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A View on Statistical Schools

by Jordi Vallverdu


Abstract

There are two main opposing schools of statistical reasoning, 'Frequentist' and 'Bayesian' approaches. Until recent days, the Frequentist or classical approach has dominated the scientific research, but Bayesianism has reappeared with a strong impulse that is starting to change the situation. Recently the controversy about the primacy of either of the two approaches seems to be unfinished at a philosophical level, but scientific practice is giving an increasingly important position to the Bayesian approach. This paper focuses on the pragmatic point of view of scientists' day-to-day practices, in which Bayesian methodology is very useful. Several facts and operational values are described as the core-set for understanding the change.

I. What does 'Bayesian' or 'Frequentist' mean?

The Bayesian and Frequentist theories are the leading ways to understand the various uses of statistics:

The Bayesian perspective on probabilities says that a probability is a measure of a person's degree of belief in an event, given the information available (Dale, 2003). Thus, probabilities refer to a state of knowledge held by an individual, rather than to the properties of a sequence of events. The use of subjective probability in calculations of the expected value of actions is called subjective expected utility. There has been a renewal of interest for Bayesianism since 1954, when L.J. Savage wrote Foundations of Statistics. There are a large number of types of Bayesians (ironically, Good I.J. (1971) talked about '46656 kinds of Bayesians', Amer. Statist., 25: 62-63.), depending on their attitude towards subjectivity in postulating priors. Recent Bayesian books are: Earman (1992), Howson & Urbach (1989), Bernardo & Smith (1996).

Frequentists understand probability as a long-run frequency of a 'repeatable' event, and have developed a notion of confidence intervals. Probability would be a measurable frequency of events determined from repeated experiments. Reichenbach, Giere or Mayo have defended that approach from a philosophical point of view, referred to by Mayo (1997) as the 'error statistical view' (as opposed to the Bayesian or 'evidential-relation view'). As Giere (1988: 189) asks: 'Are Scientists Bayesian Agents?... The overwhelming conclusion is that humans are not Bayesian agents' (where Knill et al 1996 defend the opposite view).

How to choose?

One of the recurrent arguments against/ in favor of one of the two positions (Frequentist or Bayesian) consists in saying that, 'A true scientist is always/ never a Frequentist/ Bayesian' (you can choose between the two possibilities, Cousins, 1995, Efron 1986). It seems to be an epistemological law about statistical practices: 'A true scientist never belongs to the opposite statistical school.' What I can say is that this is a usual metatheoretical thought about Frequentist/ Bayesian approaches and their ontological fitting with reality, which is not useful for clarifying the closure of scientific controversies, because they depend on another kind of values. We cannot know what happens exactly in scientists' minds, but we can know how they act and, therefore, infer from their actions how they think.

Obviously, we suppose and accept cognitive activity for scientists. The question is: at what cost can we introduce cognitive arguments inside the statistical techniques embedded in scientific practices? And when we talk about 'cognition' me must include not only rational aspects of cognition, but also irrational ones (Thagard, 1992). In addition, Causality is a complex and ancient problem: the Philosopher of Science Paul Thagard (1999) has offered a very powerful conceptual framework to understand scientific causal explanations (specially, of diseases) with his idea of 'causal network instantiation' (p. 114). According to him: 'causal networks are not simple schemas that are used to provide single causes for effects, but they instead describe complex mechanisms of multiple interacting factors', p. 115-116. But Thagard is no Bayesian: he pursues another line of explanation which he considers better suited to psychological reasoning: explanatory coherence. (Ibid. p. 65-66).

But we must go to the core set of the question: what are the new values, beyond scientific values from Merton, implied in the choice between the two schools?

2.1. Formational values

We will use the words of Bland & Altman (1998): 1160, to illustrate these kinds of values:

'Most statisticians have become Bayesians or Frequentists as a result of their choice of university. They did not know that Bayesians and Frequentists existed until it was too late and the choice had been made.' There have been subsequent conversions. Some who were taught the Bayesian way discovered that when they had huge quantities of medical data to analyze the Frequentist approach was much quicker and more practical, although they remain Bayesian at heart. The same idea is repeated in a different way by a High Energy physicist, D'Agostini (1998:1): 'The intuitive reasoning of physicists in conditions of uncertainty is closer to the Bayesian approach than to the Frequentist ideas taught at university and which are considered the reference framework for handling statistical problems.' — One thing is the theory taught at university, and another one is the true scientific practice.

Some Frequentists have had Damascus road conversions to the Bayesian view (like Harrell, Frank E. Jr, 2000 Practical Bayesian data Analysis from a Former Frequentist, downloadable PDF document at http://hesweb1.ed.virginia.edu/biostat/teaching/bayes.short.course.pdf). Many practicing statisticians, however, are fairly ignorant of the methods used by the rival camp and too busy to have time to find out. As the epidemiologist Berger says (2003): 'Practising epidemiologists are given little guidance in choosing between these approaches apart from the ideological adherence of mentors, colleagues and editors.' Giles (2002), talking about members of the Intergovernmental Panel on Climatic Change (IPCC), says that those researchers were suspicious of Bayesian statistics because 'these attitudes also stem from the authors' backgrounds', p. 477.

So, the arguments go beyond the ethereal philosophical arena and closer to practical ones. Better opportunities to find a good job is an important argument, and the value of a Bayesian academic training is now accepted: 'Where once graduate students doing Bayesian dissertations were advised to try not to look too Bayesian when they went on the job market, now great numbers of graduate students try to include some Bayesian flavor in their dissertations to increase their marketability', Wilson (2003): 372.

2.2. Metaphysical values

By their writings, we can extract some information about scientist's thoughts. Knowledge is framed by feelings, emotions, facts, and even faiths. How to consider, then, classical and continuous disputes among the full range of possible positions between realists and subjectivists? (Savage, 1954; Suppes, 1970; Weatherford, 1982). All scientists believe for different reasons, that the constituents of the world have certain dispositions that can be discovered under certain investigative conditions. As expressed by Hacking (1972): 133: 'Euler at once retorted that this advice is metaphysical, not mathematical. Quite so! The choice of primitive concepts for inference is a matter of 'metaphysics'. The orthodox statistician has made one metaphysical choice and the Bayesian another.'

2.3. Simplicity and cheapness: computerizing statistical thought

One of the arguments against Bayesian methods says that the Bayesian approach is too complex to apply in day-to-day research. And simplicity is one of the best values for scientific activity. But during the past few years a large amount of Bayesian software programs have appeared which have changed the situation: now it is easy, fast and cheap to implement the Bayesian approach in experimental practices (Escoto 2003; Gigerenzer 1998). Programs like BACC, [B/ D], BOA, BUGS (Bayesian inference using Gibbs sampling, and WinBUGS), MINITAB, EPIDAT, FIRST BAYES, HYDRA, STATA, SAS, S-Plus and others, some of them available as freeware, make possible an efficient use of Bayesian methods in several scientific fields. Their flexibility helps to incorporate multiple sources of data and of uncertainty within a single coherent composite model.

Until the 1980's, the potential for the application of Bayesian methods was limited by the technical demands placed on the investigator. Over the past fifteen years these limitations have been substantially reduced by innovations in scientific computing (faster computer processors, according to NAS, 1991) and drastic drops in the cost of computing (Editorial BMJ, 1996). These changes and an increase in the number of statisticians trained in Bayesian methodology are encouraging the new status of Bayesianism (Tan, 2001).

Medicine is, perhaps, the scientific field in which Bayesian analysis is being more intensively applied (Szolovits, 1995; Grunkemeir & Payne, 2002). Two trends, evidence-based medicine and Bayesian statistics are changing the practice of contemporary medicine. As Ashby & Smith (2000) tells us: 'Typically the analysis from such observational studies [those of epidemiology] is complex, largely because of the number of covariates. Probably for this reason, Bayesian applications in epidemiology had to wait for the recent explosion in computer power, but are now appearing in growing numbers', p. 3299 (see also Breslow, 1990 and Ashby & Hutton, 1996.

The development of Markov Chain Monte Carlo (MCMC) computation algorithms, now permit fitting models with incredible realistic complexity. When we study models for multiple comparisons, we can see that Frequentists adjust Multiple Comparison Procedures (MCP) considering intersection of multiple null hypotheses. They also advocate for a control of the familywise error-rate (FWE). So, 'Bayesians will come closer to a Frequentist per-comparison or to a FEW approach depending on the credibility they attach to the family of (null) hypotheses being tested... the Bayesian is closer to the per-comparisonist', Berry & Hochberg (1999): 216. The Bayesian approach has received a great impulse from MCMC models (Dunson, 2001; Carlin & Louis, 2000; Gelman et al 1996). MCMC procedures are also extremely flexible and constitute the primary factor responsible for the increased use and visibility of Bayesian methods in recent years.

2.4. Ethical values

We can find an appeal to ethical values as parts of arguments about both schools. Wilson (2003) affirms that Bayesian methods are a more ethical approach to clinical trials and other problems. On the contrary, Fisher (1996) affirms that 'Ethical difficulties may arise because of the differing types of belief', especially during Randomized Clinical Trials (the Phase III Trials in the FDA model).

From the history of standard literature on ethics in medical research, one can infer the great value of prior beliefs in clinical trials. And the key concept is 'uncertainty': 'Subjective opinions are typically not included in the background material in a clinical trial protocol, but as they are often a driving force behind the existence of a protocol, and as uncertainty is deemed to be ethically important, documentation will be useful. Without documentation it may be difficult to determine whether uncertainty exists... There are compelling ethical reasons that uncertainty should be present before a clinical trial is undertaken' (Chaloner & Rhame, 2001: 591 and 596). When uncertainty is introduced in the reasoning procedures, the quantification of prior beliefs and, therefore, the use of Bayesian methodologies, seems to be an operationally and ethically better decision.

2.5. Better fitting for results

Berger (2003), proposes using both models and studying case by case their possibilities: 'Based on the philosophical foundations of the approaches, Bayesian models are best suited to addressing hypotheses, conjectures, or public-policy goals, while the Frequentist approach is best suited to those epidemiological studies which can be considered 'experiments', i.e. testing constructed sets of data.' Usually, we find no such equitable position. But this is not a theoretical question but a practical one: Bayesian methods work better than Frequentist. Therefore, Bayesian methods are increasing their application range, although it does not always mean that there are more 'true Bayesians'.

As Wilson (2003) explains: 'their methodological successes [from Bayesian] have indeed impressed many within the field and without, but those who have adopted the Bayesian methods have often done so without adopting the Bayesian philosophy'. As the Editorial from British Medical Journal (1996) states, 'most people find Bayesian probability much more akin to their own thought processes... The areas in which there is most resistance to Bayesian methods are those were the Frequentist paradigm took root in the 1940s to 1960s, namely clinical trials and epidemiology. Resistance is less strong in areas where formal inference is not so important, for example during phase I and II trials, which are concerned mainly with safety and dose finding.'

Therefore, Popper or Lakatos could say: 'Bayesian methods solve problems better than Frequentist ones'. And practical success usually means the theory's success. Look to the history of science: Copernicus astronomical tables were better than those of Ptolomeus and if at first, were accepted as an instrument, in a later they were considered as a true representation of reality.

The Scientific Information and Computing Center at CIBA-GEIGY's Swiss headquarters in Basle moved towards the systematic use of Bayesian methods not so much as a result of theoretical conviction derived from philosophical debates, but rather as a pragmatic response to the often experienced inadequacy of traditional approaches to deal with the problems with which CIBA-GEIGY statisticians were routinely confronted (Racine et al, 1986). An example: clinical trials made by pharmaceutical industries are usually Bayesian (Estey & Thall, 2003) although such methods are not easily implemented (Wang et al, 2002).

Bayesian methods are ideally suited to dealing with multiples sources of uncertainty, and risk assessment must include a lot of them: one experiment can be affected by several terms like sex, age, occupation, skill of technician, number of specimens, time of sampling, genetic background, source of intake. So, according to an epidemiologist, Dunson (1991): 1225: 'Bayesian approaches to the analysis of epidemiological data represent a powerful tool for interpretation of study results and evaluation of hypotheses about exposure-disease relations. These tools allow one to consider a much broader class of conceptual and mathematical models than would have been possible using non-Bayesian approaches'.

Grunkmeier & Payne (2002: 1901), talking about surgery enumerate several advantages of Bayesian statistics applied to it: '(1) providing direct probability statements — which are what most people wrongly assume they are getting from conventional statistics; (2) formally incorporating previous information in statistical inference of a data set, a natural approach which follows everyday reasoning; and (3) flexible, adaptive research designs allowing multiple examination of accumulating study data.' The Bayesian approach is more efficient at unifying and calculating multilevel causal relationships.

2.6. Diffusion of science: guidelines

At the core of science lies information communication. By the process of writing and communicating his/ her results, a scientist is at the same time evaluated (through peer review) and judged (by his/ her colleagues). All the norms implied in the guidelines, define a trend in 'good' scientific practices. And those groups who control the communication channels can make sure that special kinds of ideas are never allowed. Therefore, design and control of communication channels is something crucial for the interest of a community.

The Frequentist approach has dominated statistics journals all through 20th Century but, recently, Bayesians are gaining more and more power. As Wilson (2003): 372, says: 'Bayesians have successfully and extensively published in JASA and other prominent journals, bringing their methods into the spotlight where they cannot be ignored'. It is not only a question of general perception but also of radical changes in the bases of the epistemic frame

The International Committee of Medical Journal Editors, wrote the Uniform Requirements for Manuscripts Submitted to Biomedical Journals, available at http://www.icmje.org, where they specified for statistical norms: 'Avoid relying solely on statistical hypothesis testing, such as the use of P values, which fail to convey important quantitative information.' Nevertheless, we must also recognize that the use of statistical methodologies in medical research is highly controversial, beyond the Bayesian-Frequentist dilemma (Altman et al, 2002). Spiegelhalter (1999) reflects that: 'Current international guidelines for statistical submissions to drug regulatory authorities state that 'the use of Bayesian and other approaches may be considered when the reasons for their use are clear and when the resulting conclusions are sufficiently robust.'

So, these new trends 'accepted' as the new axiological frame for statistical research have changed the weight of both schools: while Frequentist models are decreasing their expansion, Bayesian ones are being employed in an increasing number of situations. Basanez (2004) has explained the reasons for this gradual shift: practical, theoretical and philosophical.

3. Framing values: conclusions about theories and uses

My old Webster's Dictionary has its own definition of 'dilemma': '1. a situation requiring a choice between equally undesirable alternatives. 2. any difficult or perplexing situation or problem. 3. Logic. a form of syllogism in which the major premise is formed of two or more hypothetical propositions and the minor premise is an exhaustive disjunctive proposition, as: If A, then B; if C then D. Either A or C. Therefore, either B or D.'

It seems clear that we have not been talking about logic relationships inside statistical controversies. Therefore, the third definition is not of interest. The second one seems to be closer to the aims of this paper: the analysis of a complex problem for which there is no obvious solution. However, the first definition is the core of this paper: Does the Bayesian vs. Frequentist dilemma constitute a difficult choice 'between equally undesirable alternatives'? Are we forced to die for our rational criteria like Buridan's ass?

At a metatheoretical level, that is philosophy, the debate is still open and more and more complex. But that is not the level of analysis we have considered as crucial for the solution of the debate. We have talked about scientific practices in which are involved both statistical approaches. And when scientists work, they take decisions continuously.

We have shown a new range of values that constitute part of the statistical axiology. These are non-epistemic values, but shape the underlying framework of research epistemology. Academic training, ease of use, powerful infrastructures, cognitive fitting, ethics, metaphysical options, cheapness, and better results, are the arguments to decide in favor of either one of the two approaches. Perhaps these are not the values which theoreticians would have chosen, but are the real values which appear when we look at scientists' practices and reflections.

We don't know if the prediction made by Bruno de Finetti (1937), that it would take until the year 2020 for the Bayesian view of statistics to completely prevail will be accurate. This is another question, far from our interests and methodology. I have indicated several values that make it possible to choose between both approaches.

A clear fact is that Bayesian analysis is widely used in a variety of fields, from the pioneering field of medicine to engineering, image processing, expert systems, decision analysis, psychological diagnoses (Meehl & Rosen, 1955), criminal investigations (Sullivan & Delaney, 1982), for presenting evidence in court (Feinberg & Schervish, 1986; Matthew, 1994; Mossman, 2000), gene sequencing, financial predictions, neural networks or epidemiological studies. If we return to the classic paper of Winkler (1974) 'Why are experimental psychologists (and others) reluctant to use Bayesian inferential procedures in practice?', we read: 'This state of affairs appears to be due to a combination of factors including philosophical conviction, tradition, statistical training, lack of 'availability', computational difficulties, reporting difficulties, and perceived resistance by journal editors.' Well, all these factors (non-epistemic values) are now not against but in favor of the Bayesian approach.

Is the solution to unify as a synthesis both approaches (Berger et al, 1997), like a synthesis solution to a dualistic problem? Could a hybrid method of inference satisfy both camps? Is the 'Likelihood' approach a third alternative? (Senn, 2003). But this is, once more, a philosophical question.

Finally, we must consider the existence of a really fundamental question: how to make decisions based on evidence. And we find a basic problem: there are several decision levels with their own individual exigencies regarding what is considered as evidence. If we talk about decision making in health controversies, we should consider several levels like: decision making for patients (diagnosis), decision making for individual patients (interventions), decision making about studies (start from prior beliefs and data monitoring), decision making for pharmaceutical companies and public policy decision making (Ashby & Smith, 2000). But these multi-criteria analyses can be found in other scientific fields, such as forestry (Kangas & Kangas, 2004). And we find another set of problems present in both approaches when they are applied to controversial scientific practices, such as those of risk assessment: difficulties in establishing clear relationships, the significant sample, data interpretation, cognitive paradoxes (Simpson, Ellsberg, St. Petersbourg, Base-Rate Fallacy, Kahneman & Tversky, 1982), the idea of evidence at multiple levels (Jasanoff 1994).

Considering the previous arguments, we must admit that the dilemma, understood as a choice between equally undesirable alternatives, is a false dilemma. We have enough judgment elements to decide rationally between one of two approaches, and so do scientists from diverse fields, whose words we have reproduced here. To understand these decisions better we have enumerated a new set of values that needs to be included in a richer and sounder scientific axiology.


Bibliography

Altman, D.G. et al (2002, June 5th) How Statistical Expertise is Used in Medical Research', JAMA, 287(21): 2817 2820.

Ashby, D. & Hutton, J.L. (1996) 'Bayesian epidemiology', in Bayesian biostatistics, Berry D., Stangl D. (eds.), New York: Marcel Dekker.

Ashby, D. & Smith A.F.M. (2000) 'Evidence-based medicine as Bayesian decision- making', Statistics in Medicine, 19: 3291-3305.

Basanez, Maria Gloria et al (2004) 'Bayesian statistics for parasitologists', TRENDS in Parasitology, 20 (2): 85-91.

Berger, J.A. et al (1997) 'Unified Frequentist and Bayesian testing of a precise hypothesis', Stat. Sci., 12(3): 133-160.

Berger, Z.D. (2003, September) 'Bayesian and Frequentist Models: Legitimate Choices for Different Purposes', AEP, Vol. 13, No. 8: 583.

Bernardo, J.M. & Smith, A.F.M. (1996) Bayesian theory, USA: John Wiley & Sons Ltd. Berry, Donald A. & Hochberg, Yosef (1999) 'Bayesian perspectives on multiple comparisons', Journal of Statistical Planning and Inference, 82: 215-227.

Bland, Martin J. & Altman, Douglas G. (1998, October 24th ) 'Bayesian and Frequentists', British Medical Journal, 317: 1151-1160.

Breslow, N. (1990) 'Biostatics and Bayes', Statistical Science, 5(3): 269-298.

Carlin, J.B. & Louis, T.A. (2000) Bayes and Empirical Bayes Methods for data Analysis, Boca Raton: Chapman & Hall.

Chaloner, Kathryn & Rhame, Frank S. (2001) Quantifying and documenting prior beliefs in clinical trials', Statistics in Medicine, 20: 581-600.

Cousins, R.D. (1995), 'Why isn't every physicist a Bayesian?' Am. J. Phys. 63: 398.

D'Agostini, G. (1998, November 24th) 'Bayesian Reasoning Versus Conventional Statistics in High Energy Physics', invited talk at the XVIII International Workshop on Maximum Entropy and Bayesian Methods, Garching/ Munchen.

Dale, Andrew I. (2003) Most Honourable Remembrance. The Life and Work of Thomas Bayes, USA: Springer-Verlag.

De Finetti, Bruno (1937) 'La Prevision: Ses Lois Logiques, Ses Sources Subjectives', Annals de l'Institute Henri Poincare, 7: 1-68.

Dunson, David B. (2001) 'Commentary: Practical Advantages of Bayesian Analysis of Epidemiologic Data', American Journal of Epidemiology, 153 (12): 1222-1226.

Earman, John (1992) Bayes or Bust?, Cambridge, MA: MIT Press.

Editorial (1996, September 7th) 'Bayesian statistical methods. A natural way to assess clinical evidence', British Medical Journal, 313: 569-570.

Efron, B. (1986), 'Why isn't everyone a Bayesian?' American Statistician. 40: 1-5.

Escoto, Ben (2003, September) Bayesianism and Simplicity, USA: Stanford University, Ph.D. Thesis.

Estey, E.H. & Thall, P.F. (2003, July) 'New Designs for Phase 2 Clinical Trials', Blood, 102(2): 442-448.

Feinberg, S.E. & Schervish, M.J. (1986) The relevance of Bayesian inference for the preservation of evidence and for legal decision making', Boston University Law Review, 66: 771-798.

Fisher, Lloyd D. (1996) 'Comments on Bayesian and Frequentist Analysis and Interpretation of Clinical Trials', Controlled Clinical Trials, 17: 423-434.

Gelman, A. et al (1996) Bayesian Data Analysis, Boca Raton: Chapman & Hall.

Giere, Ronald (1988) Understanding Scientific Reasoning, USA: The University of Chicago.

Gigerenzer, Gerd et al (1990) The Empire of Chance. How probability changed science and everyday life, UK: Cambridge University Press.

Gigerenzer, Gerd (1998) 'We Need Statistical Thinking, Not Statistical Rituals', Behavioral and Brain Sciences, 21, 2: 199-200.

Giles, Jim (2002, August 1st) 'When doubt is a sure thing', Nature, 418: 476-478.

Grunkemeier, G.L. & Payne, N. (2002) 'Bayesian Analysis: A New Statistical Paradigm for New Technology', Annals of Thoracic Surgery, 74: 1901-1908.

Hacking, Ian (1972) 'Likelihood', British Journal of Philosophy of Science, 23: 132-137.

Howson, C. & Urbach, P. (1989) Scientific Reasoning: The Bayesian Approach, La Salle, IL: Open Court.

Jasanoff, Sheila (1994) The Fifth Branch, USA: Harvard University Press.

Kahneman, D. & Tversky, A. (1982), 'Subjective probability: A judgment of representativeness', in Kahneman, D., Slovic, P. & Tversky, A. (eds.) Judgement Under Uncertainty: Heuristics and Biases, NY: Cambridge University Press.

Kangas, A.S. & Kangas, Jyrki (2004) 'Probability, possibility and evidence: approaches to consider risk and uncertainty in forestry decision analysis', Forest Policy and Economics, 6: 169-188.

Knill, D.C., Kersten, D. & Yuille, A. (1996) 'A Bayesian formulation of visual perception' in Perception as Bayesian Inference (eds. Knill, D.C. & Richards, R.W.) 1-21, USA: Cambridge University Press.

Matthew, R. (1994) 'Improving the odds of justice?', New Scientist, 142: 12-13.

Mayo, Deborah G. (1997) 'Error Statistics and Learning From Error: Making a Virtue of Necessity', Philosophy of Science (Proceedings), 64: 195-212.

Meehl, P.E. & Rosen, A. (1955) 'Antecedent probability and the efficiency of psychometric signs, patterns, or cutting scores', Psychological Bulletin, 52: 194- 216.

Mossman, Douglas (2000) 'The Meaning of Malingering Data: Further Applications of Bayes' Theorem', Behav. Sci. Law, 18: 761-779.

NAS (1991) The Future of Statistical Software: Proceedings of a Forum (Panel on Guidelines for Statistical Software, National Research Council), USA: The National Academies Press.

Racine, A. et al (1986) 'Bayesian Methods in Practice. Experiences in the Pharmaceutical Industry', Applied Statistics, 35 (2): 93-150.

Savage, Leonard J. (1954) The Foundations of Statistics, NY: John Wiley.

Senn, Stephen (2003, August) 'Bayesian, Likelihood and Frequentist Approaches to Statistics', Applied Clinical Trials — actmagazine.com, 35-38.

Spiegelhalter, D.J. (1999, August 21st) 'An introduction to Bayesian methods in health technology assessment (Statistical Data Included)', British Medical Journal.. Sullivan, R.C. & Delaney, H.R. (1982) 'Criminal investigations: A decision-making process', Journal of Police Science and Administration, 10: 335-343.

Suppes, Patrick (1970) A Probabilistic Theory of Causality, Helsinki: North-Holland Publishing Company.

Szolovits, Peter (1995) 'Uncertainty and Decisions in Medical Informatics', Methods of Information in Medicine, 34: 111-121.

Tan, S.B. (2001, July) 'Bayesian Methods for Medical Research', Annals Academy of Medicine, 30(4): 444-446.

Thagard, Paul (1992) Conceptual Revolutions, USA: Princeton University Press

Thagard, Paul (1999) How Scientists Explain Disease, Princeton, NJ: Princeton University Press.

Wang, Jixian et al (2002) An approximate Bayesian risk-analysis for the gastro- intestinal safety of ibuprofen', Pharmacoepidemiology and Drug Safety, 11: 695-701.

Weatherford, Roy (1982) Philosophical Foundations of Probability Theory, USA: Routledge & Kegan Paul Ltd.

Wilson, Greg (2003) 'Tides of Change: is Bayesianism the new paradigm in statistics?', Journal of Statistical Planning and Inference, 113: 3171- 374.

© Jordi Vallverdu 2007

E-mail: Jordi.vallverdu@uab.es

Jordi Vallverdu, Ph.D.
Philosophy Department
Universitat Autonoma de Barcelona
E-08193 Bellaterra (BCN)
Catalonia
Spain