Transcript
Math76 Summer 2020
Introduction to Bayesian Computation
Lecture 3:Bayesian Inference1 July 2020
Recap of Previous Lecture
X Y R R
A B A ER BER
P A prob of event A
RX c A
ProductJoint probability
PCA and B PCAA B PCA B
PLA 1B PCBpCBIA PCA
MarginalPCA PCA B t PCA a B
PCA PCAI B PCB PCA l B PC B
IndependenceJ
A B are independent iff
PCA B PLA PCB
PCALB PCA iff A and B are independent
m X ly m x
BaYes'RpB pcaipgpcpg PCBM.lk
PCA1B prior
RBI evidence
likelihoodBayes Rule
Case Study – Smoking and Lung Cancer
In the early 1950s, an NIH scientist named Jerome Cornfield was faced with an important question:
Question:
Why were so many people dying of lung cancer in 1950?
For more background, see
https://blogs.sas.com/content/iml/2013/03/18/biography-of-jerome-cornfield.html
Formulation
PCHagan smoker Pcc is
P acsanY.FI smoker p Cc I s
PC ClsP PCc pCslc p
PCs pcsldp.cc PCs l c pfc
Pccl sPC
stp.cc P PC slc pccpCns1c pcc pC sl c pGc
The Data
o
r atotal total Of
of cancerpatients i healthy patients
interviewed interviewed
Likelihood Function – Breakout Exercise
Instructions:
1. As always, everyone should introduce themselves. To helpbreak the ice, tell everyone the probability that you’d havepancakes if you had to choose between pancakes and wa✏es.
2. As a group, use the Cornfield data on the previous slide todevelop a likelihood function. (i.e. P(S |C ) and P(¬S |C )).
3. Do you see any downfalls of using the data this way?
Likelihood Function
PL s I c z ttsipatientategarytotal in patient category
5437 165543T Z G 97
nonsmokers in cancer patien catPC Slc
total in patient cat
165543 I 0.03
Prior ProbabilitiesLung Cancer mortality rate in 1950: 40.1
100000
Chart courtesy of
https://canceratlas.cancer.org/the-burden/lung-cancer/
P c
PC c
o
Posterior Probabilities
p.cc s PCSlc p
pCslc pCc tpCsl c pCnc
0.97 1040 00
7949 14010.97 l ooo
p Slac 794,9491401 ol
total in control
pccls
Results
More history at
https://www.cdc.gov/tobacco/data˙statistics/sgr/history/index.htm
Yes lung cancer is partiallycaused by smoking
Sequential Updates
Let's say I have 2 observations B and C
Bayes rule gives me
P A 1B cPCB CIA pea
I can then split the likelihood into two parts usingthe product rule
PCAI B cPCalB PCB
collecting terms we get
PCA1B c PCctAB2pCBlAjpcaNfnteioesajheatasthIsneisposxera.r
PLA1Bobtained in
PCCIB PCB heusual form of Baye's
PCCIA B p AIB ruleThis is like Bayes rule using
pcc I gPCAIB as the prnor whenincorporating C
More on this next week
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