WIXLIB

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATIONDOI: 10.37394/232010.2020.17.12prior probability P(A). The greater the valueof P(A), the higher the creditability of thetest.The next example examines whathappens, if the test is negative.Example 4: Assume that Mr. X makes adiagnostic test, which is negative. Find theprobability to be a carrier of the virus:i) Under the conditions of Example 2, andii) under the conditions of Example 3.Solution: Consider the following events: A: The subject is a carrier of thevirus. B: The test is negative.i) In this case we have P(A) 0.02 andP(B/A) 0.03. Assuming that 100 peoplemake the test, we should have on average98x97% 95.06 negative tests from thenoncarriers and 2x3% 0.06 from thecarriers of the virus, i.e. an average of 95.12in total negative tests. Therefore,P(B) 0.9512.Replacing the values of P(A), P(B) andP(B/A) to equation (1) one finds thatP(A/B) 0.0006. Therefore, the probabilityfor Mr. X to be a carrier of the virus is only0.06%.ii) Here we have P(A) 0.85 andP(B/A) 0.03. Further, assuming that 100people make the test, we shall have onaverage 15x97% 14.55 negative tests fromthe noncarriers and 85x3% 2.55 from thecarriers of the virus, i.e. an average of 17.1in total negative tests. Therefore,P(B) 0.171.Replacing the values of P(A), P(B) andP(B/A) to equation (1) one finds thatP(A/B) 0.1491. Therefore, the probabilityfor Mr. X to be a carrier of the virus is14.91%. One observes here that the greaterthe value of the prior probability P(A), thelower the creditability of the test.Remark: The outcomes of the previousthree examples support the view of manyepidemiologists that, at the initial stage, the“blind” diagnostic tests for COVID-19performed on the general population are noteffective, burdening purposeless thehealthcare system of the correspondingcountry.To check this from another optical angle,one has to take into account the statisticalestimation that the existing diagnostic testsfor COVID-19 give 30% incorrectlynegative (IN) results and 10% incorrectlypositive (IP) results. Assume that 2% of thepopulation of a country has been infectedE-ISSN: 2224-3410Michael Gr. Voskoglouby the coronavirus of COVID-19 and thatthe government decides to undergo theheavy cost of performing one million“blind“ tests on the general population.Among those people, 20000 on averageshould be carriers and 980000 noncarriersof the virus. Therefore, we should have20000x30% 6000 on average IN resultsand 14000 correctly positive (CP) resultsfrom the carriers and 980000x10% 98000IP results from the noncarriers of the virus.This means that 6000 people infected by thevirus with IN tests will not take the requiredprecautions, therefore transmitting easilythe virus to the other people.Further, denote, for simplicity, by CPand IP the numbers of CP and IP results ofthe tests respectively. Then, the probabilityP(CP) of a positive test to be correct isequal toP(CP) CP : (CP IP) (4)Inourcase,P(CP) 14000:(14000 98000) 0.125, i.e. only12.5%! Therefore, there is an urgent needfor the 112000 in total people with positivetests to make a second test in order to checktheir real health condition, etc.Equation (4) shows that P(CP) increases,either if the number CP increases or if thenumber IP decreases. The former happens ifmore people are infected by the virus,whereas the latter will happen if the qualityof the diagnostic tests will be improved.When, for example, 20% of thepopulation is infected by the virus, it isstraightforward to check that the probabilityP(CP) will be approximately equal to63.6%. Consequently, the more people areinfected by the virus, the higher thecreditability (and therefore the usefulness)of the diagnostic tests in detecting thepositive cases.Ourlastexampleconcernsacombination of the Bayes’ rule and thetheorem of total probability for the solutionof the corresponding problemExample 5: A country consists of threeconfederate districts, say D1, D2 and D3,where it lives the 20%, 25% and 55% of itstotal population respectively. A percentageof 60%, 45% and 10% respectively of thepopulation of each one of those districts isagainst the confederation wanting for itsdistrict to be an independent country. Whatis the probability that one of those people,chosen randomly, lives in district D3?95Volume 17, 2020

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATIONDOI: 10.37394/232010.2020.17.12Solution: Consider the events Ai: A person lives in district Di, i 1,2, 3, and B: A person is against theconfederationOn the basis of the given data it turns outthat P(A1) 0.2, P(A2) 0.25, P(A3) 0.55and P(B/A1) 0.6, P(B/A2) 0.45, P(B/A3) 0.1. We want to calculate the probabilityreasoning, is graphically represented inFigure 2, retrieved from [10]. The wholeprocess for explaining a phenomenon startswith the humans’ observations a1, a2, , anof the real world connected to it, which laidbyinduction(intuitively)tothedevelopment of theory T1 about thisphenomenon. Theory T1 is verified bydeductivereasoningandadditionaldeductive inferences K1, K2, ., Ks arreobtained. Next, a new series of observationsb1, b2, ,bm follow. If some of thoseobservations are not compatible to the lawsof theory T1, a new theory T2 is developedto replace/extend T1, and so on. In each casethe new theory extends or rejects theprevious one approaching more and more tothe objective truth related to thecorresponding phenomenon.This procedure is known as the scientificmethod. The term was introduced during the19th century, when significant terminologiesappeared establishing clear boundariesbetween science and non science. However,the scientific method characterizes thedevelopment of science since at least the17th century. Aristotle (384-322 BC) isrecognized as the inventor of the scientificmethod due to his refined analysis of thelogicalimplicationscontainedindemonstrative discourse. The first book inthe history of human civilization written onthe basis of the principles of the scientificmethod is, according to the existingwitnesses, the “Elements” of Euclid (365300 BC) addressing the axiomaticfoundation of Geometry.The scientific method is highly based onthe Trial and Error procedure, a termintroduced by C. Lloyd Morgan (18521936) [14]. This procedure is characterizedby repeated attempts, which are continueduntil success or until the subject stopstrying.As an example, the geocenrtic theory(Almagest) of Ptolemy of Alexandria (100170), being able to predict satisfactorily themovements of the planets and the moon,was considered to be true for centuries.However, it was finally proved to be wrongand has been replaced by the heliocentrictheory of Copernicus (1473-1543). TheCopernicus’ theory was supported andenhanced a hundred years later by theobservations/studies of Kepler and Galileo,but it faced many obstacles for a longP(A3/B) [P(B/A3)P(A3)] : P(B) (4)The Ai’s are obviously pairwise disjointevents and their union is equal to the samplespace X of the inhabitants of the country(mathematically speaking the Ai’s form apartition of X). Therefore, by the theoremof total probability [6] one finds thatP(B) P(A1 B) P(A2 B) P(A3 B) and,by the Bayes’ theorem,P(B) P(B/A1)P(A1) P(B/A2)P(A2) P(B/A3)P(A3) (5).Replacing the values of the probabilitiesinvolved in equation (5) one finds thatP(B) 0.2875. Therefore, equation (4) givesthat P(A3/B) 0.0628 or 6.28%4, BayesianScienceReasoninginMany scientists and philosophers of scienceargue nowadays that the whole sciencecould be considered as a Bayesian process[9-11]. In this section we are going tosupport and justify this view.Figure 2: The scientific methodThe process of scientific thinking, being asynthesis of inductive and deductiveE-ISSN: 2224-3410Michael Gr. Voskoglou96Volume 17, 2020

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATIONDOI: 10.37394/232010.2020.17.12period, especially from the church, beforeits final justification [15].Another characteristic example is theEinstein’sgeneralrelativitytheorydeveloped at the beginning of the 20thcentury. This theory has replaced theNewton’s classical gravitational theory,which was believed to be true for more thantwo centuries. The Einstein’s new approachwas based on the fact that, according to hisspecial theory of relativity (1905) thedistance (r) and the time (t) are changing ina different way with respect to a motionlessand to a moving observer.To support his argument Einsteinintroduced the concept of the 4-dimensionaltime-space and after a series of intensiveefforts (1908-1915) he finally managed toprove that the geometry of this space is nonEuclidean. This can be physically explainedby the distortion created to the time-spacedue to the presence of mass or of anequivalent amount of energy, which looksanalogous to the distortion created by a ballof bowling on the level of a trampoline.Einstein’s theory was experimentallyverified by the irregularity of the Hermes’orbit around the sun and later by themagnitude of the light’s divergence, whichwas calculated during the eclipse of the sunon May 29, 1919. In fact, the eclipse letsome stars, which normally should bebehind the sun, to appear besides it on thesky [16].The previous discussion about thescientific method reveals the importance ofinductive reasoning for scientific thinking.In fact, the premises of all the scientifictheories (with possible exception only forpure mathematics), expressed by axioms,basic principles, etc., are based on humanintuitionandinductivereasoning.Therefore, a deductive inference developedon the basis of a scientific theory, is trueunder the CONDITION that the premises ofthe corresponding theory are true. In otherwords, if H denotes the hypothesis imposedby those premises and I denotes thedeductive inference, then the conditionalprobability P(I/H), which can be calculatedby the Bayes’ rule, expresses the degree oftruth of the deductiveinference.Consequently, the argument that theWHOLE SCIENCE is characterized byBayesian reasoning seems to be true.It must be emphasized that the error ofthe inductive reasoning is transferred to aE-ISSN: 2224-3410Michael Gr. Voskogloudeductive inference through its premises.Therefore, the scientific error in its finalform is actually a deductive and not aninductive error! This means that none of theexisting scientific theories could beconsidered as been absolutely true; it simplycould be considered as approaching thetruth in a better way than the previoustheories, that has replaced, did.5. ConclusionIn the present study was shown thatBayesian Reasoning could be considered asan interface between bivalent and fuzzylogic. Its usefulness to everyday lifesituations was also illustrated by suitableexamples and its importance for the wholescience was studied.References:[1] Moor, J. The Dartmouth CollegeArtificial Intelligence conference: Thenext fifty years, AI Magazine, Vol. 27,2006, pp.87–91.[2] Hodges, A., Alan Turing: The Enigma,CentenaryEdition,PrincetonUniversity Press: Princeton, NJ, USA,2012.[3] Janes, E.T., Probability Theory: TheLogicofScience,CambridgeUniversity Press, UK, 8th Printing,2011 (first published, 2003).[4] Mumford, D., The Dawning of the Ageof Stochasticity, in V. Amoid, M.Atiyah, P. Laxand & B. Mazur (Eds.),Mathematics:FrontiersandPerspectives, AMS, 197-218, 2000.[5] Kosko, B. Fuzzy Thinking: The NewScience of Fuzzy Logic, Hyperion NewYork 1993[6] Shahbazova, S.N., Sugeno, M.,Kacpzyk, J. (Eds.),RecentDevelopments in Fuzzy Logic andFuzzy Sets (dedicated to L.A.Zadeh),Spinger,NY,2020.[7] Schuler, J. & Lipschutz, S., Schaum’sOutline of Probability, 2nd Edition,McGraw-Hill, NY, USA, 2010.[8] Shiryaev, A.N., Probability-1, 3dEdition, Spinger,NY,2016.[9] Horgan, J., Bayes’ Theorem: What isthe Big Deal?”, January m 10] Athanassopoulos, E. and Voskoglou,M.Gr., A Philosophical Treatise on the97Volume 17, 2020

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATIONDOI: 10.37394/232010.2020.17.12Creative Commons Attribution License 4.0(Attribution 4.0 International, CC BY 4.0)Connection of Scientific Reasoningwith Fuzzy Logic, Mathematics, 8,article 875, 2020.[11] Bertsch McGrayne, S., The Theory thatwould not die, Yale University Press,New Haven and London, 2012.[12] What do you think about machines -detail/26871[13] Jeffreys, H., Scientific Inference, 3dEdition, Cambridge University Press,UK, 1973.[14] Thrope, W.H., The origins and rise ofethology: The science of the naturalbehavior of animals, Praeger, LondonNY, 1979.[15] Gingerich, O., The Eye of the Heaven –Ptolemy,Copernicus,Kepler,American Institute of Physics, NY,1993.[16] Singh, S., Bing Bang - The Origin ofthe Universe, Harper PerennianPublishers, NY, 2005.E-ISSN: 2224-3410Michael Gr. VoskoglouThis article is published under the terms of the CreativeCommons Attribution License d.en US98Volume 17, 2020

probability, Bayesian reasoning has been proved to be very important to everyday life situations [9] and for the whole science as well [10]. Recent researches give evidence that even the mechanisms under which the human brain works are Bayesian [11]! Consequently, Bayesian reasoning is very useful for Machine Learning, the sector of