From diagnosis to therapy: algorithmic methodologies for precision medicine

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(19) p (Ω) = 1 0 ≤ p (E) ≤ 1. ∀E ⊆ Ω. ∀E, F ⊆ Ω TVDI UIBU E ∩ F = ∅, p (E ∪ F ) = p (E) + p (F ) ∀E, F ⊆ Ω TVDI UIBU E ∩ F = ∅, p (E ∪ F ) = p (E) + p (F ) − p (E ∪ F ) 4VQQPTF XF XBOU UP EFUFSNJOF UIF QSPCBCJMJUZ PG BO FWFOU E BTTVNJOH UIBU BOPUIFS FWFOU F IBQQFOFE ɨJT RVBOUJUZ JT DBMMFE UIF DPOEJUJPOBM QSPCBCJMJUZ PG E HJWFO F BOE JT EFOPUFE CZ p (E|F ) *U DBO CF FTUJNBUFE BT p (E|F ) =. p (E ∩ F ) . p (F ). . 8F TBZ UIBU UXP FWFOUT E BOE F BSF JOEFQFOEFOU JG p (E|F ) = p (E) ɨJT JNQMJFT UIBU JG UXP FWFOUT BSF JOEFQFOEFOU UIFO p (E ∩ F ) = p (E) p (F ) *G UXP FWFOUT BSF JOEFQFOEFOU PODF XF LOPX UIF PVUDPNF PG B UIJSE FWFOU XF BSF JO UIF QSFTFODF PG DPOEJUJPOBM JOEFQFO EFODF .PSF GPSNBMMZ XF TBZ UIBU UXP FWFOUT E BOE F BSF DPOEJUJPOBMMZ JOEFQFOEFOU HJWFO G JG p (E|F ∩ G) = p (E|G) /PX HJWFO UXP FWFOUT E BOE F TVDI UIBU p (E) = 0 BOE p (F ) = 0 UIF #BZFT ɧFPSFN BċSNT UIBU p (E|F ) =. p (F |E) p (E) . p (F ). . 3ĒğĕĠĞ 7ĒģĚĒēĝĖĤ (JWFO B QSPCBCJMJUZ TQBDF (Ω, p) XF DBMM SBOEPN WBSJBCMF X B GVODUJPO XIPTF EPNBJO JT Ω ɨF SBOHF PG WBMVFT UBLFO CZ X JT DBMMFE UIF TQBDF PG X 8IFO EFBMJOH XJUI SBOEPN WBSJBCMFT XF EFOPUF XJUI X = x UIF FWFOU {e ∈ Ω|X (e) = x} 8F DBMM UIF WBMVFT PG p (X = x) GPS BMM WBMVFT x PG X UIF QSPCBCJMJUZ EJTUSJCVUJPO PS QSPCBCJMJUZ NBTT GVODUJPO PS QNG PG UIF SBOEPN WBSJBCMF X 8IFO XF BSF SFGFSSJOH UP UIF QSPCBCJMJUZ EJTUSJCVUJPO PG X XF XSJUF p (X) -FU X BOE Y CF UXP SBOEPN WBSJBCMFT EFmOFE PO UIF TBNF TBNQMF TQBDF Ω 8F. 'PS TJNQMJDJUZ XF XJMM VTF UIF OPUBUJPO p (x) JOTUFBE PG p (X = x). .

(20) DBMM p (X = x, Y = y) UIF KPJOU QSPCBCJMJUZ EJTUSJCVUJPO PG X BOE Y ɨFO UIF GPMMPXJOH UXP FRVBUJPOT IPMET p (X = x) =. . p (X = x, Y = y) ,. . p (X = x|Y = y) p (Y = y) .. . y. p (X = x) =. y. *O FRVBUJPO UIF QSPCBCJMJUZ EJTUSJCVUJPO p (X = x) JT BMTP DBMMFE UIF NBSHJOBM QSPCBCJMJUZ EJTUSJCVUJPO PG X SFMBUJWF UP UIF KPJOU EJTUSJCVUJPO p (X = x, Y = y) (JWFO B TFU PG n SBOEPN WBSJBCMFT X1 , . . . , Xn EFmOFE PO UIF TBNF TBNQMF TQBDF Ω XF DBMM DIBJO SVMF UIF GPMMPXJOH FRVBUJPO p (x1 , . . . , xn ) = p (xn |xn−1 , xn−2 , . . . , x1 ) × . . . × p (x2 |x1 ) × p (x1 ) .. . "OPUIFS TQFDJBM GVODUJPO BTTPDJBUFE XJUI BOZ EJTDSFUF SBOEPN WBSJBCMF JT DBMMFE DVNVMBUJWF EJTUSJCVUJPO GVODUJPO PS DEG ɨJT GVODUJPO JT EFmOFE GPS BOZ SFBM OVNCFS x BT UIF QSPCBCJMJUZ UIBU UIF SBOEPN WBSJBCMF XJMM CF MFTT UIBO PS FRVBM UP x ɨF GVODUJPO OPUBUJPO F JT PGUFO VTFE UP SFQSFTFOU UIF DEG PG B SBOEPN WBSJBCMF #Z EFmOJUJPO XF DBO XSJUF FX (x) = p (X ≤ x) .. ɨF BCPWF BSHVNFOUT BTTVNFT UIBU UIF TFU PG QPTTJCMF WBMVFT PG B SBOEPN WBSJBCMF JT mOJUF BOE DPVOUBCMF *G UIF TFU PG QPTTJCMF WBMVFT JT VODPVOUBCMZ JOmOJUF JF SFBM OVNCFST XF TQFBL JOTUFBE PG DPOUJOVPVT SBOEPN WBSJBCMFT 6OMJLF UIFJS EJTDSFUF DPVOUFSQBSUT UIF BTTPDJ BUFE QSPCBCJMJUZ GVODUJPO DBMMFE QSPCBCJMJUZ EFOTJUZ GVODUJPO PS QEG JT B DPOUJOVPVT GVODUJPO UIBU EFTDSJCFT UIF SFMBUJWF MJLFMJIPPE GPS UIF SBOEPN WBSJBCMF UP UBLF PO B HJWFO WBMVF (JWFO B DPOUJOVPVT SBOEPN WBSJBCMF X JUT QSPCBCJMJUZ EFOTJUZ GVODUJPO JT UZQJDBMMZ JOEJDBUFE CZ fX (x) ɨF BSFB EFmOFE CZ UIF GVODUJPO JO B QSFDJTF SBOHF PG WBMVFT DPSSFTQPOET UP UIF QSPCBCJMJUZ PG. .

(21) PCTFSWJOH TVDI WBMVFT .PSF GPSNBMMZ . b. p (a ≤ X ≤ b) =. fx (x) dx.. a. ɨJT JNQMJFT UIBU p (a) =. a a. fx (x) dx = 0 GPS FBDI x JO UIF TQBDF PG UIF WBSJBCMF. *O PSEFS GPS UIF BSFB PG B QEG UP DPSSFTQPOE XJUI B QSPCBCJMJUZ TPNF SFTUSJDUJPOT BSF OFDFTTBSZ fX (x) ≥ 0 . +∞ −∞. ∀x. fx (x) dx = 1. ɨF DVNVMBUJWF EJTUSJCVUJPO GVODUJPO GPS B DPOUJOVPVT SBOEPN WBSJBCMF JT UIF DPOUJOVPVT GVODUJPO EFmOFE CZ FX (x) = p (X ≤ x) , GPS BOZ SFBM OVNCFS x 4JODF UIF GVODUJPO F JT EFmOFE BT B QSPCBCJMJUZ JU POMZ SFUVSOT WBMVFT JO UIF SBOHF [0, 1] 1MVHHJOH BOZ SFBM OVNCFS x JOUP FX (x) SFUVSOT UIF QSPCBCJMJUZ UIBU UIF SBOEPN WBSJBCMF X XJMM IBWF B WBMVF MFTT UIBO PS FRVBM UP UIF OVNCFS x /PUF UIBU FX (x) BOE fX (x) BSF DMPTFMZ SFMBUFE CZ UIJT EFmOJUJPO . t=x. FX (x) =. fX (t) dt,. t=−∞. XIFSF t IBT CFFO VTFE BT UIF EVNNZ WBSJBCMF PG JOUFHSBUJPO UP CF QFSGFDUMZ DPSSFDU GSPN B DBMDVMVT TUBOEQPJOU 'SPN UIF PUIFS EJSFDUJPO XF IBWF fX (x) =. dFX (x) dx. BOE TP JU JT DMFBS UIBU HJWFO POF PG UIFTF GVODUJPOT XF DBO EFUFSNJOF UIF PUIFS /PX TVQQPTF XF IBWF UXP SBOEPN WBSJBCMFT X BOE Y EFmOFE PO UIF TBNF QSPCBCJMJUZ TQBDF Ω *G GPS BMM WBMVFT x PG X BOE y PG Y UIF FWFOUT X = x BOE Y = y BSF JOEF. .

(22) QFOEFOU p (X = x|Y = y) = p (X = x) UIFO XF TBZ UIBU X BOE Y BSF JOEFQFOEFOU BOE XSJUF I (X, Y ) *G XF IBWF B UIJSE SBOEPN WBSJBCMF Z BOE XIFOFWFS p (z) = 0 UIF FWFOUT X = x BOE Y = y BSF DPOEJUJPOBMMZ JOEFQFOEFOU HJWFO Z = z p (X = x|Y = y ∩ Z = z) = p (X = x|Z = z) UIFO XF TBZ UIBU X BOE Y BSF DPOEJUJPOBMMZ JOEFQFOEFOU HJWFO Z BOE XSJUF I (X, Y |Z) 'ĦğĕĒĞĖğĥĒĝĤ Ġė 4ĥĒĥĚĤĥĚĔĤ " QNG HJWFT BMM UIF JOGPSNBUJPO XF OFFE BCPVU B EJTDSFUF SBO EPN WBSJBCMF XIJMF B QEG EPFT UIF TBNF GPS B DPOUJOVPVT SBOEPN WBSJBCMF 8F DBO DBMDVMBUF UIF QSPCBCJMJUZ PG BOZ FWFOU XF XBOU GSPN UIFN )PXFWFS UP TVNNBSJ[F UIF EJTUSJCVUJPO NPSF DPODJTFMZ UXP DPNNPO DBMDVMBUJPOT DBO CF NBEF GPS B SBOEPN WBSJBCMF UIF FYQFDUFE WBMVF PS NFBO BOE UIF WBSJBODF ɨF NFBO PG B SBOEPN WBSJBCMF JT EFmOFE BT UIF BWFSBHF WBMVF UIBU XPVME CF PCTFSWFE GPS UIF SBOEPN WBSJBCMF JG JU DPVME CF PCTFSWFE PWFS BOE PWFS BHBJO BO JOmOJUF OVNCFS PG UJNFT *U JT EFmOFE BT E [X] =. . x · p (X = x). GPS B EJTDSFUF SBOEPN WBSJBCMF BOE E [X] =. x · f (x) dx. GPS B DPOUJOVPVT SBOEPN WBSJBCMF ɨF TVN PS JOUFHSBM JT UBLFO PWFS BMM QPTTJCMF WBMVFT PG X *O FJUIFS DBTF JU JT SFBTPOBCMF UP JOUFSQSFU UIJT DBMDVMBUJPO BT UBLJOH UIF XFJHIUFE BWFSBHF PG UIF QPTTJCMF WBMVFT PG X XIFSF UIF XFJHIUT BSF UIF QSPCBCJMJUJFT ɨF WBSJBODF PG B SBOEPN WBSJBCMF JT B NFBTVSF PG IPX TQSFBE PVU JUT QPTTJCMF WBMVFT BSF " SBOEPN WBSJBCMF XJMM IBWF B TNBMM WBSJBODF JG JUT QPTTJCMF WBMVFT BMM GBMM JO B TNBMM UJHIU SBOHF XJUI IJHI QSPCBCJMJUZ *U XJMM IBWF B MBSHF WBSJBODF JG JUT QPTTJCMF WBMVFT BSF WFSZ TQSFBE PVU PWFS B XJEF SBOHF BOE UIFSF JT B SFBTPOBCMF DIBODF UIBO BOZ PG UIPTF WBMVFT DPVME CF PCTFSWFE ɨF WBSJBODF. .

(23) DBMDVMBUJPO JT V [X] =. . x2 p (X = x) − E [X] .. 'PS B DPOUJOVPVT SBOEPN WBSJBCMF XF KVTU SFQMBDF TVNNBUJPO XJUI JOUFHSBUJPO "OPUIFS FRVJW BMFOU GPSNVMB GPS UIF WBSJBODF XIJDI JT TPNFUJNFT FBTJFS UP EFBM XJUI JT V [X] = E X 2 − (E [X])2. " DBMDVMBUJPO SFMBUFE UP UIF WBSJBODF JT DBMMFE UIF TUBOEBSE EFWJBUJPO PG B SBOEPN WBSJBCMF *U JT TJNQMZ EFmOFE BT UIF QPTJUJWF TRVBSF SPPU PG UIF WBSJBODF ɨJT JT BDUVBMMZ VTFE NPSF PGUFO JO QSBDUJDF UIBO UIF WBSJBODF CFDBVTF UIF VOJUT PG UIF TUBOEBSE EFWJBUJPO DBMDVMBUJPO BSF UIF TBNF BT UIF VOJUT PG UIF PSJHJOBM WBSJBCMF XIJMF UIF VOJUT PG UIF WBSJBODF BSF UIF TRVBSF PG UIF VOJUT PG UIF PSJHJOBM WBSJBCMF. . σX =. V [X]. 4VQQPTF OPX XF IBWF UXP SBOEPN WBSJBCMFT X BOE Y ɨFJS SFMBUJPOTIJQ DBO CF PCUBJOFE UISPVHI B NFBTVSF DBMMFE DPWBSJBODF PS Cov (X, Y ) *U JT EFmOFE BT Cov (X, Y ) = E [(X − E [X]) (Y − E [Y ])] = E [XY ] − E [X] E [Y ] .. ɨF DPWBSJBODF JUTFMG EPFT OPU DPOWFZ NVDI NFBOJOH DPODFSOJOH UIF SFMBUJPOTIJQ CFUXFFO X BOE Y 5P BDDPNQMJTI UIJT XF DPNQVUF UIF DPSSFMBUJPO DPFċDJFOU GSPN JU 4VQQPTF XF IBWF UXP EJTDSFUF OVNFSJD SBOEPN WBSJBCMFT X BOE Y ɨFO UIF DPSSFMBUJPO DPFċDJFOU ρ (X, Y ) PG X BOE Y JT HJWFO CZ ρ (X, Y ) =. Cov (X, Y ) . V [X] V [Y ]. ɨF DPSSFMBUJPO DPFċDJFOU JT BMXBZT CFUXFFO −1 BOE +1 " WBMVF HSFBUFS UIBO 0 JOEJDBUFT UIF WBSJBCMFT BSF QPTJUJWFMZ DPSSFMBUFE BOE B WBMVF MFTT UIBO 0 JOEJDBUFT UIFZ BSF OFHBUJWFMZ DPSSFMBUFE #Z QPTJUJWFMZ DPSSFMBUFE XF NFBO BT POF JODSFBTFT UIF PUIFS JODSFBTFT XIJMF CZ OFHBUJWFMZ DPSSF. .

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