Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Features
Speaker Deck
PRO
Sign in
Sign up for free
Search
Search
Bayesian Statistical Analysis: A Gentle Introduction
Search
Chris Fonnesbeck
December 05, 2011
Research
4
560
Bayesian Statistical Analysis: A Gentle Introduction
Get to know the Reverend Bayes.Reverend
Chris Fonnesbeck
December 05, 2011
Tweet
Share
More Decks by Chris Fonnesbeck
See All by Chris Fonnesbeck
Statistical Thinking for Data Science
fonnesbeck
5
930
Structured Decision-making and Adaptive Management For The Control Of Infectious Disease
fonnesbeck
3
100
Estimating Microbial Diversity
fonnesbeck
0
110
Other Decks in Research
See All in Research
People Driven Transformation / 人が起点の、社会の変え方
dmattsun
0
150
Accurate Method and Variable Tracking in Commit History
tsantalis
0
230
Generative Spoken Dialogue Language Modeling [対話論文読み会@電通大]
yuta0306
1
130
精神疾患患者のアクティビティデータを利用したリハビリテーションのためのシステムに関する研究
comfortdesignlab
0
140
20240127_熊本から今いちど真面目に都市交通~めざせ「車1割削減、渋滞半減、公共交通2倍」~ 全国路面電車サミット2024宇都宮
trafficbrain
1
650
SSII2023 医療支援における画像処理研究の動向と展望
moda0
0
100
センサデータを活用した 肌質改善への支援システムに関する研究
comfortdesignlab
0
150
Source Code Diff Revolution (JetBrains Open Reading Club)
tsantalis
0
240
説明可能AI:代表的手法と最近の動向
yuyay
1
570
第4回ナレッジグラフ勉強会:ISWC2023論文読み会
kg_wakate
1
200
FMP L3 Year 1 Project Proposal
haiinya
0
150
AIを前提とした体験の実現に向けて/toward_ai_based_experiences
monochromegane
1
220
Featured
See All Featured
Let's Do A Bunch of Simple Stuff to Make Websites Faster
chriscoyier
501
140k
No one is an island. Learnings from fostering a developers community.
thoeni
14
2.1k
RailsConf & Balkan Ruby 2019: The Past, Present, and Future of Rails at GitHub
eileencodes
124
32k
A better future with KSS
kneath
230
16k
Java REST API Framework Comparison - PWX 2021
mraible
PRO
18
6.9k
Designing with Data
zakiwarfel
95
4.8k
CoffeeScript is Beautiful & I Never Want to Write Plain JavaScript Again
sstephenson
153
14k
A Tale of Four Properties
chriscoyier
150
22k
Pencils Down: Stop Designing & Start Developing
hursman
115
11k
Building Effective Engineering Teams - LeadDev
addyosmani
26
1.8k
The Cost Of JavaScript in 2023
addyosmani
13
3.8k
The Cult of Friendly URLs
andyhume
73
5.6k
Transcript
Bayesian Statistical Analysis A Gentle Introduction Center for Quantitative Sciences
Workshop 18 November 2011 Christopher J. Fonnesbeck Monday, December 5, 11
What is Bayesian Inference? Monday, December 5, 11
Practical methods for making inferences from data using probability models
for quantities we observe and about which we wish to learn. Gelman et al., 2004 Monday, December 5, 11
Rev. Thomas Bayes Monday, December 5, 11
Rev. Thomas Bayes Simon Laplace Monday, December 5, 11
Conclusions in terms of probability statements p( |y) unknowns observations
Monday, December 5, 11
Classical inference conditions on unknown parameter p(y| ) unknowns observations
Monday, December 5, 11
Classical vs Bayesian Statistics Monday, December 5, 11
Frequentist Monday, December 5, 11
Frequentist observations random Monday, December 5, 11
Frequentist model, parameters fixed Monday, December 5, 11
Frequentist Inference Monday, December 5, 11
Choose an estimator ˆ µ = P xi n based
on frequentist (asymptotic) criteria Monday, December 5, 11
Choose a test statistic based on frequentist (asymptotic) criteria t
= ¯ x µ s/ p n Monday, December 5, 11
Bayesian Monday, December 5, 11
Bayesian observations fixed Monday, December 5, 11
Bayesian model, parameters “random” Monday, December 5, 11
Components of Bayesian Statistics Monday, December 5, 11
Specify full probability model 1 Pr(y| )Pr( |⇥)Pr(⇥) Monday, December
5, 11
data y Monday, December 5, 11
data y covariates X Monday, December 5, 11
data y covariates X parameters ✓ Monday, December 5, 11
data y covariates X parameters ✓ missing data ˜ y
Monday, December 5, 11
2 Calculate posterior distribution Pr( |y) Monday, December 5, 11
3Check model for lack of fit Monday, December 5, 11
Why Bayes? ? Monday, December 5, 11
“... the Bayesian approach is attractive because it is useful.
Its usefulness derives in large measure from its simplicity. Its simplicity allows the investigation of far more complex models than can be handled by the tools in the classical toolbox.” Link and Barker (2010) Monday, December 5, 11
coherence X ˜ y y ✓ Monday, December 5, 11
Interpretation Monday, December 5, 11
Pr( ¯ Y 1.96 ⇥ ⇥ n < µ <
¯ Y + 1.96 ⇥ ⇥ n ) = 0.95 Confidence Interval Pr(a(Y ) < ✓ < b(Y )|✓) = 0.95 Monday, December 5, 11
Credible Interval Pr(a(y) < ✓ < b(y)|Y = y) =
0.95 Monday, December 5, 11
Uncertainty Monday, December 5, 11
C alpha N z b_psi beta a_psi pi mu psi
Ntotal occupied a b Ndist psi z alpha pi N beta mu occupied N alpha beta N alpha beta Complex Models Monday, December 5, 11
Probability Monday, December 5, 11
Pr(A) = m n A = an event of interest
m = no. of favourable outcomes n = total no. of possible outcomes (1) classical Monday, December 5, 11
all elementary events are equally likely Monday, December 5, 11
Pr(A) = lim n→∞ m n n = no. of
identical and independent trials m = no. of times A has occurred (2) frequentist Monday, December 5, 11
Between 1745 and 1770 there were 241,945 girls and 251,527
boys born in Paris Monday, December 5, 11
A = “Chris has Type A blood” Monday, December 5,
11
A = “Titans will win Superbowl XLVI” Monday, December 5,
11
A = “The prevalence of diabetes in Nashville is >
0.15” Monday, December 5, 11
(3) subjective Pr(A) Monday, December 5, 11
Measure of one’s uncertainty regarding the occurrence of A Pr(A)
Monday, December 5, 11
Pr(A|H) Monday, December 5, 11
A = “It is raining in Atlanta” Monday, December 5,
11
Pr(A|H) = 0.5 Monday, December 5, 11
Pr( A|H ) = ⇢ 0 . 4 if raining
in Nashville 0 . 25 otherwise Monday, December 5, 11
Pr(A|H) = 1, if raining 0, otherwise Monday, December 5,
11
S A Pr(A) = area of A area of S
Monday, December 5, 11
S A B A ∩ B Pr(A ⇥ B) =
Pr(A) + Pr(B) Pr(A ⇤ B) Monday, December 5, 11
A A ∩ B Pr(B|A) = Pr(A B) Pr(A) Monday,
December 5, 11
A A ∩ B conditional probability Pr(B|A) = Pr(A B)
Pr(A) Monday, December 5, 11
Independence Pr(B|A) = Pr(B) Monday, December 5, 11
S A B A ∩ B Pr(B|A) = Pr(A B)
Pr(A) Monday, December 5, 11
S A B A ∩ B Pr(A|B) = Pr(A B)
Pr(B) Pr(B|A) = Pr(A B) Pr(A) Monday, December 5, 11
Pr(A B) = Pr(A|B)Pr(B) = Pr(B|A)Pr(A) Monday, December 5, 11
Bayes Theorem Pr(B|A) = Pr(A|B)Pr(B) Pr(A) Monday, December 5, 11
Bayes Theorem Pr( |y) = Pr(y| )Pr( ) Pr(y) Posterior
Probability Prior Probability Likelihood of Observations Normalizing Constant Monday, December 5, 11
Bayes Theorem Pr( |y) = Pr(y| )Pr( ) R Pr(y|
)Pr( )d Monday, December 5, 11
“proportional to” Pr( |y) Pr(y| )Pr( ) Monday, December 5,
11
Pr( |y) Pr(y| )Pr( ) Posterior Prior Likelihood Monday, December
5, 11
information p( |y) p(y| )p( ) Monday, December 5, 11
“Following observation of , the likelihood contains all experimental information
from about the unknown .” θ y y L(✓|y) Monday, December 5, 11
binomial model data parameter sampling distribution of X p(X|✓) =
✓ N n ◆ ✓x (1 ✓)N x Monday, December 5, 11
binomial model likelihood function for θ L(✓|X) = ✓ N
n ◆ ✓x (1 ✓)N x Monday, December 5, 11
prior distribution p(θ|y) ∝ p(y|θ)p(θ) Monday, December 5, 11
Prior as population distribution Monday, December 5, 11
Monday, December 5, 11
Prior as information state Monday, December 5, 11
Monday, December 5, 11
All plausible values Monday, December 5, 11
Between 1745 and 1770 there were 241,945 girls and 251,527
boys born in Paris Monday, December 5, 11
Bayesian analysis is subjective Monday, December 5, 11
Statistical analysis is subjective Monday, December 5, 11
“... all forms of statistical inference make assumptions, assumptions which
can only be tested very crudely and can almost never be verified.” - Robert E. Kass Monday, December 5, 11
3 Model checking Monday, December 5, 11
1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 x
p(x) separation Monday, December 5, 11
source: Gelman et al. 2008 Monday, December 5, 11
weakly-informative prior -4 -2 0 2 4 0.0 0.1 0.2
0.3 0.4 xrange Pr(x) Monday, December 5, 11
source: Gelman et al. 2008 Monday, December 5, 11
example: genetic probabilities Monday, December 5, 11
X-linked recessive Monday, December 5, 11
Monday, December 5, 11
affected carrier no gene unknown Woman Husband Brother Mother is
the woman a carrier? Monday, December 5, 11
Pr(θ = 1) = Pr(θ = 0) = 1 2
Pr(θ = 1) Pr(θ = 0) = 1 prior odds Monday, December 5, 11
affected carrier no gene unknown Woman Husband Brother Son Son
Mother Monday, December 5, 11
Pr(y1 = 0, y2 = 0|θ = 1) = (0.5)(0.5)
= 0.25 Monday, December 5, 11
Pr(y1 = 0, y2 = 0|θ = 1) = (0.5)(0.5)
= 0.25 Pr(y1 = 0, y2 = 0|θ = 0) = 1 Monday, December 5, 11
Pr(y1 = 0, y2 = 0|θ = 1) = (0.5)(0.5)
= 0.25 Pr(y1 = 0, y2 = 0|θ = 0) = 1 “likelihood ratio” p(y1 = 0, y2 = 0|θ = 1) p(y1 = 0, y2 = 0|θ = 0) = 0.25 1 = 1/4 Monday, December 5, 11
what about Mom? Monday, December 5, 11
what about Mom? y = {y1 = 0, y2 =
0} Pr( = 1|y) = Pr(y| = 1)Pr( = 1) Pr(y) = Pr(y| = 1)Pr( = 1) P ✓ Pr(y| )Pr( ) Monday, December 5, 11
y = {y1 = 0, y2 = 0} Monday, December
5, 11
Pr( = 1|y) = p(y| = 1)Pr( = 1) p(y|
= 1)Pr( = 1) + p(y| = 0)Pr( = 0) y = {y1 = 0, y2 = 0} Monday, December 5, 11
Pr( = 1|y) = p(y| = 1)Pr( = 1) p(y|
= 1)Pr( = 1) + p(y| = 0)Pr( = 0) = (0.25)(0.5) (0.25)(0.5) + (1.0)(0.5) = 0.125 0.625 = 0.2 y = {y1 = 0, y2 = 0} Monday, December 5, 11
3rd unaffected son? Pr( = 1|y3 ) = (0.5)(0.2) (0.5)(0.2)
+ (1)(0.8) = 0.111 posterior from previous Monday, December 5, 11
Hierarchical Models Monday, December 5, 11
effectiveness of cardiac surgery example Monday, December 5, 11
Hospital Operations Deaths A 47 0 B 148 18 C
119 8 D 810 46 E 211 8 F 196 13 G 148 9 H 215 31 I 207 14 J 97 8 K 256 29 L 360 24 Monday, December 5, 11
clustering induces dependence between observations Monday, December 5, 11
parameters sampled from common distribution j hospital j survival rate
Monday, December 5, 11
population distribution j f(⇥) hyperparameters Monday, December 5, 11
θ1 θ2 θk y1 y2 yk ... ... deaths parameters
Monday, December 5, 11
θ1 θ2 θk y1 y2 yk ... ... deaths parameters
µ, σ2 hyperparameters Monday, December 5, 11
, ϕµ ϕσ θ1 θ2 θk y1 y2 yk ...
... deaths parameters µ, σ2 hyperparameters Monday, December 5, 11
non-hierarchical models of hierarchical data can easily be underfit or
overfit Monday, December 5, 11
“experiments” j = 1, . . . , J likelihood
∼ Binomial( , ) deaths j operations j θj logit( ) ∼ N(µ, ) θi σ2 population model µ ∼ , ∼ Pµ σ2 Pσ priors Monday, December 5, 11
0/47 = 0 18/148 = 0.12 8/119 = 0.07 46/810
= 0.06 Monday, December 5, 11
Monday, December 5, 11
Monday, December 5, 11