The Mathematical Formula Of The Causal Relationship k

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1 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh The Mahemacal Formula Of The Causal elaonsh Ilja Baručć Horandsrasse D-644 Jever Germany Corresondng auhor: Manuscr submed o vra on ednesday June 4 5 Absrac: The deermnsc relaonsh beween cause and effec s deely conneced wh our undersandng of he hyscal scences and her exlanaory ambons Though rogress s beng made he lac of heorecal redcons and exermens n quanum gravy maes dffcul o use emrcal evdence o jusfy a heory of causaly a quanum level n normal crcumsances e by redcng he value of a well-confrmed exermenal resul For a varey of reasons he roblem of he deermnsc relaonsh beween cause and effec s relaed o basc roblems of hyscs as such Dese he common belef s a remarable fac ha a heory of causaly should be conssen wh a heory of everyhng and s because of hs lned o roblems of a heory of everyhng Thus far solvng he roblem of causaly can hel o solve he roblems of he heory of everyhng (a quanum level) oo Key words: Cause ffec Cause and effec causal relaonsh causaly Inroducon On he one hand as already menoned above and hs may no come as a surrse s hghly desrable o formulae a quanum mechancal verson of he relaonsh beween cause and effec Bu a leas one of he dffcul quesons ha chaos heory rases for he esemology of deermnsm of he relaonsh beween cause and effec can here exs a deermnsc relaonsh beween a cause and an effec a all In oher words wha s necessy wha s randomness? Quanum gravy for nsance can rovde us a comleely new vew concernng he mos fundamenal of all relaonshs he deermnsc relaonsh beween he cause and he effec Alhough numerous aems have been made n hs oc here s no commonly acceed soluon of quanum gravy u o he resen day esearch n quanum gravy exremely dffcul due o he mssng close relaonsh beween heory and exermen s owng boh a echncal and a conceual dffculy oo A non-neglgble mnory of he hyscs focus her aenon on wha s now called loo quanum gravy whle he majory of he hyscss s worng n he feld called srng heory Thus far here s no sngle generally agreed heory n quanum gravy However s sll que unclear n rncle and even n racce how o mae any concree redcons n hese heores nder hese condons quanum gravy and he deermnsc relaonsh beween a cause and an effec aear o be nmaely conneced wh one anoher The soluon of he roblems of causaon can hel o solve he roblems of quanum gravy oo Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9

2 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh Defnons Defnon The xecaon Value And The Varance Of A andom Varable Le denoe a random varable whch can ae he value wh he robably () he value wh he robably () and so on u o value wh he robably () Then he execaon of a sngle random varable s defned as () whle he execaon value of he oulaon () s defned as () More moran all robables add u o one (+ + + = ) Que naurally he execed value can be vewed somehng le he weghed average wh s beng he weghs (3) nder condons where all oucomes x are equally lely (ha s = = = ) he weghed average urns fnally no a smle average Le * denoe he comlex conjugae of he random varable The comlex conjugae of a random varable * s defned as * hus ha * (4) Le ()² denoe he varance of he random varable The varance of he random varable a a sngle Bernoull ral s defned as (5) or as (6) where denoes somehng le an execaon value of an Somemes hs s called he hdden varable Le () denoe he sandard devaon of he random varable The sandard devaon of he random varable s defned as (7) Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9

3 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh Defnon Varable The Logcal Conradcon And The Inner Conradcon Of A andom Le ()² denoe he logcal conradcon e defne (8) Le () denoe he nner conradcon e defne (9) Scholum nder condons of secal heory of relavy can denoe he execaon value as deermned by he saonary observer whle can denoe he value ( e afer he collase of he wave funcon) as deermned by he movng observer O The Cause 3 Defnon xecaon Value Of The Cause () A A Ceran Bernoull Tral In general we defne he execaon value of he cause a one sngle Bernoull ral ( e a a ceran on n sace-me e ceera) as () where denoes he robably a one sngle Bernoull ral ha he random varable s he cause of he effec he random varable 4 Defnon Tral xecaon Value Of The Cause squared (²) A A Ceran Bernoull In general we defne he execaon value of he cause squared a one sngle Bernoull ral ( e a a ceran on n sace-me e ceera) as () 3 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9

4 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh 4 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 where denoes he robably a one sngle Bernoull ral ha he random varable s he cause of he effec he random varable Defnon The Varance ()² Of The Cause A A Ceran Bernoull Tral 5 In general we defne he varance ()² of he cause a one sngle Bernoull ral ( e a a ceran on n sace-me e ceera) as () The sandard devaon () Of The Cause A one sngle Bernoull ral follows as (3) The ffec Defnon xecaon Value Of The ffec () A A Ceran Bernoull Tral 6 In general we defne he execaon value he effec as (O) a one sngle Bernoull ral ( e a a ceran on n sace-me e ceera) as (4) where denoes he robably a one sngle Bernoull ral ha he random varable s he cause of he effec he random varable Defnon xecaon Value Of The ffec squared ( ²) A A Ceran Bernoull 7 Tral In general we defne he execaon value of he effec squared a one sngle Bernoull ral ( e a a ceran on n sace-me e ceera) as (5)

5 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh 5 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 where denoes he robably a one sngle Bernoull ral ha he random varable s he cause of he effec he random varable Defnon The Varance ()² Of The ffec A A Ceran Bernoull Tral 8 In general we defne he varance ()² of he cause a one sngle Bernoull ral ( e a a ceran on n sace-me e ceera) as (6) The sandard devaon of () of he effec a one sngle Bernoull ral follows as (7) The Cause And The ffec Defnon xecaon Value Of The Cause And ffec A A Ceran 9 Bernoull Tral In general we defne he execaon value of cause and he effec O a one sngle Bernoull ral ( e a a ceran on n sace-me e ceera) as (8) where denoes he jon robably dsrbuon of cause and effec a one sngle Bernoull ral Defnon The Co-Varance Of The Cause And The ffec A A Ceran Bernoull Tral In general we defne he varance of he cause and effec as ( ) a one sngle Bernoull ral ( e a a ceran on n sace-me e ceera) as (9)

6 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh 6 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 Defnon The Mahemacal Formula Of The Causal elaonsh In general we defne he mahemacal formula of he causal relaonsh (nsen s elformel) from he sandon of hlosohy classcal logc and robably heory as C () or somehng as () or as () Scholum I s moran o noe ha he mahemacal formula of he causal relaonsh s no dencal wh Pearson s coeffcen of correlaon Axoms The followng heory s based on he followng axoms Axom I (Axom I) Axom II (Axom II) Axom III (Axom III) Consequenly s

7 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh 7 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 3 Theorems Theorem The Cause 3 Clam In general under some crcumsances he effec s deermned as (3) Proof Sarng wh Axom I s (4) Mullyng hs equaon by sandard devaon of () s (5) Due o our defnon of he sandard devaon of () of he cause a a ceran Bernoull ral as s (6) Afer dvson follows ha (7) Quod era demonsrandum Scholum Such a defnon of a cause s useful under condons where here s a robably and a sandard devaon e ceera

8 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh 8 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 Theorem The ffec 3 Clam In general under some crcumsances he effec s deermned as (8) Proof Sarng wh Axom I s (9) Mullyng hs equaon by sandard devaon of () s (3) Due o our defnon of he sandard devaon of ()² of he effec a a ceran Bernoull ral as s (3) Afer dvson follows ha (3) Quod era demonsrandum Scholum Such a defnon of he effec s useful under condons where here s a robably and a sandard devaon e ceera

9 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh 9 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 Theorem The Cause And The ffec 33 Clam In general under some crcumsances he cause and he effec are deermned as (33) Proof Sarng wh Axom I s (34) Mullyng hs equaon by he co-varance ( ) of he cause and he effec s (35) Due o our defnon of he co-varance of he cause and he effec a a ceran Bernoull ral as we oban (36) Afer Dvson follows ha (37) Quod era demonsrandum Scholum I s necessary o mae a dfference beween one sngle Bernoull ral and he whole oulaon ( e samle) of he sze

10 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 Theorem The Mahemacal Formula Of The Causal elaonsh 34 Clam In general he mahemacal formula of he causal relaonsh s deermned as (38) Proof Sarng wh Axom I s (39) Mullyng hs equaon by he cause s (4) nder he assumon of commuavy he mullcaon by he effec o yelds (4) Due o our heorem above s Thus far follows ha (4) Due o our heorem above s Consequenly s (43) Due o our heorem concernng he co-varance of cause and effec s

11 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 Thus far we oban n he followng he nex relaonsh as (44) earrangng equaon yelds (45) whch s equvalen o our formula of he causal relaonsh a each Bernoull ral as (46) Quod era demonsrandum Scholum The followng llusraon may be of hel somehow Fg ffec yes no Cause yes no The above formula of he causal relaonsh s ensurng he deermnsc relaonsh a every sngle Bernoull ral nder he assumon ha he robables from ral o ral are consan and no changng ( e condons of secal heory of relavy v=consan) e oban he followng cure whle s he oulaon sze or he number of rals

12 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 Theorem The Formula Of The Causal elaonsh Of A Bnomal andom 35 Varable Clam nder condons where he causal relaonsh s consan from ral o ral he mahemacal formula of he causal relaonsh can be smlfed as (47) Proof Sarng wh Axom I s (48) Mullyng hs equaon by he causal relaonsh s (49) whch s equvalen o (5) or o (5) where denoes he oal number of Bernoull rals he samle sze e ceera Due o our heorem above hs equaon s equvalen wh (5) or wh (53)

13 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh e defne and smlfed as Quod era demonsrandum The formula above can be (54) Scholum nder he condons above he sgnfcance of he causal relaonsh can be esed usng he Ch-Square dsrbuon wh one degree of freedom The followng x able may rovde an overvew Fg Cause yes no ffec yes no In sascs he h coeffcen nroduced by Karl Pearson s one of he nown measures of assocaon for wo bnomal random varables There are suaons where he h coeffcen s dencal wh he mahemacal formula of he causal relaonsh bu boh are no dencal n general 36 Theorem The Ch-Square Dsrbuon And The Formula Of The Causal elaonsh Clam nder some assumons he mahemacal formula of he causal relaonsh s deermned by he ch-square dsrbuon as (55) Proof Sarng wh Axom I s (56) Mullyng hs equaon by he causal relaonsh s 3 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9

14 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh 4 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 (57) whch s equvalen o (58) where denoes he execaon value of he causal relaonsh a each sngle Bernoull ral and denoes he devaon of he causal relaonsh a each Bernoull ral A each Bernoull ral he normal random varable of a sandard normal dsrbuon (called a sandard score or a z-score) a each sngle Bernoull ral s deermned as Z Thus far we oban Z (59) nder condons [] where and we oban Z (6) or Z (6) Afer he square roo oeraon s

15 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh 5 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9 Z (6) Summarzng yelds Z (63) whch s equvalen wh Z ² (64) In oher words s ² Z (65) nder condons where he causal relaonsh s consan from ral o ral we oban ² Z (66) In sascs s nown ha Z ² wh degrees of freedom we oban ² Z (67) here denoes he average value of he causal relaonshs afer Bernoull rals e re-wre hs equaon above as (68) where ² denoes he ch-squared dsrbuon (also ch-square dsrbuon) wh degrees of freedom A he end follows ha (69) Quod era demonsrandum

16 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh Scholum Ths roof enable us o use he mahemacal formula of he causal relaonsh for hyohess esng wh he ossbly o calculae he -values he ß-value e ceera even under condons where he s no consan from ral o ral nder hese condons lease recall he relaonsh Z where denoes he -dsrbuon wh degrees of freedom and Z denoes he Z value For more deals on hs oc I mus refer he reader o rmary leraure 4 Dscusson There s a long radon of dualsm [] beween causaly and sascs For reasons no relevan here sascs seemed o exclude causaly and vce versa secally due o some quanum mechancal osons (Hesenberg s uncerany Bell s heorem CHSH-Inequaly) he deermnsc relaonsh beween a cause and s own effec became an mossbly Meanwhle he quanum mechancal no-go-heorems whch excluded he deermnsc relaonsh beween cause and effec are already refued [3] [4] [5] hle he mahemacal mehodology o exrac cause and effec relaonsh ou of (non-) exermenal daa s already ublshed [6] [7] [8] [9] eer-revewed [] and resened o he scenfc communy hs hghly orgnal aroach gves a new unnown and exac mahemacal dervaon of he mahemacal formula of he causal relaonsh from a urely mahemacal sarng on In general daa can be gahered hrough an observaonal sudy hrough an exermen e ceera Aferwards sascal nference allows he researcher o asses evdence n favor or some hyoheses abou he oulaon from whch a samle has been drawn The mahemacal formula of he causal relaonsh can be used as a es of sgnfcance o suor or o rejec hyoheses/clams based on daa gahered secally we are enabled o es wheher here s a causal relaonsh beween random varables nvesgaed or no For examle n a clncal ral he null hyohess mgh be ha here s no causal relaonsh beween a random varable ( e Helcobacer ylor) and an effec ( e human gasrc cancer) In oher words we would wre H: = In oher words Helcobacer ylor and human gasrc cancer are ndeenden of each oher In he same clncal ral he alernave hyohess HA s a saemen of wha a sascal hyohess es s se u o esablsh The oosng hyohess s he alernave hyohess (HA) For examle n a clncal ral he alernave hyohess HA mgh be ha here s a causal relaonsh beween a random varable ( e Helcobacer ylor) and an effec ( e human gasrc cancer) In oher words we would wre HA: # The fnal concluson s always gven n erms of he null hyohess as eher "rejec H n favor of Ha" or "do no rejec H" xamle Helcobacer ylor has been dscussed [] [] as beng assocaed wh human gasrc cancer In several revous (edemologc) sudes and mea-analyss has been reored ha here s a close relaon beween H ylor nfecon and gasrc cancer Sll he cause of human gasrc cancer s no denfed aom emura e al [] conduced a long-erm rosecve sudy of =56 Jaanese aens 46 had H ylor nfecon and 8 dd no 6 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9

17 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh (mean follow u 78 years endoscoy a enrollmen and hen beween one and hree years afer enrollmen) one of he unnfeced aens develoed gasrc cancer Le us show hs daa n he followng --able Fg Helcobacer ylor nfecon of human somach Human gasrc cancer yes no yes no H: = o sgnfcan causal relaonsh beween Helcobacer ylor nfecon of human somach and human gasrc cancer Alha = 5 % HA: # Sgnfcan causal relaonsh beween Helcobacer ylor nfecon of human somach and human gasrc cancer xermen An observaonal sudy or an exermen s erformed daa are gahered Daa analyss Calculaon of he causal relaonsh (7) Calculaon of Pearson ch-square sasc uncorreced for connuy one degree of freedom and of he -value The Pearson ch-square sasc uncorreced for connuy s calculaed as follows: ² Calculaed a d bc (7) a bc d a cb d The followng conngency able may rovde some dealed nformaon abou hs formula Fg Cause ffec yes no yes a b a+b no c d c+d a+c b+d 7 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9

18 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh Due o he daa of he sudy above he Ch-Square can be calculaed as ² (7) Calculaed The Pearson ch-square sasc uncorreced for connuy s 8853 The P value s Ths resul s sgnfcan a < 5 Concluson The daa above do no suor he null hyohess HO we mus rejec H n favor of HA (-value 3997) In oher words here s a hghly sgnfcan causal relaonsh beween an nfecon of human somach wh Helcobacer ylor and he develomen of human gasrc cancer (= value 3997) Helcobacer ylor s he cause of human gasrc cancer The mehods above where already demonsraed and used o analyze he relaonsh beween Helcobacer ylor and he gasrc cancer A III rd Inernaonal Bomerc Conference scheduled from July 6-6 n Monréal Canada a sgnfcan causal relaonsh beween Helcobacer ylor and human gasrc cancer usng he mehods above was resened [3] o he scenfc communy Helcobacer ylor s he cause of gasrc cancer 5 Concluson Ths ublcaon rovdes an exac mahemacal dervaon of he relaonsh beween he cause and he effec A new mahemacal mehodology for mang causal nferences on he bass of (non-) exermenal daa for evaluang causal relaonshs from (non-) exermenal daa s resened n he smles and mos nellgble form Anyone who wshes o elucdae cause effec relaonshs from (non-) exermenal daa wll fnd hs ublcaon useful Fnally a unfed mahemacal and sascal model of he relaonsh beween he cause and he effec s avalable Acnowledgmen one Aendx one 8 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9

19 Ilja Baručć The Mahemacal Formula Of The Causal elaonsh eferences [] Ilja Baručć "Causaon And The Law Of Indeendence" Symosum on Causaly 6 Frday July 7 - Sunday July 9 6 Jena nversy Germany [] Ilja Baručć "The Deermnsc elaonsh Beween Cause And ffec Inernaonal Bomerc Conference Kobe JAPA 6 3 Augus h://secrearanej/bc/rogramme/-5/p-/49-p--3df [3] Ilja Baručć "An Hesenberg efuaon Of Hesenberg s ncerany elaon" Amercan Insue of Physcs - Conf Proc 37 3 () [4] Ilja Baručć "An-Bell - efuaon of Bell's heorem" Amercan Insue of Physcs - Conf Proc () [5] Ilja Baručć "An Hesenberg efuaon of Hesenberg s ncerany Prncle" Inernaonal Journal of Aled Physcs and Mahemacs vol 4 no [6] Ilja Baručć De Kausalä ssenschafsverlag Hamburg [7] Ilja Baručć Causaly ew Sascal Mehods Second don Hamburg: Boos on Demand [8] Ilja Baručć Causaly I A heory of energy me and sace 5h ed Morrsvlle: Lulu; 648 [9] Ilja Baručć Causaly II A heory of energy me and sace 5h ed Morrsvlle: Lulu; 376 [] M Thomson evews Causaly ew Sascal Mehods I Barucc Shor Boo evew ( dor: Prof Dr Agnes M Herzberg ) Inernaonal Sascal Insue ( Drecor: Prof Dr Danel Berze ) Vol 6 o - Arl 6 6 h://scbsnl/sbr/mages/v6-_ar6df [] HO: Inernaonal Agency for esearch on Cancer IAC monograhs on he evaluaon of he carcnogenec rss o humans Schsosomes lver flues and Helcobacer ylor F-6937 Lyon France Vol [] aom emura MD Shro Oamoo MD Sochro Yamamoo MD obuosh Masumura MD Shuj Yamaguch MD Mcho Yamado MD Kyom Tanyama MD aom Sasa MD and onald J Schlemer MD "Helcobacer ylor Infecon and he Develomen of Gasrc Cancer" The ew ngland Journal of Medcne Vol 345 o [3] Ilja Baručć "ew Mehod for Calculang Causal elaonshs" III rd Inernaonal Bomerc Conference Monréal Québec Canada July 6 6 h://wwwmedcnemcgllca/edemology/hanley/bc6rogramdf 49 9 Manuscr submed o vra (ednesday June 4 5) Ilja Baručć Jever Germany All rghs reserved ednesday June 4 5 9:57:9