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Rock, PhD, PE, CIH", FontSize->24]], "Subtitle", CellChangeTimes->{{3.485609136120798*^9, 3.4856091511532907`*^9}, { 3.4856091945334663`*^9, 3.485609199379443*^9}, {3.4951031489375*^9, 3.49510314984375*^9}, {3.495106455296875*^9, 3.495106455453125*^9}, { 3.5143083846926413`*^9, 3.514308395249558*^9}, {3.521200786392234*^9, 3.5212007955962496`*^9}}, TextAlignment->Center], Cell[TextData[StyleBox["", FontSize->18]], "Subsubtitle", CellChangeTimes->{ 3.483202458953512*^9, {3.495105345328125*^9, 3.495105347890625*^9}, { 3.49510644571875*^9, 3.495106448390625*^9}, {3.5143083980990458`*^9, 3.514308409442589*^9}, {3.5212008014306602`*^9, 3.521200810447476*^9}, 3.527875323726525*^9}] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Outline of Presentation", "Section", CellChangeTimes->{{3.5217121357823687`*^9, 3.5217121389803743`*^9}}], Cell["\<\ Why we need Probability Theory Axioms of Probability Theory Bayes Rule, Marginalization and Evidence The Binary Decision Problem, - Deduction: Probability of M Passes Given N trials and prob = p. - Induction: pdf[p] given M passes observed in N trials. Confidence Interval (frequentist) vs Credible Region (Bayesian) Deductive v Inductive Reasoning Summary \ \>", "Subsection", CellChangeTimes->{{3.5217121493699923`*^9, 3.5217122947154474`*^9}, { 3.5217435388116617`*^9, 3.5217435977329655`*^9}, 3.5221605713366623`*^9, { 3.522160664359625*^9, 3.5221608108438826`*^9}, {3.5278803277171264`*^9, 3.527880338947172*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Why We need Probability Theory", "Section", CellChangeTimes->{{3.5213587493144965`*^9, 3.5213587591737137`*^9}, { 3.521528887592781*^9, 3.5215288884351826`*^9}, 3.5217121577472067`*^9}], Cell[TextData[{ StyleBox["\[OpenCurlyDoubleQuote]The actual science of logic is conversant \ at present only with things either certain, impossible, or entirely doubtful, \ none of which (fortunately) we have to reason on. Therefore the true logic \ for this world is the calculus of Probabilities, which takes account of the \ magnitude of the probability which is, or ought to be, in a reasonable man\ \[CloseCurlyQuote]s mind.\[CloseCurlyDoubleQuote] ", FontSize->36, FontWeight->"Bold", FontColor->GrayLevel[0], Background->RGBColor[0.87, 0.94, 1]], "\n\tJames Clerk Maxwell (1850), quoted by Edwin Jaynes (1994)." }], "Text", CellChangeTimes->{ 3.5213586939967995`*^9, {3.5213588068629975`*^9, 3.521358820715822*^9}, { 3.521387693806424*^9, 3.5213877234776764`*^9}, {3.521528898871601*^9, 3.5215288996984024`*^9}}, FontSize->24] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Deductive vs. Inductive Logic", "Section", CellChangeTimes->{{3.5219746193287735`*^9, 3.521974627581188*^9}}], Cell["\<\ You can reason by deduction if you have axioms to start with, and you can \ reason by inference if you have data to start with. Deduction finds Lemmas, Theorems and Rules from a few axioms. \tAbstract Mathematicians live in this world Induction finds a cause from measurements of several effects. \tOthers who learn from mistakes live here \ \>", "Subsection", CellChangeTimes->{{3.521974638969208*^9, 3.5219748048754997`*^9}, { 3.5278753959933214`*^9, 3.5278754440328608`*^9}, {3.5278791735323143`*^9, 3.527879202517542*^9}, {3.527880390892935*^9, 3.52788043209451*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Probability Theory - LaPlace (1812) \ \>", "Section", CellChangeTimes->{ 3.483202458955147*^9, {3.51430857274755*^9, 3.514308578875259*^9}, { 3.521214270017158*^9, 3.5212142766159697`*^9}, {3.5213269011529207`*^9, 3.5213269102789364`*^9}, {3.5213270974324656`*^9, 3.5213271191321034`*^9}, {3.5213569441188393`*^9, 3.521356949235648*^9}, { 3.521357224373331*^9, 3.5213572276337366`*^9}, {3.5213595213910522`*^9, 3.5213595285670652`*^9}, {3.5213615662426443`*^9, 3.521361568348648*^9}, { 3.521387826110257*^9, 3.521387835127073*^9}}], Cell[CellGroupData[{ Cell["\<\ La th\[EAcute]orie des probabiliti\[EAcute]s n\[CloseCurlyQuote]est que le \ bon sens reduit au calcul\ \>", "Subsection", CellChangeTimes->{{3.521357576369498*^9, 3.521357611869548*^9}, { 3.521359543917492*^9, 3.5213597260634117`*^9}, {3.521361573824258*^9, 3.5213615806726694`*^9}, 3.52138782228825*^9}], Cell["1. Represent probabilities with real numbers.", "Text", CellChangeTimes->{{3.521356965022876*^9, 3.5213570504486256`*^9}, 3.5213571423327875`*^9, {3.521357174094443*^9, 3.5213572052788973`*^9}, { 3.521357552139464*^9, 3.5213575708994904`*^9}, {3.521359795795534*^9, 3.521359831332397*^9}}, FontSize->24], Cell["2. Qualitative agreement with common sense.", "Text", CellChangeTimes->{{3.5213566083474493`*^9, 3.5213566549759307`*^9}, 3.52135707565827*^9, {3.5213598196947765`*^9, 3.5213598346396027`*^9}}, FontSize->24], Cell["3. Consistent Answers. ", "Text", CellChangeTimes->{{3.5213567130704327`*^9, 3.521356820707022*^9}, { 3.521356892420348*^9, 3.521356920859198*^9}, {3.52135712117915*^9, 3.521357121881151*^9}, {3.5213598477904263`*^9, 3.521359878163679*^9}}, FontSize->24] }, Open ]], Cell["\<\ Probability Theory is nothing but good sense reduced to calculations\ \>", "Subsection", CellChangeTimes->{{3.52135974791905*^9, 3.5213597699774895`*^9}, { 3.521361499552527*^9, 3.5213615035929337`*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Probability Theory - Cox Desiderata (1946)\ \>", "Section", CellChangeTimes->{ 3.483202458955147*^9, {3.51430857274755*^9, 3.514308578875259*^9}, { 3.521214270017158*^9, 3.5212142766159697`*^9}, {3.5213269011529207`*^9, 3.5213269102789364`*^9}, {3.5213270974324656`*^9, 3.5213271191321034`*^9}, {3.5213569441188393`*^9, 3.521356949235648*^9}, { 3.521357224373331*^9, 3.5213572276337366`*^9}, {3.521359887898096*^9, 3.521359893420506*^9}, {3.5221693704319487`*^9, 3.522169374791198*^9}}], Cell[CellGroupData[{ Cell["\<\ Cox (1946) sought a complete probability theory \ \>", "Subsection", CellChangeTimes->{{3.521357576369498*^9, 3.521357611869548*^9}, { 3.5213599327325754`*^9, 3.5213599858194685`*^9}, {3.521360063024004*^9, 3.521360070995618*^9}}], Cell["\<\ 1. If there is more than one way to estimate the probability that a \ proposition is true, all observers with the same information should obtain the same estimate.\ \>", "Text", CellChangeTimes->{{3.521356965022876*^9, 3.5213570504486256`*^9}, 3.5213571423327875`*^9, {3.521357174094443*^9, 3.5213572052788973`*^9}, { 3.521357552139464*^9, 3.5213575708994904`*^9}, {3.5213599996254926`*^9, 3.5213600001246934`*^9}}, FontSize->18], Cell["\<\ 2. When estimating the probability that a proposition is true, one also estimates the probability that it is false.\ \>", "Text", CellChangeTimes->{{3.5213566083474493`*^9, 3.5213566549759307`*^9}, 3.52135707565827*^9, {3.521360002995098*^9, 3.5213600036034994`*^9}}, FontSize->18], Cell["\<\ 3. An estimate for the probability that a proposition, Y, is true, plus an estimate that the probability another proposition, X, is true when Y \ is true, quantifies an estimate for the probability that both X and Y are true.\ \>", "Text", CellChangeTimes->{{3.5213567130704327`*^9, 3.521356820707022*^9}, { 3.521356892420348*^9, 3.521356920859198*^9}, {3.52135712117915*^9, 3.521357121881151*^9}, {3.5213600090479093`*^9, 3.52136000956271*^9}}, FontSize->18] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Probability Axioms Lead to Probability Theory\ \>", "Section", CellChangeTimes->{ 3.483202458955147*^9, {3.51430857274755*^9, 3.514308578875259*^9}, { 3.521214270017158*^9, 3.5212142766159697`*^9}, {3.5213269011529207`*^9, 3.5213269102789364`*^9}, {3.521327122220909*^9, 3.5213271888018255`*^9}, { 3.5278792312939987`*^9, 3.527879245051707*^9}}], Cell[CellGroupData[{ Cell["\<\ Cox (1946) showed his desiderata lead to two unique axioms\ \>", "Subsection", CellChangeTimes->{{3.5213604051794047`*^9, 3.5213604290474467`*^9}, { 3.5213606481654315`*^9, 3.521360649257434*^9}, {3.5213607862256737`*^9, 3.5213607870680757`*^9}, {3.521361880021595*^9, 3.5213618969008245`*^9}, { 3.5214127576637588`*^9, 3.521412757991359*^9}}], Cell[TextData[{ "SUM RULE: \n\t", StyleBox["prob[ X | C ] + prob[ ", FontSize->24], Cell[BoxData[ FormBox[ OverscriptBox["X", StyleBox["-", FontWeight->"Bold"]], TraditionalForm]], FontSize->24], StyleBox[" | C ] = 1", FontSize->24] }], "Text", CellChangeTimes->{{3.521360541804445*^9, 3.5213606372454123`*^9}, { 3.5213606707854714`*^9, 3.5213607316567783`*^9}, {3.521361075325382*^9, 3.52136108008339*^9}, {3.521361133513484*^9, 3.5213611338410845`*^9}}], Cell[TextData[{ "PRODUCT RULE: \n\t", StyleBox["prob[ X, Y | C ] = prob[ X | Y, C ] * prob[ Y | C ]", FontSize->24] }], "Text", CellChangeTimes->{{3.521360541804445*^9, 3.5213606372454123`*^9}, { 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\[CloseCurlyDoubleQuote] means Boolean \ \[OpenCurlyDoubleQuote]AND\[CloseCurlyDoubleQuote] \[OpenCurlyDoubleQuote] prob[ X, Y | C ] \[CloseCurlyDoubleQuote] means \ probability that both X AND Y are true, given C is true \[OpenCurlyDoubleQuote]PDF\[CloseCurlyDoubleQuote] means probability \ distribution function \[OpenCurlyDoubleQuote]LKF\[CloseCurlyDoubleQuote] means likelihood function\ \>", "Text", CellChangeTimes->{{3.5213619178204613`*^9, 3.521362066785123*^9}, { 3.521362164987296*^9, 3.5213622314590125`*^9}, {3.5213867939360437`*^9, 3.521386822546494*^9}, {3.521388184411686*^9, 3.5213881908700976`*^9}, { 3.521412827629882*^9, 3.5214128297202854`*^9}, {3.5215289447824817`*^9, 3.52152897236333*^9}}, FontSize->24], Cell["\<\ Note that all probabilities are conditional on some identifiable information.\ \>", "Text", CellChangeTimes->{{3.52136209342997*^9, 3.5213621190296154`*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", 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"for a pass, and LKF =", StyleBox[" ", FontSlant->"Italic"], StyleBox["(", FontColor->RGBColor[0, 0, 1]], StyleBox["1", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["-", FontColor->RGBColor[0, 0, 1]], StyleBox["p", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[")", FontColor->RGBColor[0, 0, 1]], " for a fail\nBy Bayes Rule, the posterior after 2 trials = LKF * priorLKF \n\ postLKF2 = ", Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox["p", "2"], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], TraditionalForm]]], " if both trials resulted in pass\n\t =", StyleBox[" ", FontSlant->"Italic"], StyleBox["p", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" (", FontColor->RGBColor[0, 0, 1]], StyleBox["1", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["-", FontColor->RGBColor[0, 0, 1]], StyleBox["p", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[") ", FontColor->RGBColor[0, 0, 1]], "if {1st, 2nd) trials were {P, F} or {F, P} \n\t = ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"1", " ", "-", " ", "p"}], ")"}], "2"], TraditionalForm]], FormatType->"TraditionalForm", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " if both trials were fail" }], "Text", CellChangeTimes->{{3.521402931941512*^9, 3.5214029700835795`*^9}, { 3.5214030950397987`*^9, 3.521403099829007*^9}, 3.5214031859723587`*^9, { 3.5215309694558516`*^9, 3.5215309933082933`*^9}, {3.521626164842035*^9, 3.5216262058701067`*^9}, {3.5216262694870186`*^9, 3.5216263285487223`*^9}}], Cell[TextData[{ "By induction, if data are ", StyleBox["m", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " passes after ", StyleBox["n", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " trials, then:\n\n\tpostLKFn =", StyleBox[" ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ FormBox[ SuperscriptBox["p", "m"], 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