform the current model.

"The Bayesian approach delivers the answer to the right question in the sense that Bayesian inference provides answers conditional on the observed data and not based on the distribution of estimators or test statistics over imaginary samples not observed" (Rossi et al., 2005). It is remarkable and one of the most interesting ideas of bayesianism, the way of have priors and the use of likelihood inside the formula is a significant thing, moreover, it can generate degree of beliefs and it is a decision theoretic foundation (Bernardo & Smith, 2000; Roberts, 2007).

The purpose of most of statistical inference is to facilitate decision-making (Roberts, 2007). The optimal decision is the Bayesian decision.         

The likelihood principle, by itself, is not sufficient to build a method of inference but should be regarded as a minimum requirement of any viable form of inference. (Rossi et al., 2005).

So, Bayesianism is a complete method of inference with prior probabilities, it integrates the likelihood principle and with it, you can obtain a result or posterior probabilities with a degree of belief… then you can take an optimal decision about your data and hypothesis.

In conclusion, I think that the ideas of Popper are not wrong and are useful in some aspects of sciences but the Bayesianism, even when is contradictory with Popper ideas is a relevant method of inference and I can say that is the best method to phylogenetic analysis at the moment.
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Bernardo J, Smith A (2000). Bayesian Theory. John Wiley & Sons, West Sussex, England.
 

Helfenbein, K. G., & DeSalle, R. (2005). Falsifications and corroborations: Karl Popper’s influence on systematics. Molecular phylogenetics and evolution, 35(1), 271-280.

Mayo, D. G. (1996). Error and the growth of experimental knowledge. University of Chicago Press.

Robert, C. (2007). The Bayesian Choice. 2nd edition. Springer, Paris, France.

Rossi, P, Allenby, G, McCulloch, R. (2005). Bayesian Statistics and Marketing. John Wiley & Sons, West Sussex, England.

Choose one...

Always in all life moment we place in the situation of choose between two or more options, maybe for social reasons in every area always tell us that we must take a decision, what religion we must believe, whether we are pro-yankees or not, what political party we belong, even in Biology we must choose. whether we are Botanist, Primatologist, Ornithologist, etc. In Compared Biologist class my professor asked me choose one philosophical current which I must use to make and approach my questions. I can choose between three different philosophical currents: Bayesianism, Likelihood Ism, Frequentist and Popperian  Falsificationis

We're going to talk each of the three streams. Although the Bayesianism has a origin very old, until only a few years had a major resurgence in science. This stream applies the probability that some events occur given certain observations (priors), which are changing or being updated. The probability is denoted Pr(H|O), and is the probability of the hypothesis given the data, observations or evidence (Sober, 2008; Kruschke, 2011). On the other hand is the Likelihoodism, which lacks of priors and the logic is contrary to the Bayesianism, looks for the probability how well the hypothesis fits to the data Pr(O|H) (Sobber, 2008; Royal, 1998). Both branches are extremely powerful to make inferences and contrasting hypotheses, the only difference is the concept used for making comparisons (Sober 2008, Kruschke, 2011). The likelihoodism uses the concept of favoring to show that the evidence says regarding the comparison of two hypotheses, while the Bayesianism adopts the concept of confirmation to show that the evidence says regarding a hypotheses and it's negation (Sober, 2008. Pag 34). Finally, the frecuentism, which has dominated the last century, is based on the probability that an evidence or event occurs depending on a set of experiments and N (Johansson, 2011), that gives a ratio and the value P is calculated and compared with a null model or null hypotheses(Sober, 2008; Johansson, 2011; Wagenmakers, 2007). The main difference (i think) is it's philosophical perspective on the comparasion of hypotheses and the use of priors in Bayesianism (Sober, 2008), in other topics are very similar. As a first approach to this affirmation I put the following exercise: Suppose that we have our hypothesis (H) and data (T), now when we do the Bayesian analysis we seek the probability of H given T(Pr(T|H)), in contrast to Likelihood we want to look at how well H fits to our data T(Pr(H|T)), when we apply the Bayes theorem to our example we have: p(T|H)p(H) = p(H|T)p(T). Then = Pr(T|H)=[p(H|T)p(T)]/p(H). From a value of Likelihood we can get the posterior values, the example is somewhat crude and simplist but implies the idea. I don't pretend to fill this with formulas and derivations that even I can't explain but Branden Fitelson from page 7 of his article "Likelihoodism, Bayesianism, and Realtional Confirmation" shows us some examples of how some Bayesian measures are more Likelihoodians than Bayesians and vice versa, if someone wants deepen in the topic.

One of the main problems in Bayesianism and Frecuentism is little objectivity when the data are managed (Ayacaguer, 2000). On one side are the priors of the Bayesian analysis, and have more influence in the analysis and their value can be altered to benefit any particular hypothesis, this is one reason why many people argue the unreliability of this method when is used in daily life, in government agencies because anyone can manipulate priors to the own benefit. It has used 'flat-priors' as a solution to this problem, which causes that the entire analysis falls on the Likelihood, but the Frequentism is not far behind, because you can manipulate the P values or the values of positive or negative false (the famous alpha and beta), to favor some result in special, just as the criticism is the use of a null hypothesis (Ayacaguer, 2000; Johansson, 2011). Similarly the Frequentism present the N problem, because the P values are influenced by the sample that was used, so we can know beforehand what would be the result if we use a small N or a big N and the P value also can be influenced subjectively by the amount of N that is used (Ayacaguer, 2000; Wagenmakers, 2007; Johansson, 2011). So if it's subjectivity we have a winner ¡Likelihoodismo!. But we don't get excited because Likelihoodism also has crtitics and one of these is it's restriction to some cases (Sober, 2005).

According everything written above, it appears that Likelihoodism is the best stream and therefore I'll choose it, but no. It can sound crazy but for me and after of all this timeexploring this trend I can conclude that one can't choose any stream in special, but I have to highlight that all have good and bad things and for that reason I consider they complement each other and all can be used in Bayesin analysis (obviously without declare Bayesianista). To understand this idea we must have the main components in mind of Bayesian analysis: the priors and the Likelihood. Already I denote the relationship between both Likelihood and Bayes using the theorem. On the other hand, in priors is where It would enter the Frequentism, we can consider the results from a Frequentist analys as priors in the Bayesin analysis, let me give you an example: Suppose you arrive to a new city and want to know if that month is rainy or not, throughout the month you take notes on what days is raining and is not raining, assuming that it rains 25 of 30 days. From that relationship and calculating the P value you will know that this month is rainy or not, but What could you say from this assertion on the following months?, really nothing, but from these observations you could infer how likely is that the next month it will rainy, because throughout that month we have noticed that in general before the rain come the sky is clouded, so if the next day we see the sky is clouded (O), we know that the probability of rain is going to be high (H), all thanks to the prior value we obtained from our frequentist observations. I would like to give an example that occurred to me while I played Xbox to better explain my idea. Suppose we are going to fight with the  Final Boss, at the begining, we dont know anything, how are attacks, his  moves and we have to spend one or more lives to defeat it, is in that moment where our Bayesian, Likelihoodism and Frequentism analysis arises !!. At first we do not know how to approach the enemy (flat priors) and defeat it (H) with our initial strategy (T) is very unlikely (low likeliihod, p (T | H)), which ultimately leaves a unlikely to pass the game (posterioris bayes, Pr (H | T)). As we move forward in the fight we noticed that the enemy has a particular frequency for certain attacks, then we will know what is the probability of making certain attack, these probabilities increase as you fight more and make more observations on the movements of the enemy, this, I believe, is a well frequentist analysis (we have an accumulation of knowledge and increase our priors), once we know these make a change in our strategy (T) and the probability of defeat given that our change strategy (Pr (T | H)), so that eventually the probability of passing the game increases (Pr (H | T)).

So, you will ask, ¿ where is the Popperian Falsificationism ?, well, I think that is the less critical among the 4 currents. Basically Popper says us: In science we must reject some theories and hypotheses to corroborate others ( but, It doesn't mean that these are true), something like Modus Tollens Tollens (If A is true, no mean that B are true too). So in this way the three currents use  Popper's logic: The likelihood 'favoring' one hypotheses over another, Bayesianism 'confirms' one hypotheses respect its own negation, and Frequentism compare one hypotheses against the null hypotheses, but we never corroborate that are true hypotheses

 I think choose between any of these three currents, is like choose just one phylogenetic search method is better, all three have good and bad things and often what really influences is the data, not the method. I consider more appropriate, for example what Morrone and Crisci do with the two methods of historical biogeography (Panbiogeography and Historical Cladistic), they show how each method is complementary each other, and that are necessary steps for a good biogeographic analysis (Morrone & Crisci, 1995; Morrone 2001). This (I think), allows us to look the problem at multiple ways and allows find multiples and well solutions, avoiding bias. Always tell me that extremes are not good, so, why we don't avoid the extremes, and find a intersection between them ? and take the better of each one, just imagine how world will change if religious zealots get find a middle point. At the end these are only methods and it seems to me more crucial and critical the objectivity with which the researcher will  analyze the results.

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References.

• Branden Fitelson. Likelihoodism, Bayesianism, and Relational Confirmation. Syntheses (2007).
• Tobias Johansson. Hail the imposible: p-values, evidence and likelihood. Scandinavian Journal of Psychology (2011).
• L.C. Silva Ayacaguer & A. Muñoz Villegas. Debate sobre métodos frecuentistas vs bayesianos. Gac Sanit (2000).
• Eric-Jan Wagernmakers. A practical solution to the pervaise problems of p values. Psychonomic Bulletin & Review (2007).
• Silvio Pinto. El Bayesianismo y la Justificación de la inducción. Principia (2002).
• Royall, R.  Statistical Evidence: A Likelihood Paradigm, Boca Raton, Fla.:Chapman and Hall.(1997).
• Elliot Sober.  Evidence adn Evolution: The Logic Behind The Science. Cambridge University Press, United States of America. (2008).
•  Kruschke, J.K.. Bayesian data analysis: A tutorial with R and BUGS. Amsterdam: Elsevier. 2011.
• Juan J. Morrone. Homology, biogeography and areas of endemism. Diversity and Distribution (2001).
• Sober, E.: 2005, ‘Is Drift a Serious Alternative to Natural Selection as an Explanation of Complex Adaptive Traits?’. In: A. O’Hear (ed.): Philosophy, Biology and Life. Cambridge: Cambridge University Press.