Introduction

The description of how the scientific knowledge is justified, organized and changes with time is one of the intentions of the philosophy. One of its central questions is the issue of the empirical evidence in favor of the theoretical hypotheses.

There has been a long-standing debate among philosophers on whether science is based on critical falsification as suggested by Karl Popper, or on inductive generalizations. Although both models have had a significant impact on the field of science, none of them offers a completely satisfactory explanation of the way modern scientific reasoning works in practice.

Indeed, inductivism and falsificationism have severe flaws. The problem of induction is that there is no a circular way to justify inductive inferences (Hume's problem) and falsificationism cannot describe the empirical dynamics of confirmation in modern science, which does not usually rely on a single decisive experiment.

It is against this context that Bayesianism has come out as a strong alternative. It offers a quantitative and flexible model to allow scientists to rationally modify their confidence in hypotheses as new data is obtained by interpreting reasoning as a process of probabilistic belief updating on the basis of the Bayes theorem.

The following essay discusses Bayesianism as a scientific confirmation theory, with a case study of the discovery of the Higgs boson in CERN in 2012. It states that Bayesianism is an effective way to model the rational dynamics of belief change when new evidence is presented, and also admits that it has its own weaknesses when it comes to theory change and paradigm shifts.

The Problem of Induction

According to the problem of induction that was argued by David Hume, past observation cannot logically ensure that the future will follow the same pattern. Inductive reasoning makes use of a finite number of observations to formulate a general statement.

Hume said that reasoning alone cannot be used to justify inductive, without falling into circularity: we would have to have assumed the homogeneity of nature, that the future will resemble the past, to justify induction.

Efforts to improve induction through a probabilistic reformulation, asserts that most general laws are probably true if there is a sufficient amount of evidences. But this fails mathematically too, because the probability with finite evidence that a statement of general laws is true is near to zero.

It is out of this crisis that scientists followed an alternative model that can explain rational belief revision by not involving certainty, but rather embraces uncertainty as a fundamental part of scientific method.

The Problem of Falsificationism

Falsificationism by Popper replaced the concept of inductive confirmation with the concept of conjecture and refutation: scientific theories cannot be proven true, but falsified by counter-evidence. A scientific theory should be falsifiable which means that it has possible observations that can make it wrong.

However, this model has its problems, too. First, there are auxiliary assumptions, background conditions, measurement models and instrumentation hypotheses, which are always involved in testing a theory as demonstrated by the Duhem-Quine thesis and the research programmes of Imre Lakatos. When an experiment goes against a prediction, it is not always clear which assumption is to fault.

For instance, science does not give up about a theory once it is proven to be wrong. Scientists, as Thomas Kuhn and Lakatos suggested, usually work within paradigms or research programmes that allow to have discrepancies until a superior alternative is found.

Falsificationism therefore does not reflect the incremental and probabilistic aspect of confirmation in actual science. A more realistic description is provided by a probabilistic model such as Bayesianism, which does not falsify a hypothesis but rather raises or lowers the level of belief in the hypothesis.

Bayesianism

Bayesian reasoning is described mathematically by Bayes' theorem, which was formulated by Thomas Bayes:

$$P(H|E) = \frac{P(E|H),P(H)}{P(E)}$$

where:

• $P(H)$: prior probability of hypothesis $H$;
• $P(E|H)$: likelihood of observing evidence $E$ if $H$ is true;
• $P(E)$: overall probability of observing $E$;
• $P(H|E)$: posterior probability, the updated degree of belief in $H$ after observing $E$.

This model implicates that in case of new evidence there is no a certain conclusion, instead, the weight of the probabilities of hypotheses is changed. When $P(E|H)$ is high, then we will have a high confirmation of H.

A logic of scientific justification is then demonstrated by Bayesianism to a problem of updating of beliefs. Scientific reasoning turns into a learning process without end and every bit of evidence contributes to a progressive modification of credence.

Bayesianism has two main interpretations:

• Objective Bayesianism: we have to give probabilities in the basis of objective principles, like the Principle of Indifference. We give them equal chances when we do not know that any particular hypothesis is better than another;
• Subjective Bayesianism: probabilities are degrees of belief which in fact are held by individual scientists. Every scientist initially makes his own priors in terms of the background knowledge and rationalizes them by means of the Bayes theorem.

However, the two perspectives have one thing in common; with evidences shared to the scientific community, the beliefs are drawn to a fixed posterior distribution that we refer to as scientific consensus.

Bayesianism considers the background knowledge. In fact, the probability of the likelihood $P(E|H)$, should not be simply on H alone but on a set of additional hypotheses on instruments, theoretical models and measurement accuracy. This is the truth of modern day science where there is no hypothesis that is ever tested on its own.

The Higgs Boson

The Higgs boson plays a central role in the Standard Model of particle physics: it shows how elementary particles obtain a mass through the process of spontaneous symmetry breaking in the electroweak sector. Peter Higgs, Francois Englert and others proposed it first in 1964, independently.

The theory states that the universe is filled with a scalar field, referred to as Higgs field, which interacts with particles, including W and Z bosons, to give them a mass. The quantum excitation of this field is a new particle, called Higgs boson.

The Higgs boson was the long missing puzzle of the Standard Model: this was true in theory, but the direct observation of it was technically absent. With the help of the data of the previous CERN accelerators, Large Electron-Positron Collider (LEP) and Tevatron, experimental physicists successively narrowed the range of possible masses of the Higgs, with evidence suggesting that should it exist, it would be lighter than 200 GeV.

Therefore, the pre-2012 probability of the Higgs hypothesis being true known as $P(H)$ was large but not definite: even though the Model strongly favored it, the absence of its observation continued to leave the possibilities of having new alternatives.

At one point, CERN constructed the Large Hadron Collider (LHC): a ring of superconducting magnets with a circumference of 27 km with the ability to accelerate protons to 13 TeV and collide them. The detectors ATLAS and CMS were two massive detectors in charge of searching the Higgs boson independently.

Case Study: The Discovery of the Higgs Boson

An example of Bayesian confirmation in modern physics is the discovery of the Higgs boson in 2012 at CERN.

The LHC experiments (ATLAS and CMS) examined events involving billions of proton-proton collisions and reassembled the resultant products to identify excesses that would be associated with the Higgs signal. The Higgs would manifest itself as a narrow resonance at a certain value of the mass which is essentially a bump in the distribution of energies.

Suppose that the hypothesis is as follows; $H$: "The Higgs boson exists and has a mass of less than 200 GeV" and $E$: "an excess number of events are seen". Then the likelihood $P(E|H)$ is the probability of such an excess, had the particle really existed, and the complementary likelihood $P(E|\neg H)$ is the probability of the background fluctuating in the same way to provide precisely the same excess.

These experiments demonstrated that $P(E|\neg H)$ was quite low, of the order of one in 3.5 million, which relates to the $5\sigma$ level of significance. That is, assuming that the null hypotheses were correct, that no Higgs boson existed, then such an excess would occur almost never by chance.

In terms of Bayesian, that is, the probability of $P(E|\neg H)$ was very small, while $P(E|H)$ was immense. Thus, posterior probability that the Higgs exists, $P(H|E)$ being the probability that the Higgs exists given data seen, was also close to one. The Higgs hypothesis made such strong predictions of experimental data, and under the condition of its negation turned them into such improbabilities, that rational belief of the existence of the particle has become almost but not absolutely certain.

This was discovered in bits: initial data of significance of $2\sigma$ or $3\sigma$ statistic slightly raised the posterior probability, but next data gradually confirmed the belief. This incremental updating of confidence is what fits the model of rational learning of Bayesian model.

Moreover, the $5\sigma$ rule of the particle physics community can be interpreted as a decision level that is associated with a sufficiently large posterior probability that is acceptable. Although physicists usually develop their discoveries using frequentist tests, such as using the $p$-values and $\sigma$-values, their structure of inference remains bayesian.

Philosophically, this demonstrates how implicitly even nominally frequentist methodologies justify themselves by Bayesian reasoning. Physicists believe, not only the numerical value of the answer but also its theoretical reasonableness and consistency with previous models and the danger of systematic bias of Bayesian inference.

Also, the ATLAS and CMS results are combined, which proves the Bayesian model averaging practice. The independent likelihood functions that were provided by each experiment; the combination of them allowed the scientists to create a global posterior that represented the totality of available evidence.

This led to the announcement of the discovery on July 4, 2012, of the observation of a new particle that was in agreement with the Higgs boson, the combined significance of which was over $5\sigma$.

What took place here is that some high prior prediction was revised to a almost maximal posterior conviction, a paradigmatic illustration of scientific confirmation.

The Scientific Confirmation

The Higgs case demonstrates several philosophical points of view on the Bayesian concept of science.

To begin with, confirmation is not a binary concept. Scientific belief is a development, by means of the posterior revisions and by the increment of evidence. The monitored boost in confidence of the Higgs is a perfect example of the Bayesian learning curve.

The second one is related to the strong interdependence of the theory and evidence. The data analysis of the LHC relied on complex simulations, with the Standard Model, detector calibration and background models, all aspects that Bayesianism formally formalizes, using conditional probabilities.

Third, Bayesianism clarifies the position of discovery. Physicists never established the Higgs in some absolute sense; they gained a posterior probability that was so likely to be certain that it was irrational to doubt this further. The discovery then not only is not logical demonstration, but is a great probabilistic confirmation.

But Bayesianism is too has its weaknesses. It provides the assumption of a given hypothesis space and unchanging background conditions. In practice, when a scientific revolution appears, as the Newtonian mechanics to the quantum one, the hypothesis-evidence model has to evolve. Bayesian updating has difficulty in responding to these paradigm shifts, as between mutually incompatible conceptual systems there are priors and likelihoods whose meaning has been lost.

In this regard, Bayesianism has a very good explanation of science, but it needs the addition of historical and sociological explanations to comprehend revolutionary science.

However, the Bayesian model is invaluable: it represents the most accurate formal model of confirmation, coherence and rationality existing in the modern science.

Conclusion

Bayesianism constitutes one of the most consistent, realistic philosophical frameworks of scientific reasoning. It regards belief constantly revised with evidence as grades of knowledge and in this way reconciles the empirical experiments of science with their theoretical model.

The Higgs boson was actually found demonstrating this process of gradual learning, where the experimental data gradually changed the theoretical prediction of its existence into a near-certainty, with each new dataset greatly narrowing the posterior belief of the hypothesis until it was discovered to be nearly certain plausible.

However, Bayesianism cannot solve all the difficulties alone: it fails concerning theory change. But Bayesianism has the success that inductivism and falsificationism fail to achieve: it describes the rational processes of belief in the uncertainty of empirical experience.

Science is never successful in achieving any absolute truth, it is successful in dealing with uncertainty, in handling the degrees of our beliefs in the light of the continually increasing data of the world.