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30 THI S Y HC 07/2011 - S 62

GII THIU PHNG PHP PHN TCH BAYES Phn 1: Din gii kt qu chn on

Nguyn Vn Tun*

Tm tt: Phng php phn tch d liu c in v ph bin da vo l thuyt phn chng, nhng l mt phng php c nhiu khim khuyt vn c ch ra ngay t khi phng php ny ra i vo khong 100 nm trc. Trong vi thp nin gn y, phng php phn tch Bayes cng ngy cng ph bin v t c nhiu thnh cng ngon mc trong di truyn hc, nghin cu khoa hc v nghin cu lm sng. Bi vit ny gii thiu ng dng phng php phn tch Bayes trong vi trng hp lm sng.

Summary: The traditional frequentist methodology of data analysis relies on the concept of proof by contradiction. This methodology has several flaws and deficiencies which have been pointed out since its inception about 100 years ago. In recent decades, the Bayesian methodology has regained its prominence as an alternative approach to scientific inference and data analysis. The Bayesian methodology has proven a great success in genetics, scientific research, and clinical research. In this article, I introduce the basic concept of Bayesian inference via two cases of clinical diagnosis.

Hai trng hp lm sng Trng hp 1. N, 47 tui, t pht hin mt u

nh bn v tri v ch quan ngi l bu ung th nn n gp bc s. Sau khi khm tng qut v xem xt tin s gia nh, bc s ngh ch i chp nh nh. Kt qu nh nh l dng tnh. Ch mun bit nguy c ch mc bnh ung th v l bao nhiu. Bc s nn tr li nh th no?

Trng hp 2. Nam bnh nhn, 60 tui, sc khe bnh thng, BMI 23 kg/m2, khng c tin cn gia nh vi bnh tiu ng. Tuy nhin kt qu xt nghim glucose trong mu l 127 mg/dL. Theo tiu chun do ADA ngh, ng c chn on l tiu ng, nhng ng khng tin. ng mun bit nguy c tht mnh mc bnh tiu ng. Bc s nn c li khuyn no cho bnh nhn?

C th ni hai trng hp trn rt tiu biu trong lm sng. Trc mt xt nghim nh nh nh vi kt qu dng tnh, v bit rng phng php xt nghim c nhng hn ch v chnh xc, ngi bc s cng nh bnh nhn mun bit kh nng mnh mc bnh l cao hay thp. Tng t, i vi cc kt qu xt nghim khng c gi tr nh phn m l mt dy gi tr nh nng glucose trong mu, v kt qu xt nghim gn ngng chn on bnh, ngi thy thuc phn vn khng bit nn ra mc bnh hay

khng mc bnh, bi v ai cng bit rng bt c xt nghim no cng khng hon ho. Hai trng hp tiu biu trn cng ni ln mt c im ca y hc hin i: l tnh bt nh trong bt c o lng no, bt c xt nghim no, v do bt c chn on no. Ch th m ng t y khoa William Osler tng ni y khoa l mt khoa hc bt nh v mt ngh thut xc sut (medicine is a science of uncertainty and an art of probability).(1)

X l tnh trng bt nh i hi n khoa hc. X l xc sut cn phi c ngh thut. Trong vi nm gn y, mt phng php phn tch mi ra i v ang dn tr thnh ph bin trong nghin cu khoa hc v nghin cu lm sng c th p ng hai nhu cu v khoa hc v ngh thut. Ni l mi nhng trong thc t th khng mi, bi v c s l thuyt ca phng php ny ra i t th k 18. l suy lun theo trng phi Bayes (Bayesian inference) do Thomas Bayes xut vo nm 1763.(2) Thomas Bayes l mt linh mc, nhng cng l mt nh ton hc ti t. Tuy l ti t nhng di sn ca ng li (ch mt bi bo duy nht) lm thay i c th gii khoa hc, thay i cch suy ngh v s bt nh trong khoa hc, v ch ra mt phng php suy lun hon ton logic. Ngy nay, phng php Bayes c ng dng trong hu ht tt c lnh vc khoa hc, k c trong cng ngh thng tin (ng dng Bayes trong vic ngn chn nhng th rc in t), tin lng kinh t, phn tch cc mi quan h x hi, v l gii qui trnh suy ngh ca con ngi. Ngy nay, suy lun theo trng phi Bayes c nhc n trn bo ch i chng ch khng ch trong bo khoa hc. Nhng t bo ln nh New York Times, Economist, Guardian, v.v. u thng xuyn nhc n phng php suy lun Bayes.

Suy lun Bayes da vo nh l Bayes (Bayesian theorem). C th pht biu nh l Bayes theo ngn ng hng ngy nh sau: nhng g chng ta bit l tng hp nhng g chng ta bit cng vi chng c thc t. C th ni rng nh l Bayes th hin cch suy ngh rt ph bin ca tt c chng ta: l chng ta tip thu kin thc theo kiu tch ly. Trong hai trng hp trn, trc khi gi bnh nhn i xt nghim, chng ta bit c kh nng bnh nhn mc bnh nh th no (qua cc thng tin v t l *Vin nghin cu y khoa Garvan Sydney, Australia

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THI S Y HC 07/2011 - S 62 31

hin hnh trong cng ng), sau khi c kt qu xt nghim chng ta c thm chng c thc t, v hai thng tin ny gip cho chng ta nh gi li kh nng mc bnh ca bnh nhn.

nh l Bayes d nhin cng c th m t mt cch n gin qua xc sut. Gi H l bnh trng, v D l chng c (c th l kt qu xt nghim hay d liu), nh l Bayes pht biu rng xc sut H vi iu kin D xy ra k hiu P(H | D) l:

DP

HPHDPDHP || [1]

trong P(H) l kh nng mc bnh trc khi xt

nghim; v P(D | H) l xc sut kt qu dng tnh vi

iu kin c bnh H; P(D) l phn b ca d liu.

Nhn qua nh l trn, chng ta thy suy lun Bayes c 3 thng tin. Thng tin th nht l thng tin m chng ta mun bit, thut ng ting Anh gi l posterior information thng tin hu nh. Thng tin th hai l thng tin chng ta bit, ting Anh l prior information thng tin tin nh. V, thng tin th ba l thng tin thc t, thut ng ting Anh l likelihood. y, thng tin c ngha l kh nng hay xc sut. Chng ta mun bit kh nng bnh nhn tht s mc bnh. Do , 3 yu t trn thng c gi l posterior probability, prior probability, v likelihood, c th th hin qua cng thc chung nh sau:

Xc sut hu nh = Xc sut tin nh + D liu thc t

Chng ta th xt qua 3 thng tin trong cng thc trn mt cch chi tit hn nh sau:

Thng tin tin nh Trc khi thc hin mt cng trnh nghin cu,

chng ta c vi nim v mc nh hng ca mt liu php can thip. Trc khi c kt qu xt nghim, chng ta thng bit kh nng mt c nhn mc bnh cao c no. Nhng thng tin chng ta bit trc nh th c gi l thng tin tin nh. Trong phn tch v suy lun Bayes, c th ni thng tin tin nh ng mt vai tr quan trng. Quan trng l v kt qu phn tch c th thay i ty theo cch chng ta cung cp thng tin tin nh. Quay li hai trng hp t ra trn y, chng ta th xt qua thng tin tin nh.

i vi trng hp 1, thng tin tin nh l kh nng mc bnh. Kh nng ny c th l t l hin hnh (prevalence) ung th v trong cng ng thuc tui ca c nhn. Thng tin tin nh cng

c th l gi tr tin lng qua m hnh Gail.(3) Chng hn nh i vi ph n 47 tui, khng c tin s ung th v, khng c nhng yu t nguy c khc, th s liu dch t hc cho bit t l hin hnh ung th v l khong 1%. Do , chng ta c mt thng tin tin nh: P = 0,01.

Thng tin tin nh trong trng hp 1 cn l chnh xc ca phng php chp nh nh. Thng thng, c hai ch s c th s dng phn nh chnh xc ca mt phng php xt nghim: nhy (sensitivity) v c hiu (specificity). Tht ra, hai thut ng ting Anh ny khng hn thch hp, nhng v gii y khoa s dng qu lu nn chng ta tm chp nhn hai thut ng . Ti s gii thch hai ch s ny nh sau:

nhy l xc sut c kt qu dng tnh nu c nhn tht s mc bnh. Ni cch khc, nhy tr li cu hi: nu 100 ngi mc bnh ung th v u i chp nh nh th c bao nhiu ngi c kt qu dng tnh. Nu phng php nh nh hon ton chnh xc, chng ta k vng tt c 100 ngi s c kt qu dng tnh. Nhng trong thc t, khng c phng php no hon chnh, nn nhy ca nh nh thng khong 90% hoc thp hn (nhng chng ta s lc quan vi 90%).(4)

c hiu l xc sut c kt qu m tnh nu c nhn tht s khng mc bnh. Thng thng c hiu ca nh nh khong 80%. c hiu 80% c th hiu nh sau: nu 100 ngi khng mc bnh u i chp nh nh th s c 80 ngi c kt qu m tnh. Ni cch khc, s c 20 ngi c kt qu dng tnh, v y l trng hp dng tnh gi (false positive).

i vi trng hp 2, thng tin tin nh l s phn b v nng glucose trong cng ng. Trong trng hp 1, thng tin tin nh l t l hin hnh mc bnh ung th v. Nhng trong trng hp 2, thng tin tin nh l s phn b glucose trong cng ng, nht l nhng ngi cng gii tnh v cng tui. Hai thng s phn nh mt phn b l s trung bnh v lch chun (hoc phng sai). Nghin cu ca chng ti (cha cng b) trong cng ng ngi Vit cho thy nam 60 tui, nng glucose trung bnh l 105 mg/dL v phng sai l 860 mg/dL2. C th hnh dung s phn b ca glucose trong cng ng qua biu 1.

Mt thng tin tin nh khc l tin cy ca phng php o lng glucose. Chng ta bit rng nng glucose trong mu dao ng trong mi c nhn ngay trong iu kin bnh thng (khng c can thip sinh hc). Chng hn nh trng hp

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32 THI S Y HC 07/2011 - S 62

bnh nhn c o glucose 5 ln lin tip trong 5 ngy, v kt qu nh sau (mg/dL): 127, 124, 125, 120, 126

Nu ch da vo kt qu ngy th nht v ngy th nm, c nhn ny c phn vo nhm tiu ng (v nng glucose cao hn hay bng 126 mg/dL). Nhng nu da vo kt qu ngy th hai, th ba v th t th c nhn khng c chn on tiu ng. y l mt tnh trng bt nh rt ph bin trong lm sng v chn on. Khc vi s bt nh trong chn on ung th v (khi chnh xc c nh lng bng nhy, c hiu hay t l dng tnh gi), s bt nh trong cc o lng mang tnh lin tc (continuous variable) nh nng glucose c nh lng bng h s tin cy m thut ng ting Anh l coefficient of reliability.

H s tin cy cng ging nh h s tng quan (coefficient of correlation). H s tng quan o lng mc tng quan gia hai bin s. hiu h s tin cy, chng ta cn n khi nim gi tr tht (true values) v gi tr quan st (observed values hay measured values). Chng hn trong trng hp trn, chng ta khng bit nng glucose tht ca c nhn l bao nhiu, m ch bit cc gi tr o c dao ng trong khong 120 n 127 nmg/dL. Tuy khng bit gi tr tht l bao nhiu, chng ta c th c tnh t gi tr o lng c, v c lng s (estimate) n gin nht l gi tr trung bnh. H s tin cy o lng mc tng quan gia gi tr quan st c v gi tr tht. H s tin cy dao ng t 0 (hon ton khng tin cy) n 1 (tin cy tuyt i).

c tnh h s tin cy, ngi ta thng lm nhng nghin cu ngn hn. Trong , mt nhm

c nhn c ly mu v o hn 2 ln (thng l 3 ln). T d liu , c th c tnh phng sai (variance) nng glucose. Phng sai ny thc cht gm c 2 thnh phn: phng sai do dao ng trung bnh trong mi c nhn (vit tt l W) v phng sai do dao ng gia cc cc nhn (B). H s tin cy (vit tt l R) c tnh bng cch ly B chia cho tng phng sai:

WBBR

H s tin cy v c bit l phng sai W l nhng thng tin tin nh rt quan trng trong vic i n mt chn on chnh xc v lu v di. Nghin cu trc y ca chng ti cho thy phng sai W l 815 mg/dL2, v h s tin cy l 0.86.

Tm li, thng tin tin nh ca hai trng hp c th tm lc trong bng s liu sau y: Thng tin tin nh

Trng hp 1 Trng hp 2

V tn s mc bnh hay phn b trong cng ng

Tn s mc bnh l 1%, hay P=0.01

Phn b glucose trong cng ng, vi trung bnh 105 mg/dL v phng sai 860 mg/dL2.

chnh xc v tin cy ca phng php xt nghim

nhy: 0.90 c hiu: 0.80 Dng tnh gi: 0.20

H s tin cy: R=0.86

Phng sai trung bnh trong mi c nhn W = 815

D liu thc t D liu thc t, trong bi cnh hai trng hp

trn, l kt qu xt nghim. i vi trng hp 1 (nghi ng ung th v), kt

qu xt nghim n gin l dng tnh. i vi trng hp 2 (tiu ng) th d liu thc

t c th m t bng lut phn phi. Chng ta bit rng nng glucose tun theo lut phn phi chun (normal distribution). Lut phn phi chun, nh cp trn, c xc nh bng hai thng s: trung bnh v phng sai. Lut phn phi c th p dng m t mt bin s cho mt nhm i tng hay cho mt c nhn.

Trong trng hp 2, chng ta ch c mt kt qu xt nghim glucose vi gi tr 127 mg/dL. Nhng chng ta bit rng nu nng glucose ca c nhn ny c o nhiu ln th chc chn nng khng phi l 127 nhng c th dao ng theo lut phn phi chun. Do , chng ta c th pht biu rng i vi c nhn 2, nng glucose tun theo lut phn phi chun vi trung bnh 127 mg/dL v phng sai 815 mg/dL2 (da vo thng tin tin nh). Cc gi tr glucose ca c nhn ny c th th hin bng biu 2.

0 50 100 150 200

0.00

00.

002

0.00

40.

006

0.00

80.

010

0.01

20.

014

Glucose (mg/dL

Den

sity

Biu 1: Phn b nng glucose trong cng ng nam

gii 60 tui, trung bnh 105 mg/dL v phng sai 860 mg/dL2

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Thng tin hu nh Da vo hai thng tin tin nh v d liu thc

t, chng ta c th tr li cu hi t ra lc ban u. Phng php c thng tin hu nh l ng dng nh l Bayes nh m t trong phn u.

i vi trng hp 1, cn nhc li rng chng ta mun bit xc sut bnh nhn mc bnh ung th v l bao nhiu sau khi c kt qu dng tnh t nh nh. hiu cng thc 1 ti s gii thch bng cch gi nh rng chng ta c mt qun th gm 1000 ph n trong tui 47. Chng ta bit rng (qua thng tin tin nh) trong s 1000 ngi s c 10 ngi mc bnh ung th v, v do 990 ngi khng mc bnh.

Cng qua thng tin tin nh, chng ta bit rng nhy l 90%. Do , trong s 10 ngi mc bnh, s c 9 ngi (10 x 0,90) c kt qu dng tnh (v 1 ngi c kt qu m tnh).

Ngoi ra, v c hiu l 80%. Do , trong s 990 ngi khng mc bnh, s c 792 ngi (990 x 0,80) c kt qu m tnh (v 198 ngi c kt qu dng tnh - y tht ra l dng tnh gi).

Nh vy, tng cng chng ta c 9 + 198 = 207 ngi c kt qu dng tnh. Tuy nhin, trong s ny, ch c 9 ngi tht s mc bnh ung th v, cn 198 ngi khng mc bnh (v do t l dng tnh gi). Do , xc sut m ngi ph n mc bnh ung th v vi iu kin c kt qu dng tnh l: 9 / 207 = 4,3%.

Tht ra, c th c tnh da vo cng thc (1) mt cch nhanh gn hn. Gi nhy l P(D | H); xc sut dng tnh gi l P(D | NoH), NoH l khng mc bnh; v t l hin hnh l P(H), xc sut mc bnh vi kt qu dng tnh l:

NoHDPHPHDPHPHDPHPDHP

|1|||

043.02.010.090.001.0

90.001.0|

DHP

C l nhiu ngi s ngc nhin ti sao xc sut

mc bnh thp. Nhng y, cn ni thm rng

phn ln bc s hiu lm rng nhy l xc sut mc bnh, nn nhiu bc s cho rng xc sut mc bnh l 90%. Nhng cch din gii sai. Sai lm l v c s nhm ln gia hai xc sut. Gi H l bnh ung th v D l kt qu xt nghim dng tnh, nhy c nh ngha l: P(D | H)

Cn xc sut mc bnh vi iu kin c kt qu dng tnh l: P(H | D)

Tuy cch vit ch khc nhau v th t ca H v D, nhng ngha th rt khc nhau! nhy phn nh chnh xc ca phng php nh nh ch khng ni ln xc sut mc bnh ung th v.

Trong trng hp 2, cu hi v thng tin hu nh c phn khc hn trng hp 1. Chng ta mun bit vi kt qu xt nghim glucose trong mu l 127 mg/dL, xc sut m c nhn mc bnh tiu ng l bao nhiu. V 126 mg/dL l ngng chn on tiu ng, nn cu hi c th vit bng ngn ng xc sut nh sau:

P(glucose > 126) = ? tr li cu hi , chng ta cn xc nh lut

phn phi ca glucose cho c nhn. Gi s trung bnh v lch chun ca glucose trong cng ng ln lt l mprior v sprior. Tng t, gi s trung bnh v lch chun ca glucose ca c nhn l mdata v sdata. Qua vi php tnh, c th chng minh rng glucose trung bnh v lch chun ca c nhn l:

22

22

11

dataprior

data

data

prior

prior

posterior

ss

sm

sm

m

50 100 150 200 250

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006

0.00

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0.01

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014

Glucose (mg/dL

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sity

Biu 2: Phn b nng glucose ca mt c nhn 60

tui, trung bnh 127 mg/dL v phng sai 815 mg/dL2

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34 THI S Y HC 07/2011 - S 62

v

2211

1

dataprior

posterior

ss

s

Quay li trng hp 2, xin nhc li, chng ta bit rng trong cng ng, nng glucose tun theo lut phn phi chun vi trung bnh 105 mg/dL v phng sai 860 mg/dL2. Chng ta cng bit nng glucose ca c nhn tun theo lut phn phi chun vi trung bnh 127 mg/dL v phng sai 815 mg/dL2. T hai thng tin ny, chng ta c th xc nh phn phi glucose ca c nhn sau khi iu chnh vi thng tin tin nh.

8151

8601

815127

860105

posteriorm = 116

8151

8601

1posteriors = 20.5

Ni cch khc, nng glucose trung bnh v lu v di (hay nng tht) ca c nhn l 116 mg/dL, nhng c th dao ng trong khong 75.8 n 156.2 mg/dL vi xc sut 95% (tc 1161.9620.5 n 116+1.9620.5). C th hnh dung phn b glucose ca c nhn ny nh Biu 3.

Nhng chng ta mun bit P(glucose > 126). Bi v nng glucose tun theo lut phn phi chun vi trung bnh 116 mg/dL v lch chun 20,5 mg/dL, nn c th m t bng cng thc sau y:

22

5.202116cos

25.201glucose

eglu

ef

Xc sut trn chnh l din tch di ng biu din ca hm s glucose cho c nhn . Theo :

126

glucose126glucose fP = 0,312 Ni cch khc, mc d nng glucose ca c

nhn ny nm trong ngng chn on tiu ng (127 mg/dL), nhng v bit c tin cy ca phng php xt nghim v thng tin tin nh trong cng ng, xc sut m bnh nhn c nng glucose trn 126 mg/dL tht ra ch 31,2%, cha thuyt phc chn on c nhn ny mc bnh tiu ng. Chi tit v l thuyt v cch ng dng c th xem qua vi nghin cu trc ca ngi vit bi ny(5,6) v ng nghip khc.(7-10)

Suy lun Bayes Phng php phn tch thng k ng vai tr rt

quan trng trong nghin cu y khoa, c bit l nghin cu lm sng. Khng c thng k, nghin cu lm sng ch l nhng con s v hn v khng c gi tr khoa hc. Trong thi gian khong 100 nm qua, khoa hc thng k pht trin c rt nhiu phng php phn tch c th ng dng cho rt nhiu tnh hung khc nhau trong nghin cu y hc. Nhng phng php ny da vo trit l phn nghim (falsificationism), m theo nh nghin cu tin hnh 3 bc.

Bc th nht, pht biu gi thuyt v hiu H0; Bc th hai, thu thp d liu D; v Bc th ba, c tnh xc sut D xy ra nu

H0 l tht: P(D | H0). l qui trnh chng minh o ngc (proof by

contradiction) hay cn gi l phn chng. Qui trnh ny c th p dng trong ton hc, nhng khi ng dng vo y hc th tr nn v duyn. Trong y khoa chng ta mun bit vi d liu thu thp c, xc sut gi thuyt H0 tht l bao nhiu, ch khng ai li t cu hi nu gi thuyt H0 l ng th xc sut D xy ra l bao nhiu! Tng t, khng ai ng ngn hi nu ti mc bnh th xc sut kt qu xt nghim dng tnh l bao nhiu (v nu mc bnh th phi iu tr, ch khng hi nh th). Ngi ta hi: nu kt qu xt nghim l dng tnh, kh nng ti mc bnh l bao

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Glucose (mg/dL

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Biu 3: Phn b nng glucose ca mt c nhn 60 tui, trung bnh 116 mg/dL v lch chun 20.5 mg/dL. Din tch di ng biu din (mu ) l xc sut c nhn c nng glucose cao hn 126 mg/dL (tc c

th chn on tiu ng)

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nhiu, ln k hoch i ph. Phng php phn tch c in khng th tr li cu hi ny (m ch tr li cu hi o ngc).

Trong y khoa, bt nh l qui lut ch khng phi l iu bt bnh thng. x l s bt nh, chng ta cn mt phng php phn tch v suy lun logic. V mt nh gi bng chng trong iu kin bt nh, ch c phng php Bayes l phng php logic nht. C th ni rng phng php suy lun Bayes khng phi xa l g vi bc s lm sng. Theo Bayes, kin thc chng ta tip thu qua kin thc bit v d liu thc t. Tng t, trong chn on bnh, xc sut mc bnh ty thuc vo thng tin lin quan n c nhn v kt qu xt nghim. Nu hai c nhn c nhng yu t nguy c ging nhau, nhng c nhn c kt qu dng tnh s c kh nng mc bnh cao hn so vi c nhn c kt qu m tnh. Nhn theo gc ny, suy lun Bayes chnh l mt phng php c nhn ha trong nh lng lm sng v chn on bnh. Do , phng php suy lun Bayes rt thch hp cho cc nh lm sng.

Tm li, suy lun Bayes l mt phng php suy lun da vo thng tin chng ta thu thp trc y cng vi d liu thc t c c tri thc mi hon chnh hn. C th ni l mt qui trnh suy lun tch ly. Tri thc khoa hc l tri thc c tch ly theo thi gian, v phng php Bayes cung cp cho chng ta mt phng tin rt c ch cho s pht trin khoa hc.

Quay li hai trng hp nu ra trong phn u bi vit. i vi trng hp 1, chng ta c th tr li rng kt qu nh nh l dng tnh, v l mt kt qu ng quan tm, nhng cn phi din gii trong iu kin bt nh ca phng php chp nh nh. C 100 ngi c kt qu dng tnh, th c khong 4 ngi tht s b ung th v, nhng bc s khng th chc chn rng c nhn ny s mc bnh. Xc sut 4% c l thp, nhng vn th hin cao gp 4 ln so vi mt ngi ph n trung bnh cng tui. Tht ra, trong thc t, kt qu nh nh ch l mt bc (c th quan trng) trong qui trnh chn on v xt nghim xc nh kh nng mc bnh.

i vi trng hp 2, mc d kt qu xt nghim cho thy nng glucose trong mu l 127 (tc ngng tiu ng), nhng v phng php xt

nghim c tin cy cha cao, nn kt qu t n cng mang tnh bt nh. Sau khi xem xt dao ng trong mi c nhn v kt hp vi thng tin trong cng ng, c th ni rng kh nng c nhn ny tht s b tiu ng ch 31%, thp hn ngng 95% kh xa. Trc kt qu v nh gi ny, c l mt xt nghim khc trong mt thi im khc s cung cp thng tin chc chn hn.

Phng php suy lun v phn tch da vo trng phi Bayes v ang tr nn ph bin trong nghin cu y khoa. Ngy nay, hu nh bt c lnh vc nghin cu no nht l chn on u cn n phng php Bayes. Trong tnh trng bt nh o lng, ngay c vi nhng phng php o lng c chnh xc cao, suy lun Bayes chng t l mt phng tin rt c ch. Hi vng rng bi ny cung cp cho bn c vi nim v cch ng dng phng php suy lun Bayes trong chn on lm sng.

Ti liu tham kho (1) Osler W. Trch trong Clinical ethics: a practical approach to ethical

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