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 Introdu...
Search
Chris Fonnesbeck
December 05, 2011
Research
4
620
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
1.1k
Structured Decision-making and Adaptive Management For The Control Of Infectious Disease
fonnesbeck
3
110
Estimating Microbial Diversity
fonnesbeck
0
120
Other Decks in Research
See All in Research
TRIPOD+AI Expandedチェックリスト 有志翻訳による日本語版 version.1.1
shuntaros
0
120
NeurIPS 2024 参加報告 & 論文紹介 (SACPO, Ctrl-G)
reisato12345
0
410
打率7割を実現する、プロダクトディスカバリーの7つの極意(pmconf2024)
geshi0820
0
420
セミコン地域における総合交通戦略
trafficbrain
0
130
コーパスを丸呑みしたモデルから言語の何がわかるか
eumesy
PRO
11
3.4k
【NLPコロキウム】Stepwise Alignment for Constrained Language Model Policy Optimization (NeurIPS 2024)
akifumi_wachi
3
570
ラムダ計算の拡張に基づく 音楽プログラミング言語mimium とそのVMの実装
tomoyanonymous
0
440
DeepSeek を利用する上でのリスクと安全性の考え方
schroneko
3
1.3k
[論文紹介] iTransformer: Inverted Transformers Are Effective for Time Series Forecasting
shiba4839
0
110
Building Height Estimation Using Shadow Length in Satellite Imagery
satai
3
280
データサイエンティストの採用に関するアンケート
datascientistsociety
PRO
0
190
Pix2Poly: A Sequence Prediction Method for End-to-end Polygonal Building Footprint Extraction from Remote Sensing Imagery
satai
3
180
Featured
See All Featured
Building an army of robots
kneath
304
45k
Why Our Code Smells
bkeepers
PRO
336
57k
The Pragmatic Product Professional
lauravandoore
33
6.5k
Building a Scalable Design System with Sketch
lauravandoore
462
33k
Embracing the Ebb and Flow
colly
85
4.6k
Designing Experiences People Love
moore
141
23k
Product Roadmaps are Hard
iamctodd
PRO
52
11k
実際に使うSQLの書き方 徹底解説 / pgcon21j-tutorial
soudai
177
52k
Making Projects Easy
brettharned
116
6.1k
How to Think Like a Performance Engineer
csswizardry
22
1.5k
RailsConf & Balkan Ruby 2019: The Past, Present, and Future of Rails at GitHub
eileencodes
135
33k
Optimizing for Happiness
mojombo
377
70k
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