machine learning summer school 2016

data science @ NYT

Chris Wiggins

Aug 8/9, 2016

Outline

1. overview of DS@NYT

Outline

1. overview of DS@NYT2. prediction + supervised learning

Outline

1. overview of DS@NYT2. prediction + supervised learning3. prescription, causality, and RL

Outline

1. overview of DS@NYT2. prediction + supervised learning3. prescription, causality, and RL4. description + inference

Outline

1. overview of DS@NYT2. prediction + supervised learning3. prescription, causality, and RL4. description + inference5. (if interest) designing data products

0. Thank the organizers!

Figure 1: prepping slides until last minute

Lecture 1: overview of ds@NYT

data science @ The New York Times

chris.wiggins@columbia.edu

chris.wiggins@nytimes.com

@chrishwiggins

references: http://bit.ly/stanf16

data science: searches

data science: mindset & toolset

drew conway, 2010

modern history:

biology: 1892 vs. 1995

biology changed for good.

biology: 1892 vs. 1995

new toolset, new mindset

genetics: 1837 vs. 2012

ML toolset; data science mindset

genetics: 1837 vs. 2012

ML toolset; data science mindset

arxiv.org/abs/1105.5821 ; github.com/rajanil/mkboost

data science: mindset & toolset

news: 20th century

church state

church

news: 20th century

church state

news: 21st century

church state

1851 1996

newspapering: 1851 vs. 1996

example:

millions of views per hour2015

"...social activities generate large quantities of potentially valuable data...The data were not generated for the

purpose of learning; however, the potential for learning is great’’

"...social activities generate large quantities of potentially valuable data...The data were not generated for the

purpose of learning; however, the potential for learning is great’’ - J Chambers, Bell Labs,1993, “GLS”

data science: the web

is your “online presence”

is a microscope

is an experimental tool

is an optimization tool

1851 1996

newspapering: 1851 vs. 1996 vs. 2008

“a startup is a temporary organization in search of a

repeatable and scalable business model” —Steve Blank

every publisher is now a startup

news: 21st century

church state

news: 21st century

church state

learnings

- predictive modeling- descriptive modeling- prescriptive modeling

(actually ML, shhhh…)

- (supervised learning)- (unsupervised learning)- (reinforcement learning)

h/t michael littman

Supervised

Learning

Reinforcement

Learning

Unsupervised

Learning

(reports)

learnings

- predictive modeling- descriptive modeling- prescriptive modeling

cf. modelingsocialdata.org

stats.stackexchange.com

from “are you a bayesian or a frequentist”

—michael jordan

ϕ (yif(xi;β)) + λ||β||

predictive modeling, e.g.,

“the funnel”

interpretable predictive modeling

super cool stuff

interpretable predictive modeling

super cool stuff

arxiv.org/abs/q-bio/0701021

optimization & learning, e.g.,

“How The New York Times Works “popular mechanics, 2015

optimization & prediction, e.g.,

(some models)

(some moneys)

“newsvendor problem,” literally (+prediction+experiment)

recommendation as inference

bit.ly/AlexCTM

descriptive modeling, e.g,

“segments”

argmax_z p(z|x)=14

“segments”

“baby boomer”

- descriptive data product

cf. daeilkim.com

cf. daeilkim.com ; import bnpy

modeling your audience

bit.ly/Hughes-Kim-Sudderth-AISTATS15

(optimization, ultimately)

also allows insight+targeting as inference

prescriptive modeling

“off policy value estimation”

(cf. “causal effect estimation”)

cf. Langford `08-`16;

Horvitz & Thompson `52;

Holland `86

“off policy value estimation”

(cf. “causal effect estimation”)

Vapnik’s razor

“ When solving a (learning) problem of interest,

do not solve a more complex problem as an

intermediate step.”

aka “A/B testing”;

Reporting

Learning

aka “A/B

testing”;

business as

predictive)

Some of the most recognizable personalization in our service is the collection of “genre” rows. …Members connect with these rows so

well that we measure an increase in member retention by placing the most tailored rows higher on the page instead of lower.

prescriptive modeling: from A/B to….

real-time A/B -> “bandits”

GOOG blog:

prescriptive modeling, e.g,

leverage methods which are predictive yet performant

NB: data-informed, not data-driven

predicting views/cascades: doable?

KDD 09: how many people are online?

WWW 14: FB shares

predicting views/cascades: features?

WWW 16: TWIT RT’s

Reporting

Learning

Optimizing

Exploredescriptive:

predictive:

prescriptive:

Reporting

Learning

Optimizing

Exploredescriptive:

predictive:

prescriptive:

things:

what does DS team deliver?

- build data product- build APIs- impact roadmaps

chris.wiggins@columbia.edu

chris.wiggins@nytimes.com

@chrishwiggins

references: http://bit.ly/stanf16

Lecture 2: predictive modeling @ NYT

desc/pred/pres

Figure 2: desc/pred/pres

caveat: difference between observation and experiment. why?

blossom example

Figure 3: Reminder: Blossom

blossom + boosting (‘exponential’)

Figure 4: Reminder: Surrogate Loss Functions

tangent: logistic function as surrogate loss function

define f (x) ≡ log p(y = 1|x)/p(y = −1|x) ∈ R

p(y = 1|x) + p(y = −1|x) = 1→ p(y |x) = 1/(1 + exp(−yf ))

− log2 p(yN1 ) =∑

i log2

1 + e−yi f (xi ))

≡∑

i ℓ(yi f (xi))

p(y = 1|x) + p(y = −1|x) = 1→ p(y |x) = 1/(1 + exp(−yf ))

− log2 p(yN1 ) =∑

i log2

1 + e−yi f (xi ))

≡∑

i ℓ(yi f (xi))

ℓ′′ > 0, ℓ(µ) > 1[µ < 0] ∀µ ∈ R.

p(y = 1|x) + p(y = −1|x) = 1→ p(y |x) = 1/(1 + exp(−yf ))

− log2 p(yN1 ) =∑

i log2

1 + e−yi f (xi ))

≡∑

i ℓ(yi f (xi))

ℓ′′ > 0, ℓ(µ) > 1[µ < 0] ∀µ ∈ R.

∴ maximizing log-likelihood is minimizing a surrogate convexloss function for classification (though not strongly convex,cf. Yoram’s talk)

p(y = 1|x) + p(y = −1|x) = 1→ p(y |x) = 1/(1 + exp(−yf ))

− log2 p(yN1 ) =∑

i log2

1 + e−yi f (xi ))

≡∑

i ℓ(yi f (xi))

ℓ′′ > 0, ℓ(µ) > 1[µ < 0] ∀µ ∈ R.

∴ maximizing log-likelihood is minimizing a surrogate convexloss function for classification (though not strongly convex,cf. Yoram’s talk)

but∑

i log2

1 + e−yi wT h(xi )

not as easy as∑

i e−yi wT h(xi )

boosting 1

L exponential surrogate loss function, summed over examples:

L[F ] =∑

i exp (−yiF (xi))

boosting 1

L[F ] =∑

i exp (−yiF (xi)) =

i exp(

∑tt′ wt′ht′(xi)

≡ Lt(wt)

boosting 1

L[F ] =∑

i exp(

≡ Lt(wt) Draw ht ∈ H large space of rules s.t. h(x) ∈ −1, +1

boosting 1

L[F ] =∑

i exp(

≡ Lt(wt) Draw ht ∈ H large space of rules s.t. h(x) ∈ −1, +1 label y ∈ −1, +1

boosting 1

Lt+1(wt ; w) ≡∑

i d ti exp (−yiwht+1(xi))

Punchlines: sparse, predictive, interpretable, fast (to execute), andeasy to extend, e.g., trees, flexible hypotheses spaces, L1, L∞

1, . . .

boosting 1

Lt+1(wt ; w) ≡∑

y=h′ d ti e−w +

y 6=h′ d ti e+w ≡ e−w D+ + e+w D−

1, . . .

boosting 1

Lt+1(wt ; w) ≡∑

y=h′ d ti e−w +

y 6=h′ d ti e+w ≡ e−w D+ + e+w D−

∴ wt+1 = argminw Lt+1(w) = (1/2) log D+/D−

1, . . .

boosting 1

Lt+1(wt ; w) ≡∑

y=h′ d ti e−w +

y 6=h′ d ti e+w ≡ e−w D+ + e+w D−

Lt+1(wt+1) = 2√

D+D− = 2√

ν+(1− ν+)/D, where0 ≤ ν+ ≡ D+/D = D+/Lt ≤ 1

1, . . .

boosting 1

Lt+1(wt ; w) ≡∑

y=h′ d ti e−w +

y 6=h′ d ti e+w ≡ e−w D+ + e+w D−

Lt+1(wt+1) = 2√

D+D− = 2√

ν+(1− ν+)/D, where0 ≤ ν+ ≡ D+/D = D+/Lt ≤ 1

update example weights d t+1i = d t

i e∓w

1, . . .

1Duchi + Singer “Boosting with structural sparsity” ICML ’09

predicting people

“customer journey” prediction

predicting people

fun covariates

predicting people

fun covariates observational complication v structural models

predicting people (reminder)

Figure 5: both in science and in real world, feature analysis guides futureexperiments

single copy (reminder)

Figure 6: from Lecture 1

example in CAR (computer assisted reporting)

Figure 7: Tabuchi article

cf. Friedman’s “Statistical models and Shoe Leather”2

2Freedman, David A. “Statistical models and shoe leather.” Sociologicalmethodology 21.2 (1991): 291-313.

Takata airbag fatalities

Takata airbag fatalities 2219 labeled3 examples from 33,204 comments

3By Hiroko Tabuchi, a Pulitzer winner

Takata airbag fatalities 2219 labeled3 examples from 33,204 comments cf. Box’s “Science and Statistics”4

3By Hiroko Tabuchi, a Pulitzer winner4Science and Statistics, George E. P. Box Journal of the American Statistical

Association, Vol. 71, No. 356. (Dec., 1976), pp. 791-799.

computer assisted reporting

Impact

Figure 8: impact

Lecture 3: prescriptive modeling @ NYT

the natural abstraction

operators5 make decisions

5In the sense of business deciders; that said, doctors, including those whooperate, also have to make decisions, cf., personalized medicines

operators5 make decisions faster horses v. cars

operators5 make decisions faster horses v. cars general insights v. optimal policies

maximizing outcome

the problem: maximizing an outcome over policies. . .

maximizing outcome

the problem: maximizing an outcome over policies. . . . . . while inferring causality from observation

maximizing outcome

the problem: maximizing an outcome over policies. . . . . . while inferring causality from observation different from predicting outcome in absence of action/policy

examples

observation is not experiment

examples

e.g., (Med.) smoking hurts vs unhealthy people smoke

examples

e.g., (Med.) smoking hurts vs unhealthy people smoke e.g., (Med.) affluent get prescribed different meds/treatment

examples

e.g., (Med.) smoking hurts vs unhealthy people smoke e.g., (Med.) affluent get prescribed different meds/treatment e.g., (life) veterans earn less vs the rich serve less6

6Angrist, Joshua D. (1990). “Lifetime Earnings and the Vietnam DraftLottery: Evidence from Social Security Administrative Records”. AmericanEconomic Review 80 (3): 313–336.

examples

e.g., (Med.) smoking hurts vs unhealthy people smoke e.g., (Med.) affluent get prescribed different meds/treatment e.g., (life) veterans earn less vs the rich serve less6

e.g., (life) admitted to school vs learn at school?

6Angrist, Joshua D. (1990). “Lifetime Earnings and the Vietnam DraftLottery: Evidence from Social Security Administrative Records”. AmericanEconomic Review 80 (3): 313–336.

reinforcement/machine learning/graphical models

key idea: model joint p(y , a, x)

key idea: model joint p(y , a, x) explore/exploit: family of joints pα(y , a, x)

key idea: model joint p(y , a, x) explore/exploit: family of joints pα(y , a, x) “causality”: pα(y , a, x) = p(y |a, x)pα(a|x)p(x) “a causes y”

key idea: model joint p(y , a, x) explore/exploit: family of joints pα(y , a, x) “causality”: pα(y , a, x) = p(y |a, x)pα(a|x)p(x) “a causes y” nomenclature: ‘response’, ‘policy’/‘bias’, ‘prior’ above

in general

Figure 9: policy/bias, response, and prior define the distribution

also describes both the ‘exploration’ and ‘exploitation’ distributions

randomized controlled trial

Figure 10: RCT: ‘bias’ removed, random ‘policy’ (response and priorunaffected)

also Pearl’s ‘do’ distribution: a distribution with “no arrows”pointing to the action variable.

POISE: calculation, estimation, optimization

POISE: “policy optimization via importance sample estimation”

POISE: “policy optimization via importance sample estimation” Monte Carlo importance sampling estimation

aka “off policy estimation”

aka “off policy estimation” role of “IPW”

reduction

reduction normalization

reduction normalization hyper-parameter searching

reduction normalization hyper-parameter searching unexpected connection: personalized medicine

POISE setup and Goal

“a causes y” ⇐⇒ ∃ family pα(y , a, x) = p(y |a, x)pα(a|x)p(x)

“a causes y” ⇐⇒ ∃ family pα(y , a, x) = p(y |a, x)pα(a|x)p(x) define off-policy/exploration distribution

p−(y , a, x) = p(y |a, x)p−(a|x)p(x)

p−(y , a, x) = p(y |a, x)p−(a|x)p(x) define exploitation distribution

p+(y , a, x) = p(y |a, x)p+(a|x)p(x)

p+(y , a, x) = p(y |a, x)p+(a|x)p(x) Goal: Maximize E+(Y ) over p+(a|x) using data drawn from

p−(y , a, x).

notation: x , a, y ∈ X , A, Y i.e., Eα(Y ) is not a function of y

POISE math: IS+Monte Carlo estimation=ISE

i.e, “importance sampling estimation”

E+(Y ) ≡∑

yax yp+(y , a, x)

E+(Y ) ≡∑

yax yp+(y , a, x) E+(Y ) =

yax yp−(y , a, x)(p+(y , a, x)/p−(y , a, x))

E+(Y ) ≡∑

yax yp+(y , a, x) E+(Y ) =

yax yp−(y , a, x)(p+(y , a, x)/p−(y , a, x)) E+(Y ) =

yax yp−(y , a, x)(p+(a|x)/p−(a|x))

E+(Y ) ≡∑

yax yp+(y , a, x) E+(Y ) =

yax yp−(y , a, x)(p+(a|x)/p−(a|x)) E+(Y ) ≈ N−1 ∑

i yi(p+(ai |xi)/p−(ai |xi))

E+(Y ) ≡∑

yax yp+(y , a, x) E+(Y ) =

E+(Y ) ≡∑

yax yp+(y , a, x) E+(Y ) =

let’s spend some time getting to know this last equation, theimportance sampling estimate of outcome in a “causal model” (“acauses y”) among y , a, x

Observation (cf. Bottou7 )

factorizing P±(x): P+(x)P−(x) = Πfactors

P+but not−(x)P−but not+(x)

origin: importance sampling Eq(f ) = Ep(fq/p) (as invariational methods)

the “causal” model pα(y , a, x) = p(y |a, x)pα(a|x)p(x) helpshere

factors left over are numerator (p+(a|x), to optimize) anddenominator (p−(a|x), to infer if not a RCT)

unobserved confounders will confound us (later)

7Counterfactual Reasoning and Learning Systems, arXiv:1209.2355

Reduction (cf. Langford8,9,10 (’05, ’08, ’09 ))

consider numerator for deterministic policy:p+(a|x) = 1[a = h(x)]

E+(Y ) ∝∑

i(yi/p−(a|x))1[a = h(x)] ≡∑

i wi1[a = h(x)]

E+(Y ) ∝∑

i(yi/p−(a|x))1[a = h(x)] ≡∑

i wi1[a = h(x)] Note: 1[c = d ] = 1− 1[c 6= d ]

E+(Y ) ∝∑

i(yi/p−(a|x))1[a = h(x)] ≡∑

i wi1[a = h(x)] Note: 1[c = d ] = 1− 1[c 6= d ] ∴ E+(Y ) ∝ constant−

i wi1[a 6= h(x)]

E+(Y ) ∝∑

i(yi/p−(a|x))1[a = h(x)] ≡∑

i wi1[a = h(x)] Note: 1[c = d ] = 1− 1[c 6= d ] ∴ E+(Y ) ∝ constant−

i wi1[a 6= h(x)] ∴ reduces policy optimization to (weighted) classification

8Langford & Zadrozny “Relating Reinforcement Learning Performance toClassification Performance” ICML 2005

9Beygelzimer & Langford “The offset tree for learning with partial labels”(KDD 2009)

10Tutorial on “Reductions” (including at ICML 2009)

Reduction w/optimistic complication

Prescription ⇐⇒ classification L =∑

i wi1[ai 6= h(xi)]

i wi1[ai 6= h(xi)] weight wi = yi/p−(ai |xi), inferred or RCT

i wi1[ai 6= h(xi)] weight wi = yi/p−(ai |xi), inferred or RCT destroys measure by treating p−(a|x) differently than

1/p−(a|x)

normalize as L ≡

iy1[ai 6=h(xi )]/p−(ai |xi )

i1[ai 6=h(xi )]/p−(ai |xi )

1/p−(a|x)

normalize as L ≡

destroys lovely reduction

1/p−(a|x)

normalize as L ≡

destroys lovely reduction simply11 L(λ) =

i(yi − λ)1[ai 6= h(xi)]/p−(ai |xi)

11Suggestion by Dan Hsu

1/p−(a|x)

normalize as L ≡

destroys lovely reduction simply11 L(λ) =

i(yi − λ)1[ai 6= h(xi)]/p−(ai |xi) hidden here is a 2nd parameter, in classification, ∴ harder

search

11Suggestion by Dan Hsu

POISE punchlines

allows policy planning even with implicit logged explorationdata12

12Strehl, Alex, et al. “Learning from logged implicit exploration data.”Advances in Neural Information Processing Systems. 2010.

POISE punchlines

e.g., two hospital story

POISE punchlines

e.g., two hospital story “personalized medicine” is also a policy

POISE punchlines

e.g., two hospital story “personalized medicine” is also a policy abundant data available, under-explored IMHO

tangent: causality as told by an economist

different, related goal

they think in terms of ATE/ITE instead of policy

τ ≡ E0(Y |a = 1) − E0(Y |a = 0) ≡ Q(a = 1) − Q(a = 0)

CATE aka Individualized Treatment Effect (ITE)

τ ≡ E0(Y |a = 1) − E0(Y |a = 0) ≡ Q(a = 1) − Q(a = 0)

τ(x) ≡ E0(Y |a = 1, x) − E0(Y |a = 0, x)

τ ≡ E0(Y |a = 1) − E0(Y |a = 0) ≡ Q(a = 1) − Q(a = 0)

τ(x) ≡ E0(Y |a = 1, x) − E0(Y |a = 0, x) ≡ Q(a = 1, x) − Q(a = 0, x)

Q-note: “generalizing” Monte Carlo w/kernels

MC: Ep(f ) =∑

x p(x)f (x) ≈ N−1 ∑

i∼p f (xi)

MC: Ep(f ) =∑

x p(x)f (x) ≈ N−1 ∑

i∼p f (xi) K: p ≈ N−1 ∑

i K (x |xi)

MC: Ep(f ) =∑

x p(x)f (x) ≈ N−1 ∑

i∼p f (xi) K: p ≈ N−1 ∑

i K (x |xi) ⇒

x p(x)f (x) ≈ N−1 ∑

x f (x)K (x |xi)

MC: Ep(f ) =∑

x p(x)f (x) ≈ N−1 ∑

i∼p f (xi) K: p ≈ N−1 ∑

i K (x |xi) ⇒

x p(x)f (x) ≈ N−1 ∑

x f (x)K (x |xi) K can be any normalized function, e.g., K (x |xi) = δx ,xi

, whichyields MC.

MC: Ep(f ) =∑

x p(x)f (x) ≈ N−1 ∑

i∼p f (xi) K: p ≈ N−1 ∑

i K (x |xi) ⇒

x p(x)f (x) ≈ N−1 ∑

x f (x)K (x |xi) K can be any normalized function, e.g., K (x |xi) = δx ,xi

, whichyields MC.

multivariateEp(f ) ≈ N−1 ∑

yax f (y , a, x)K1(y |yi)K2(a|ai)K3(x |xi)

Q-note: application w/strata+matching, setup

Helps think about economists’ approach:

Q(a, x) ≡ E (Y |a, x) =∑

y yp(y |a, x) =∑

y yp−(y ,a,x)

p−(a|x)p(x)

Q(a, x) ≡ E (Y |a, x) =∑

y yp(y |a, x) =∑

y yp−(y ,a,x)

p−(a|x)p(x)

= 1p−(a|x)p(x)

y yp−(y , a, x)

Q(a, x) ≡ E (Y |a, x) =∑

y yp(y |a, x) =∑

y yp−(y ,a,x)

p−(a|x)p(x)

= 1p−(a|x)p(x)

y yp−(y , a, x)

stratify x using z(x) such that ∪z = X , and ∩z , z ′ = ø

Q(a, x) ≡ E (Y |a, x) =∑

y yp(y |a, x) =∑

y yp−(y ,a,x)

p−(a|x)p(x)

= 1p−(a|x)p(x)

y yp−(y , a, x)

stratify x using z(x) such that ∪z = X , and ∩z , z ′ = ø n(x) =

i 1[z(xi) = z(x)]=number of points in x ’s stratum

Q(a, x) ≡ E (Y |a, x) =∑

y yp(y |a, x) =∑

y yp−(y ,a,x)

p−(a|x)p(x)

= 1p−(a|x)p(x)

y yp−(y , a, x)

i 1[z(xi) = z(x)]=number of points in x ’s stratum Ω(x) =

x ′ 1[z(x ′) = z(x)]=area of x ’s stratum

Q(a, x) ≡ E (Y |a, x) =∑

y yp(y |a, x) =∑

y yp−(y ,a,x)

p−(a|x)p(x)

= 1p−(a|x)p(x)

y yp−(y , a, x)

x ′ 1[z(x ′) = z(x)]=area of x ’s stratum ∴ K3(x |xi) = 1[z(x) = z(xi)]/Ω(x)

Q(a, x) ≡ E (Y |a, x) =∑

y yp(y |a, x) =∑

y yp−(y ,a,x)

p−(a|x)p(x)

= 1p−(a|x)p(x)

y yp−(y , a, x)

x ′ 1[z(x ′) = z(x)]=area of x ’s stratum ∴ K3(x |xi) = 1[z(x) = z(xi)]/Ω(x) as in MC, K1(y |yi) = δy ,yi

, K2(a|ai) = δa,ai

Q-note: application w/strata+matching, payoff

y yp−(y , a, x) ≈ N−1Ω(x)−1 ∑

ai =a,z(xi )=z(x) yi

y yp−(y , a, x) ≈ N−1Ω(x)−1 ∑

p(x) ≈ (n(x)/N)Ω(x)−1

y yp−(y , a, x) ≈ N−1Ω(x)−1 ∑

p(x) ≈ (n(x)/N)Ω(x)−1

∴ Q(a, x) ≈ p−(a|x)−1n(x)−1 ∑

y yp−(y , a, x) ≈ N−1Ω(x)−1 ∑

p(x) ≈ (n(x)/N)Ω(x)−1

∴ Q(a, x) ≈ p−(a|x)−1n(x)−1 ∑

y yp−(y , a, x) ≈ N−1Ω(x)−1 ∑

p(x) ≈ (n(x)/N)Ω(x)−1

∴ Q(a, x) ≈ p−(a|x)−1n(x)−1 ∑

“matching” means: choose each z to contain 1 positive example & 1negative example,

p−(a|x) ≈ 1/2, n(x) = 2

y yp−(y , a, x) ≈ N−1Ω(x)−1 ∑

p(x) ≈ (n(x)/N)Ω(x)−1

∴ Q(a, x) ≈ p−(a|x)−1n(x)−1 ∑

p−(a|x) ≈ 1/2, n(x) = 2 ∴ τ(a, x) = Q(a = 1, x)− Q(a = 0, x) = y1(x)− y0(x)

y yp−(y , a, x) ≈ N−1Ω(x)−1 ∑

p(x) ≈ (n(x)/N)Ω(x)−1

∴ Q(a, x) ≈ p−(a|x)−1n(x)−1 ∑

p−(a|x) ≈ 1/2, n(x) = 2 ∴ τ(a, x) = Q(a = 1, x)− Q(a = 0, x) = y1(x)− y0(x) z-generalizations: graphs, digraphs, k-NN, “matching”

y yp−(y , a, x) ≈ N−1Ω(x)−1 ∑

p(x) ≈ (n(x)/N)Ω(x)−1

∴ Q(a, x) ≈ p−(a|x)−1n(x)−1 ∑

p−(a|x) ≈ 1/2, n(x) = 2 ∴ τ(a, x) = Q(a = 1, x)− Q(a = 0, x) = y1(x)− y0(x) z-generalizations: graphs, digraphs, k-NN, “matching” K -generalizations: continuous a, any metric or similarity you

like,. . .

y yp−(y , a, x) ≈ N−1Ω(x)−1 ∑

p(x) ≈ (n(x)/N)Ω(x)−1

∴ Q(a, x) ≈ p−(a|x)−1n(x)−1 ∑

like,. . .

y yp−(y , a, x) ≈ N−1Ω(x)−1 ∑

p(x) ≈ (n(x)/N)Ω(x)−1

∴ Q(a, x) ≈ p−(a|x)−1n(x)−1 ∑

like,. . .

IMHO underexplored

causality, as understood in marketing a/b testing and RCT

Figure 11: Blattberg, Robert C., Byung-Do Kim, and Scott A. Neslin.Database Marketing, Springer New York, 2008

causality, as understood in marketing a/b testing and RCT yield optimization

causality, as understood in marketing a/b testing and RCT yield optimization Lorenz curve (vs ROC plots)

unobserved confounders vs. “causality” modeling

truth: pα(y , a, x , u) = p(y |a, x , u)pα(a|x , u)p(x , u)

truth: pα(y , a, x , u) = p(y |a, x , u)pα(a|x , u)p(x , u) but: p+(y , a, x , u) = p(y |a, x , u)p−(a|x)p(x , u)

truth: pα(y , a, x , u) = p(y |a, x , u)pα(a|x , u)p(x , u) but: p+(y , a, x , u) = p(y |a, x , u)p−(a|x)p(x , u) E+(Y ) ≡

yaxu yp+(yaxu) ≈N−1 ∑

i∼p−yip+(a|x)/p−(a|x , u)

truth: pα(y , a, x , u) = p(y |a, x , u)pα(a|x , u)p(x , u) but: p+(y , a, x , u) = p(y |a, x , u)p−(a|x)p(x , u) E+(Y ) ≡

yaxu yp+(yaxu) ≈N−1 ∑

i∼p−yip+(a|x)/p−(a|x , u)

denominator can not be inferred, ignore at your peril

cautionary tale problem: Simpson’s paradox

a: admissions (a=1: admitted, a=0: declined)

a: admissions (a=1: admitted, a=0: declined) x : gender (x=1: female, x=0: male)

a: admissions (a=1: admitted, a=0: declined) x : gender (x=1: female, x=0: male) lawsuit (1973): .44 = p(a = 1|x = 0) > p(a = 1|x = 1) = .35

a: admissions (a=1: admitted, a=0: declined) x : gender (x=1: female, x=0: male) lawsuit (1973): .44 = p(a = 1|x = 0) > p(a = 1|x = 1) = .35 ‘resolved’ by Bickel (1975)13 (See also Pearl14 )

13P.J. Bickel, E.A. Hammel and J.W. O’Connell (1975). “Sex Bias inGraduate Admissions: Data From Berkeley”. Science 187 (4175): 398–404

14Pearl, Judea (December 2013). “Understanding Simpson’s paradox”. UCLACognitive Systems Laboratory, Technical Report R-414.

a: admissions (a=1: admitted, a=0: declined) x : gender (x=1: female, x=0: male) lawsuit (1973): .44 = p(a = 1|x = 0) > p(a = 1|x = 1) = .35 ‘resolved’ by Bickel (1975)13 (See also Pearl14 ) u: unobserved department they applied to

a: admissions (a=1: admitted, a=0: declined) x : gender (x=1: female, x=0: male) lawsuit (1973): .44 = p(a = 1|x = 0) > p(a = 1|x = 1) = .35 ‘resolved’ by Bickel (1975)13 (See also Pearl14 ) u: unobserved department they applied to p(a|x) =

∑u=6u=1 p(a|x , u)p(u|x)

a: admissions (a=1: admitted, a=0: declined) x : gender (x=1: female, x=0: male) lawsuit (1973): .44 = p(a = 1|x = 0) > p(a = 1|x = 1) = .35 ‘resolved’ by Bickel (1975)13 (See also Pearl14 ) u: unobserved department they applied to p(a|x) =

∑u=6u=1 p(a|x , u)p(u|x)

e.g., gender-blind: p(a|1)− p(a|0) = p(a|u) · (p(u|1)− p(u|0))

confounded approach: quasi-experiments + instruments 17

Q: does engagement drive retention? (NYT, NFLX, . . . )

we don’t directly control engagement

we don’t directly control engagement nonetheless useful since many things can influence it

Q: does serving in Vietnam war decrease earnings15?

15Angrist, Joshua D. “Lifetime earnings and the Vietnam era draft lottery:evidence from social security administrative records.” The American EconomicReview (1990): 313-336.

US didn’t directly control serving in Vietnam, either16

16cf., George Bush, Donald Trump, Bill Clinton, Dick Cheney. . .

US didn’t directly control serving in Vietnam, either16

requires strong assumptions, including linear model

16cf., George Bush, Donald Trump, Bill Clinton, Dick Cheney. . .17I thank Sinan Aral, MIT Sloan, for bringing this to my attention

IV: graphical model assumption

Figure 12: independence assumption

IV: graphical model assumption (sideways)

Figure 13: independence assumption

IV: review s/OLS/MOM/ (E is empirical average)

a endogenous