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.