# Information Theory For Data Management

## Presentation on theme: "Information Theory For Data Management"— Presentation transcript:

Information Theory For Data Management
Divesh Srivastava Suresh Venkatasubramanian

Information Theory is relevant to all of humanity...
Motivation -- Abstruse Goose (177) Information Theory is relevant to all of humanity...

Background Many problems in data management need precise reasoning about information content, transfer and loss Structure Extraction Privacy preservation Schema design Probabilistic data ?

Information Theory First developed by Shannon as a way of quantifying capacity of signal channels. Entropy, relative entropy and mutual information capture intrinsic informational aspects of a signal Today: Information theory provides a domain-independent way to reason about structure in data More information = interesting structure Less information linkage = decoupling of structures

Tutorial Thesis Information theory provides a mathematical framework for the quantification of information content, linkage and loss. This framework can be used in the design of data management strategies that rely on probing the structure of information in data.

Tutorial Goals Introduce information-theoretic concepts to VLDB audience Give a ‘data-centric’ perspective on information theory Connect these to applications in data management Describe underlying computational primitives Illuminate when and how information theory might be of use in new areas of data management.

Outline Part 1 Part 2 Introduction to Information Theory
Application: Data Anonymization Application: Data Integration Part 2 Review of Information Theory Basics Application: Database Design Computing Information Theoretic Primitives Open Problems 7 7

Histograms And Discrete Distributions
x1 x2 x4 x3 X Column of data X f(X) x1 4 x2 2 x3 1 x4 Histogram X p(X) x1 0.5 x2 0.25 x3 0.125 x4 Probability distribution aggregate counts normalize

Histograms And Discrete Distributions
X f(x)*w(X) x1 4*5=20 x2 2*3=6 x3 1*2=2 x4 reweight normalize x1 x2 x4 x3 X Column of data X f(X) x1 4 x2 2 x3 1 x4 Histogram X p(X) x1 0.667 x2 0.2 x3 0.067 x4 Probability distribution aggregate counts

From Columns To Random Variables
We can think of a column of data as “represented” by a random variable: X is a random variable p(X) is the column of probabilities p(X = x1), p(X = x2), and so on Also known (in unweighted case) as the empirical distribution induced by the column X. Notation: X (upper case) denotes a random variable (column) x (lower case) denotes a value taken by X (field in a tuple) p(x) is the probability p(X = x)

Joint Distributions Discrete distribution: probability p(X,Y,Z)
p(Y) = ∑x p(X=x,Y) = ∑x ∑z p(X=x,Y,Z=z) X Y Z p(X,Y,Z) x1 y1 z1 0.125 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 X Y p(X,Y) x1 y1 0.25 y2 x2 y3 x3 0.125 x4 X p(X) x1 0.5 x2 0.25 x3 0.125 x4 Y p(Y) y1 0.25 y2 y3 0.5 11 11 11

Entropy Of A Column Let h(x) = log2 1/p(x)
0.5 1 x2 0.25 2 x3 0.125 3 x4 Let h(x) = log2 1/p(x) h(X) is column of h(x) values. H(X) = EX[h(x)] = SX p(x) log2 1/p(x) Two views of entropy It captures uncertainty in data: high entropy, more unpredictability It captures information content: higher entropy, more information. H(X) = 1.75 < log |X| = 2

Examples X uniform over [1, ..., 4]. H(X) = 2
Y is 1 with probability 0.5, in [2,3,4] uniformly. H(Y) = 0.5 log log 6 ~= 1.8 < 2 Y is more sharply defined, and so has less uncertainty. Z uniform over [1, ..., 8]. H(Z) = 3 > 2 Z spans a larger range, and captures more information X Y Z

Comparing Distributions
How do we measure difference between two distributions ? Kullback-Leibler divergence: dKL(p, q) = Ep[ h(q) – h(p) ] = Si pi log(pi/qi) Inference mechanism Prior belief Resulting belief

Comparing Distributions
Kullback-Leibler divergence: dKL(p, q) = Ep[ h(q) – h(p) ] = Si pi log(pi/qi) dKL(p, q) >= 0 Captures extra information needed to capture p given q Is asymmetric ! dKL(p, q) != dKL(q, p) Is not a metric (does not satisfy triangle inequality) There are other measures: 2-distance, variational distance, f-divergences, …

Conditional Probability
Given a joint distribution on random variables X, Y, how much information about X can we glean from Y ? Conditional probability: p(X|Y) p(X = x1 | Y = y1) = p(X = x1, Y = y1)/p(Y = y1) X Y p(X,Y) p(X|Y) p(Y|X) x1 y1 0.25 1.0 0.5 y2 x2 y3 x3 0.125 x4 X p(X) x1 0.5 x2 0.25 x3 0.125 x4 Y p(Y) y1 0.25 y2 y3 0.5

Conditional Entropy Let h(x|y) = log2 1/p(x|y)
H(X|Y) = Ex,y[h(x|y)] = Sx Sy p(x,y) log2 1/p(x|y) H(X|Y) = H(X,Y) – H(Y) H(X|Y) = H(X,Y) – H(Y) = 2.25 – 1.5 = 0.75 If X, Y are independent, H(X|Y) = H(X) X Y p(X,Y) p(X|Y) h(X|Y) x1 y1 0.25 1.0 0.0 y2 x2 y3 0.5 x3 0.125 2.0 x4

Mutual Information Mutual information captures the difference between the joint distribution on X and Y, and the marginal distributions on X and Y. Let i(x;y) = log p(x,y)/p(x)p(y) I(X;Y) = Ex,y[I(X;Y)] = Sx Sy p(x,y) log p(x,y)/p(x)p(y) X Y p(X,Y) h(X,Y) i(X;Y) x1 y1 0.25 2.0 1.0 y2 x2 y3 x3 0.125 3.0 x4 X p(X) h(X) x1 0.5 1.0 x2 0.25 2.0 x3 0.125 3.0 x4 Y p(Y) h(Y) y1 0.25 2.0 y2 y3 0.5 1.0

I(X;Y) = H(X) + H(Y) – H(X,Y) = H(X) – H(X|Y) = H(Y) – H(Y|X) If X, Y are independent, then I(X;Y) = 0: H(X,Y) = H(X) + H(Y), so I(X;Y) = H(X) + H(Y) – H(X,Y) = 0 I(X;Y) <= max (H(X), H(Y)) Suppose Y = f(X) (deterministically) Then H(Y|X) = 0, and so I(X;Y) = H(Y) – H(Y|X) = H(Y) Mutual information captures higher-order interactions: Covariance captures “linear” interactions only Two variables can be uncorrelated (covariance = 0) and have nonzero mutual information: X R [-1,1], Y = X2. Cov(X,Y) = 0, I(X;Y) = H(X) > 0

Information-Theoretic Clustering
Clustering takes a collection of objects and groups them. Given a distance function between objects Choice of measure of complexity of clustering Choice of measure of cost for a cluster Usually, Distance function is Euclidean distance Number of clusters is measure of complexity Cost measure for cluster is sum-of-squared-distance to center Goal: minimize complexity and cost Inherent tradeoff between two

Feature Representation
Let V = {v1, v2, v3, v4} X is “explained” by distribution over V. “Feature vector” of X is [0.5, 0.25, 0.125, 0.125] v1 v2 v4 v3 X Column of data X f(X) v1 4 v2 2 v3 1 v4 Histogram X p(X) v1 0.5 v2 0.25 v3 0.125 v4 Probability distribution aggregate counts normalize

Feature Representation
V v1 v2 v3 v4 X X1 0.5 0.25 0.125 X2 0.2 0.15 p(v2|X2) = 0.2 Feature vector

Information-Theoretic Clustering
Clustering takes a collection of objects and groups them. Given a distance function between objects Choice of measure of complexity of clustering Choice of measure of cost for a cluster In information-theoretic setting What is the distance function ? How do we measure complexity ? What is a notion of cost/quality ? Goal: minimize complexity and maximize quality Inherent tradeoff between two

Measuring complexity of clustering
Take 1: complexity of a clustering = #clusters standard model of complexity. Doesn’t capture the fact that clusters have different sizes.

Measuring complexity of clustering
Take 2: Complexity of clustering = number of bits needed to describe it. Writing down “k” needs log k bits. In general, let cluster t  T have |t| elements. set p(t) = |t|/n #bits to write down cluster sizes = H(T) = S pt log 1/pt H( ) < H( )

Information-theoretic Clustering (take I)
Given data X = x1, ..., xn explained by variable V, partition X into clusters (represented by T) such that H(T) is minimized and quality is maximized

How do we describe the complexity of a clustering ?
Soft clusterings In a “hard” clustering, each point is assigned to exactly one cluster. Characteristic function p(t|x) = 1 if x  t, 0 if not. Suppose we allow points to partially belong to clusters: p(T|x) is a distribution. p(t|x) is the “probability” of assigning x to t How do we describe the complexity of a clustering ?

Measuring complexity of clustering
Take 1: p(t) = Sx p(x) p(t|x) Compute H(T) as before. Problem: H(T1) = H(T2) !! T1 t1 t2 T2 x1 0.5 0.99 0.01 x2 h(T)

Measuring complexity of clustering
By averaging the memberships, we’ve lost useful information. Take II: Compute I(T;X) ! Even better: If T is a hard clustering of X, then I(T;X) = H(T) X T1 p(X,T) i(X;T) x1 t1 0.25 t2 x2 X T2 p(X,T) i(X;T) x1 t1 0.495 0.99 t2 0.005 -5.64 x2 0.25 I(T1;X) = 0 I(T2;X) = 0.46

Information-theoretic Clustering (take II)
Given data X = x1, ..., xn explained by variable V, partition X into clusters (represented by T) such that I(T,X) is minimized and quality is maximized

Measuring cost of a cluster
Given objects Xt = {X1, X2, …, Xm} in cluster t, Cost(t) = (1/m)Si d(Xi, C) = Si p(Xi) dKL(p(V|Xi), C) where C = (1/m) Si p(V|Xi) = Si p(Xi) p(V|Xi) = p(V)

Mutual Information = Cost of Cluster
Cost(t) = (1/m)Si d(Xi, C) = Si p(Xi) dKL(p(V|Xi), p(V)) Si p(Xi) KL( p(V|Xi), p(V)) = Si p(Xi) Sj p(vj|Xi) log p(vj|Xi)/p(vj) = Si,j p(Xi, vj) log p(vj, Xi)/p(vj)p(Xi) = I(Xt, V) !! Cost of a cluster = I(Xt,V)

Cost(clustering) = Si pi I(Xi, V)
Cost of a clustering If we partition X into k clusters X1, ..., Xk Cost(clustering) = Si pi I(Xi, V) (pi = |Xi|/|X|)

Cost of a clustering Each cluster center t can be “explained” in terms of V: p(V|t) = Si p(Xi) p(V|Xi) Suppose we treat each cluster center itself as a point:

Cost(clustering) = I(X, V) – (T, V)
Cost of a clustering We can write down the “cost” of this “cluster” Cost(T) = I(T;V) Key result [BMDG05] : Cost(clustering) = I(X, V) – (T, V) Minimizing cost(clustering) => maximizing I(T, V)

Information-theoretic Clustering (take III)
Given data X = x1, ..., xn explained by variable V, partition X into clusters (represented by T) such that I(T;X) - bI(T;V) is maximized This is the Information Bottleneck Method [TPB98] Agglomerative techniques exist for the case of ‘hard’ clusterings b is the tradeoff parameter between complexity and cost I(T;X) and I(T;V) are in the same units.

Information Theory: Summary
We can represent data as discrete distributions (normalized histograms) Entropy captures uncertainty or information content in a distribution The Kullback-Leibler distance captures the difference between distributions Mutual information and conditional entropy capture linkage between variables in a joint distribution We can formulate information-theoretic clustering problems

Outline Part 1 Part 2 Introduction to Information Theory
Application: Data Anonymization Application: Data Integration Part 2 Review of Information Theory Basics Application: Database Design Computing Information Theoretic Primitives Open Problems

Data Anonymization Using Randomization
Goal: publish anonymized microdata to enable accurate ad hoc analyses, but ensure privacy of individuals’ sensitive attributes Key ideas: Randomize numerical data: add noise from known distribution Reconstruct original data distribution using published noisy data Issues: How can the original data distribution be reconstructed? What kinds of randomization preserve privacy of individuals? 39 Information Theory for Data Management - Divesh & Suresh

Data Anonymization Using Randomization
Many randomization strategies proposed [AS00, AA01, EGS03] Example randomization strategies: X in [0, 10] R = X + μ (mod 11), μ is uniform in {-1, 0, 1} R = X + μ (mod 11), μ is in {-1 (p = 0.25), 0 (p = 0.5), 1 (p = 0.25)} R = X (p = 0.6), R = μ, μ is uniform in [0, 10] (p = 0.4) Question: Which randomization strategy has higher privacy preservation? Quantify loss of privacy due to publication of randomized data Information Theory for Data Management - Divesh & Suresh 40

Data Anonymization Using Randomization
X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} Id X s1 s2 3 s3 5 s4 s5 8 s6 s7 6 s8 Information Theory for Data Management - Divesh & Suresh 41

Data Anonymization Using Randomization
X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} Id X μ s1 -1 s2 3 s3 5 1 s4 s5 8 s6 s7 6 s8 Id R1 s1 10 s2 3 s3 6 s4 s5 9 s6 s7 7 s8 Information Theory for Data Management - Divesh & Suresh 42

Data Anonymization Using Randomization
X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} Id X μ s1 s2 3 -1 s3 5 s4 1 s5 8 s6 s7 6 s8 Id R1 s1 s2 2 s3 5 s4 1 s5 9 s6 10 s7 s8 Information Theory for Data Management - Divesh & Suresh 43

Reconstruction of Original Data Distribution
X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} Reconstruct distribution of X using knowledge of R1 and μ EM algorithm converges to MLE of original distribution [AA01] Id X μ s1 s2 3 -1 s3 5 s4 1 s5 8 s6 s7 6 s8 Id R1 s1 s2 2 s3 5 s4 1 s5 9 s6 10 s7 s8 Id X | R1 s1 {10, 0, 1} s2 {1, 2, 3} s3 {4, 5, 6} s4 {0, 1, 2} s5 {8, 9, 10} s6 {9, 10, 0} s7 s8 Information Theory for Data Management - Divesh & Suresh 44

Analysis of Privacy [AS00]
X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} If X is uniform in [0, 10], privacy determined by range of μ Id X μ s1 s2 3 -1 s3 5 s4 1 s5 8 s6 s7 6 s8 Id R1 s1 s2 2 s3 5 s4 1 s5 9 s6 10 s7 s8 Id X | R1 s1 {10, 0, 1} s2 {1, 2, 3} s3 {4, 5, 6} s4 {0, 1, 2} s5 {8, 9, 10} s6 {9, 10, 0} s7 s8 Information Theory for Data Management - Divesh & Suresh 45

Analysis of Privacy [AA01]
X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} If X is uniform in [0, 1]  [5, 6], privacy smaller than range of μ Id X μ s1 s2 1 -1 s3 5 s4 6 s5 s6 s7 s8 Id R1 s1 s2 s3 5 s4 7 s5 1 s6 s7 4 s8 Id X | R1 s1 {10, 0, 1} s2 s3 {4, 5, 6} s4 {6, 7, 8} s5 {0, 1, 2} s6 s7 {3, 4, 5} s8 Information Theory for Data Management - Divesh & Suresh 46

Analysis of Privacy [AA01]
X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} If X is uniform in [0, 1]  [5, 6], privacy smaller than range of μ In some cases, sensitive value revealed Id X μ s1 s2 1 -1 s3 5 s4 6 s5 s6 s7 s8 Id R1 s1 s2 s3 5 s4 7 s5 1 s6 s7 4 s8 Id X | R1 s1 {0, 1} s2 s3 {5, 6} s4 {6} s5 s6 s7 {5} s8 Information Theory for Data Management - Divesh & Suresh 47

Quantify Loss of Privacy [AA01]
Goal: quantify loss of privacy based on mutual information I(X;R) Smaller H(X|R)  more loss of privacy in X by knowledge of R Larger I(X;R)  more loss of privacy in X by knowledge of R I(X;R) = H(X) – H(X|R) I(X;R) used to capture correlation between X and R p(X) is the prior knowledge of sensitive attribute X p(X, R) is the joint distribution of X and R Information Theory for Data Management - Divesh & Suresh 48

Quantify Loss of Privacy [AA01]
Goal: quantify loss of privacy based on mutual information I(X;R) X is uniform in [5, 6], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} X R1 p(X,R1) h(X,R1) i(X;R1) 5 4 6 7 X p(X) h(X) 5 6 R1 p(R1) h(R1) 4 5 6 7 Information Theory for Data Management - Divesh & Suresh 49

Quantify Loss of Privacy [AA01]
Goal: quantify loss of privacy based on mutual information I(X;R) X is uniform in [5, 6], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} X R1 p(X,R1) h(X,R1) i(X;R1) 5 4 0.17 6 7 X p(X) h(X) 5 0.5 6 R1 p(R1) h(R1) 4 0.17 5 0.34 6 7 Information Theory for Data Management - Divesh & Suresh 50

Quantify Loss of Privacy [AA01]
Goal: quantify loss of privacy based on mutual information I(X;R) X is uniform in [5, 6], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} X R1 p(X,R1) h(X,R1) i(X;R1) 5 4 0.17 2.58 6 7 X p(X) h(X) 5 0.5 1.0 6 R1 p(R1) h(R1) 4 0.17 2.58 5 0.34 1.58 6 7 Information Theory for Data Management - Divesh & Suresh 51

Quantify Loss of Privacy [AA01]
Goal: quantify loss of privacy based on mutual information I(X;R) X is uniform in [5, 6], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} I(X;R) = 0.33 X R1 p(X,R1) h(X,R1) i(X;R1) 5 4 0.17 2.58 1.0 0.0 6 7 X p(X) h(X) 5 0.5 1.0 6 R1 p(R1) h(R1) 4 0.17 2.58 5 0.34 1.58 6 7 Information Theory for Data Management - Divesh & Suresh 52

Quantify Loss of Privacy [AA01]
Goal: quantify loss of privacy based on mutual information I(X;R) X is uniform in [5, 6], R2 = X + μ (mod 11), μ is uniform in {0, 1} I(X;R1) = 0.33, I(X;R2) = 0.5  R2 is a bigger privacy risk than R1 X R2 p(X,R2) h(X,R2) i(X;R2) 5 0.25 2.0 1.0 6 0.0 7 X p(X) h(X) 5 0.5 1.0 6 R2 p(R2) h(R2) 5 0.25 2.0 6 0.5 1.0 7 Information Theory for Data Management - Divesh & Suresh 53

Quantify Loss of Privacy [AA01]
Equivalent goal: quantify loss of privacy based on H(X|R) X is uniform in [5, 6], R2 = X + μ (mod 11), μ is uniform in {0, 1} Intuition: we know more about X given R2, than about X given R1 H(X|R1) = 0.67, H(X|R2) = 0.5  R2 is a bigger privacy risk than R1 X R1 p(X,R1) p(X|R1) h(X|R1) 5 4 0.17 1.0 0.0 0.5 6 7 X R2 p(X,R2) p(X|R2) h(X|R2) 5 0.25 1.0 0.0 6 0.5 7 Information Theory for Data Management - Divesh & Suresh 54

Quantify Loss of Privacy
Example: X is uniform in [0, 1] R3 = e (p = ), R3 = X (p = ) R4 = X (p = 0.6), R4 = 1 – X (p = 0.4) Is R3 or R4 a bigger privacy risk? Information Theory for Data Management - Divesh & Suresh 55

Worst Case Loss of Privacy [EGS03]
Example: X is uniform in [0, 1] R3 = e (p = ), R3 = X (p = ) R4 = X (p = 0.6), R4 = 1 – X (p = 0.4) I(X;R3) = << I(X;R4) = 0.028 X R3 p(X,R3) h(X,R3) i(X;R3) e 1.0 0.0 14.29 1 X R4 p(X,R4) h(X,R4) i(X;R4) 0.3 1.74 0.26 1 0.2 2.32 -0.32 Information Theory for Data Management - Divesh & Suresh 56

Worst Case Loss of Privacy [EGS03]
Example: X is uniform in [0, 1] R3 = e (p = ), R3 = X (p = ) R4 = X (p = 0.6), R4 = 1 – X (p = 0.4) I(X;R3) = << I(X;R4) = 0.028 But R3 has a larger worst case risk X R3 p(X,R3) h(X,R3) i(X;R3) e 1.0 0.0 14.29 1 X R4 p(X,R4) h(X,R4) i(X;R4) 0.3 1.74 0.26 1 0.2 2.32 -0.32 Information Theory for Data Management - Divesh & Suresh 57

Worst Case Loss of Privacy [EGS03]
Goal: quantify worst case loss of privacy in X by knowledge of R Use max KL divergence, instead of mutual information Mutual information can be formulated as expected KL divergence I(X;R) = ∑x ∑r p(x,r)*log2(p(x,r)/p(x)*p(r)) = KL(p(X,R) || p(X)*p(R)) I(X;R) = ∑r p(r) ∑x p(x|r)*log2(p(x|r)/p(x)) = ER [KL(p(X|r) || p(X))] [AA01] measure quantifies expected loss of privacy over R [EGS03] propose a measure based on worst case loss of privacy IW(X;R) = MAXR [KL(p(X|r) || p(X))] Information Theory for Data Management - Divesh & Suresh 58

Worst Case Loss of Privacy [EGS03]
Example: X is uniform in [0, 1] R3 = e (p = ), R3 = X (p = ) R4 = X (p = 0.6), R4 = 1 – X (p = 0.4) IW(X;R3) = max{0.0, 1.0, 1.0} > IW(X;R4) = max{0.028, 0.028} X R3 p(X,R3) p(X|R3) i(X;R3) e 0.5 0.0 1.0 1 X R4 p(X,R4) p(X|R4) i(X;R4) 0.3 0.6 0.26 1 0.2 0.4 -0.32 Information Theory for Data Management - Divesh & Suresh 59

Worst Case Loss of Privacy [EGS03]
Example: X is uniform in [5, 6] R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1} R2 = X + μ (mod 11), μ is uniform in {0, 1} IW(X;R1) = max{1.0, 0.0, 0.0, 1.0} = IW(X;R2) = {1.0, 0.0, 1.0} Unable to capture that R2 is a bigger privacy risk than R1 X R1 p(X,R1) p(X|R1) i(X;R1) 5 4 0.17 1.0 0.5 0.0 6 7 X R2 p(X,R2) p(X|R2) i(X;R2) 5 0.25 1.0 6 0.5 0.0 7 Information Theory for Data Management - Divesh & Suresh 60

Data Anonymization: Summary
Randomization techniques useful for microdata anonymization Randomization techniques differ in their loss of privacy Information theoretic measures useful to capture loss of privacy Expected KL divergence captures expected loss of privacy [AA01] Maximum KL divergence captures worst case loss of privacy [EGS03] Both are useful in practice Information Theory for Data Management - Divesh & Suresh 61

Information Theory for Data Management - Divesh & Suresh
Outline Part 1 Introduction to Information Theory Application: Data Anonymization Application: Data Integration Part 2 Review of Information Theory Basics Application: Database Design Computing Information Theoretic Primitives Open Problems Information Theory for Data Management - Divesh & Suresh

Information Theory for Data Management - Divesh & Suresh
Schema Matching Goal: align columns across database tables to be integrated Fundamental problem in database integration Early useful approach: textual similarity of column names False positives: Address ≠ IP_Address False negatives: Customer_Id = Client_Number Early useful approach: overlap of values in columns, e.g., Jaccard False positives: Emp_Id ≠ Project_Id False negatives: Emp_Id = Personnel_Number Information Theory for Data Management - Divesh & Suresh 63

Opaque Schema Matching [KN03]
Goal: align columns when column names, data values are opaque Databases belong to different government bureaucracies  Treat column names and data values as uninterpreted (generic) Example: EMP_PROJ(Emp_Id, Proj_Id, Task_Id, Status_Id) Likely that all Id fields are from the same domain Different databases may have different column names A B C D a1 b2 c1 d1 a3 b4 c2 d2 b1 a4 b3 d3 W X Y Z w2 x1 y1 z2 w4 x2 y3 z3 w3 x3 z1 w1 Information Theory for Data Management - Divesh & Suresh 64

Opaque Schema Matching [KN03]
Approach: build complete, labeled graph GD for each database D Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y) Perform graph matching between GD1 and GD2, minimizing distance Intuition: Entropy H(X) captures distribution of values in database column X Mutual information I(X;Y) captures correlations between X, Y Efficiency: graph matching between schema-sized graphs Information Theory for Data Management - Divesh & Suresh 65

Opaque Schema Matching [KN03]
Approach: build complete, labeled graph GD for each database D Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y) A B C D a1 b2 c1 d1 a3 b4 c2 d2 b1 a4 b3 d3 A p(A) a1 0.5 a3 0.25 a4 B p(B) b1 0.25 b2 b3 b4 C p(C) c1 0.5 c2 D p(D) d1 0.25 d2 0.5 d3 Information Theory for Data Management - Divesh & Suresh 66

Opaque Schema Matching [KN03]
Approach: build complete, labeled graph GD for each database D Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y) H(A) = 1.5, H(B) = 2.0, H(C) = 1.0, H(D) = 1.5 A B C D a1 b2 c1 d1 a3 b4 c2 d2 b1 a4 b3 d3 A h(A) a1 1.0 a3 2.0 a4 B h(B) b1 2.0 b2 b3 b4 C h(C) c1 1.0 c2 D h(D) d1 2.0 d2 1.0 d3 Information Theory for Data Management - Divesh & Suresh 67

Opaque Schema Matching [KN03]
Approach: build complete, labeled graph GD for each database D Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y) H(A) = 1.5, H(B) = 2.0, H(C) = 1.0, H(D) = 1.5, I(A;B) = 1.5 A B C D a1 b2 c1 d1 a3 b4 c2 d2 b1 a4 b3 d3 A h(A) a1 1.0 a3 2.0 a4 B h(B) b1 2.0 b2 b3 b4 A B h(A,B) i(A;B) a1 b2 2.0 1.0 a3 b4 b1 a4 b3 Information Theory for Data Management - Divesh & Suresh 68

Opaque Schema Matching [KN03]
Approach: build complete, labeled graph GD for each database D Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y) A B D C 1.5 1.0 2.0 0.5 A B C D a1 b2 c1 d1 a3 b4 c2 d2 b1 a4 b3 d3 Information Theory for Data Management - Divesh & Suresh 69

Opaque Schema Matching [KN03]
Approach: build complete, labeled graph GD for each database D Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y) Perform graph matching between GD1 and GD2, minimizing distance [KN03] uses euclidean and normal distance metrics A B D C 1.5 1.0 2.0 0.5 W X Z Y 2.0 1.0 1.5 0.5 Information Theory for Data Management - Divesh & Suresh 70

Opaque Schema Matching [KN03]
Approach: build complete, labeled graph GD for each database D Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y) Perform graph matching between GD1 and GD2, minimizing distance A B D C 1.5 1.0 2.0 0.5 W X Z Y 2.0 1.0 1.5 0.5 Information Theory for Data Management - Divesh & Suresh 71

Opaque Schema Matching [KN03]
Approach: build complete, labeled graph GD for each database D Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y) Perform graph matching between GD1 and GD2, minimizing distance A B D C 1.5 1.0 2.0 0.5 W X Z Y 2.0 1.0 1.5 0.5 Information Theory for Data Management - Divesh & Suresh 72

Heterogeneity Identification [DKOSV06]
Goal: identify columns with semantically heterogeneous values Can arise due to opaque schema matching [KN03] Key ideas: Heterogeneity based on distribution, distinguishability of values Use Information Bottleneck to compute soft clustering of values Issues: Which information theoretic measure characterizes heterogeneity? How to set parameters in the Information Bottleneck method? Information Theory for Data Management - Divesh & Suresh 73

Heterogeneity Identification [DKOSV06]
Example: semantically homogeneous, heterogeneous columns Customer_Id Customer_Id (908) (877) Information Theory for Data Management - Divesh & Suresh 74

Heterogeneity Identification [DKOSV06]
Example: semantically homogeneous, heterogeneous columns Customer_Id Customer_Id (908) (877) Information Theory for Data Management - Divesh & Suresh 75

Heterogeneity Identification [DKOSV06]
Example: semantically homogeneous, heterogeneous columns More semantic types in column  greater heterogeneity Only versus + phone Customer_Id Customer_Id (908) (877) Information Theory for Data Management - Divesh & Suresh 76

Heterogeneity Identification [DKOSV06]
Example: semantically homogeneous, heterogeneous columns Customer_Id (877) Customer_Id (908) (877) Information Theory for Data Management - Divesh & Suresh 77

Heterogeneity Identification [DKOSV06]
Example: semantically homogeneous, heterogeneous columns Relative distribution of semantic types impacts heterogeneity Mainly + few phone versus balanced + phone Customer_Id (877) Customer_Id (908) (877) Information Theory for Data Management - Divesh & Suresh 78

Heterogeneity Identification [DKOSV06]
Example: semantically homogeneous, heterogeneous columns Customer_Id (908) (877) Customer_Id (908) (877) Information Theory for Data Management - Divesh & Suresh 79

Heterogeneity Identification [DKOSV06]
Example: semantically homogeneous, heterogeneous columns Customer_Id (908) (877) Customer_Id (908) (877) Information Theory for Data Management - Divesh & Suresh 80

Heterogeneity Identification [DKOSV06]
Example: semantically homogeneous, heterogeneous columns More easily distinguished types  greater heterogeneity Phone + (possibly) SSN versus balanced + phone Customer_Id (908) (877) Customer_Id (908) (877) Information Theory for Data Management - Divesh & Suresh 81

Heterogeneity Identification [DKOSV06]
Heterogeneity = space complexity of soft clustering of the data More, balanced clusters  greater heterogeneity More distinguishable clusters  greater heterogeneity Soft clustering Soft  assign probabilities to membership of values in clusters How many clusters: tradeoff between space versus quality Use Information Bottleneck to compute soft clustering of values Information Theory for Data Management - Divesh & Suresh 82

Heterogeneity Identification [DKOSV06]
Hard clustering X = Customer_Id T = Cluster_Id t1 (908) t2 t3 (877) Information Theory for Data Management - Divesh & Suresh 83

Heterogeneity Identification [DKOSV06]
Soft clustering: cluster membership probabilities How to compute a good soft clustering? X = Customer_Id T = Cluster_Id p(T|X) t1 0.75 t2 0.25 (908) 0.5 Information Theory for Data Management - Divesh & Suresh 84

Heterogeneity Identification [DKOSV06]
Represent strings as q-gram distributions Customer_Id (908) (877) X = Customer_Id V = 4-grams p(X,V) 987- 0.10 87-6 0.13 7-65 0.12 -65- 0.15 65-4 0.05 5-43 0.20 -432 4321 Information Theory for Data Management - Divesh & Suresh 85

Heterogeneity Identification [DKOSV06]
iIB: find soft clustering T of X that minimizes I(T;X) – β*I(T;V) Allow iIB to use arbitrarily many clusters, use β* = H(X)/I(X;V) Closest to point with minimum space and maximum quality Customer_Id (908) (877) X = Customer_Id V = 4-grams p(X,V) 987- 0.10 87-6 0.13 7-65 0.12 -65- 0.15 65-4 0.05 5-43 0.20 -432 4321 Information Theory for Data Management - Divesh & Suresh 86

Heterogeneity Identification [DKOSV06]
Rate distortion curve: I(T;V)/I(X;V) vs I(T;X)/H(X) β* Information Theory for Data Management - Divesh & Suresh 87

Heterogeneity Identification [DKOSV06]
Heterogeneity = mutual information I(T;X) of iIB clustering T at β* 0 ≤I(T;X) (= 0.126) ≤ H(X) (= 2.0), H(T) (= 1.0) Ideally use iIB with an arbitrarily large number of clusters in T X = Customer_Id T = Cluster_Id p(T|X) i(T;X) t1 0.75 0.41 t2 0.25 -0.81 (908) -1.17 0.5 -0.17 0.77 0.19 Information Theory for Data Management - Divesh & Suresh 88

Heterogeneity Identification [DKOSV06]
Heterogeneity = mutual information I(T;X) of iIB clustering T at β* Information Theory for Data Management - Divesh & Suresh 89

Data Integration: Summary
Analyzing database instance critical for effective data integration Matching and quality assessments are key components Information theoretic measures useful for schema matching Align columns when column names, data values are opaque Mutual information I(X;V) captures correlations between X, V Information theoretic measures useful for heterogeneity testing Identify columns with semantically heterogeneous values I(T;X) of iIB clustering T at β* captures column heterogeneity Information Theory for Data Management - Divesh & Suresh 90

Information Theory for Data Management - Divesh & Suresh
Outline Part 1 Introduction to Information Theory Application: Data Anonymization Application: Data Integration Part 2 Review of Information Theory Basics Application: Database Design Computing Information Theoretic Primitives Open Problems Information Theory for Data Management - Divesh & Suresh

Review of Information Theory Basics
Discrete distribution: probability p(X) p(X,Y) = ∑z p(X,Y,Z=z) X Y Z p(X,Y,Z) x1 y1 z1 0.125 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 X Y p(X,Y) x1 y1 0.25 y2 x2 y3 x3 0.125 x4 X p(X) x1 0.5 x2 0.25 x3 0.125 x4 Y p(Y) y1 0.25 y2 y3 0.5 Information Theory for Data Management - Divesh & Suresh 92

Review of Information Theory Basics
Discrete distribution: probability p(X) p(Y) = ∑x p(X=x,Y) = ∑x ∑z p(X=x,Y,Z=z) X Y Z p(X,Y,Z) x1 y1 z1 0.125 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 X Y p(X,Y) x1 y1 0.25 y2 x2 y3 x3 0.125 x4 X p(X) x1 0.5 x2 0.25 x3 0.125 x4 Y p(Y) y1 0.25 y2 y3 0.5 Information Theory for Data Management - Divesh & Suresh 93

Review of Information Theory Basics
Discrete distribution: conditional probability p(X|Y) p(X,Y) = p(X|Y)*p(Y) = p(Y|X)*p(X) X Y p(X,Y) p(X|Y) p(Y|X) x1 y1 0.25 1.0 0.5 y2 x2 y3 x3 0.125 x4 X p(X) x1 0.5 x2 0.25 x3 0.125 x4 Y p(Y) y1 0.25 y2 y3 0.5 Information Theory for Data Management - Divesh & Suresh 94

Review of Information Theory Basics
Discrete distribution: entropy H(X) h(x) = log2(1/p(x)) H(X) = ∑X=x p(x)*h(x) = 1.75 H(Y) = ∑Y=y p(y)*h(y) = 1.5 (≤ log2(|Y|) = 1.58) H(X,Y) = ∑X=x ∑Y=y p(x,y)*h(x,y) = 2.25 (≤ log2(|X,Y|) = 2.32) X Y p(X,Y) h(X,Y) x1 y1 0.25 2.0 y2 x2 y3 x3 0.125 3.0 x4 X p(X) h(X) x1 0.5 1.0 x2 0.25 2.0 x3 0.125 3.0 x4 Y p(Y) h(Y) y1 0.25 2.0 y2 y3 0.5 1.0 Information Theory for Data Management - Divesh & Suresh 95

Review of Information Theory Basics
Discrete distribution: conditional entropy H(X|Y) h(x|y) = log2(1/p(x|y)) H(X|Y) = ∑X=x ∑Y=y p(x,y)*h(x|y) = 0.75 H(X|Y) = H(X,Y) – H(Y) = 2.25 – 1.5 X Y p(X,Y) p(X|Y) h(X|Y) x1 y1 0.25 1.0 0.0 y2 x2 y3 0.5 x3 0.125 2.0 x4 X p(X) h(X) x1 0.5 1.0 x2 0.25 2.0 x3 0.125 3.0 x4 Y p(Y) h(Y) y1 0.25 2.0 y2 y3 0.5 1.0 Information Theory for Data Management - Divesh & Suresh 96

Review of Information Theory Basics
Discrete distribution: mutual information I(X;Y) i(x;y) = log2(p(x,y)/p(x)*p(y)) I(X;Y) = ∑X=x ∑Y=y p(x,y)*i(x;y) = 1.0 I(X;Y) = H(X) + H(Y) – H(X,Y) = – 2.25 X Y p(X,Y) h(X,Y) i(X;Y) x1 y1 0.25 2.0 1.0 y2 x2 y3 x3 0.125 3.0 x4 X p(X) h(X) x1 0.5 1.0 x2 0.25 2.0 x3 0.125 3.0 x4 Y p(Y) h(Y) y1 0.25 2.0 y2 y3 0.5 1.0 Information Theory for Data Management - Divesh & Suresh 97

Information Theory for Data Management - Divesh & Suresh
Outline Part 1 Introduction to Information Theory Application: Data Anonymization Application: Data Integration Part 2 Review of Information Theory Basics Application: Database Design Computing Information Theoretic Primitives Open Problems Information Theory for Data Management - Divesh & Suresh

Information Dependencies [DR00]
Goal: use information theory to examine and reason about information content of the attributes in a relation instance Key ideas: Novel InD measure between attribute sets X, Y based on H(Y|X) Identify numeric inequalities between InD measures Results: InD measures are a broader class than FDs and MVDs Armstrong axioms for FDs derivable from InD inequalities MVD inference rules derivable from InD inequalities Information Theory for Data Management - Divesh & Suresh 99

Information Dependencies [DR00]
Functional dependency: X → Y FD X → Y holds iff  t1, t2 ((t1[X] = t2[X])  (t1[Y] = t2[Y])) X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 Information Theory for Data Management - Divesh & Suresh 100

Information Dependencies [DR00]
Functional dependency: X → Y FD X → Y holds iff  t1, t2 ((t1[X] = t2[X])  (t1[Y] = t2[Y])) X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 Information Theory for Data Management - Divesh & Suresh 101

Information Dependencies [DR00]
Result: FD X → Y holds iff H(Y|X) = 0 Intuition: once X is known, no remaining uncertainty in Y H(Y|X) = 0.5 X Y p(X,Y) p(Y|X) h(Y|X) x1 y1 0.25 0.5 1.0 y2 x2 y3 0.0 x3 0.125 x4 X p(X) x1 0.5 x2 0.25 x3 0.125 x4 Y p(Y) y1 0.25 y2 y3 0.5 Information Theory for Data Management - Divesh & Suresh 102

Information Dependencies [DR00]
Multi-valued dependency: X →→ Y MVD X →→ Y holds iff R(X,Y,Z) = R(X,Y) R(X,Z) X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 Information Theory for Data Management - Divesh & Suresh 103

Information Dependencies [DR00]
Multi-valued dependency: X →→ Y MVD X →→ Y holds iff R(X,Y,Z) = R(X,Y) R(X,Z) X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 X Y x1 y1 y2 x2 y3 x3 x4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 = Information Theory for Data Management - Divesh & Suresh 104

Information Dependencies [DR00]
Multi-valued dependency: X →→ Y MVD X →→ Y holds iff R(X,Y,Z) = R(X,Y) R(X,Z) X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 X Y x1 y1 y2 x2 y3 x3 x4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 = Information Theory for Data Management - Divesh & Suresh 105

Information Dependencies [DR00]
Result: MVD X →→ Y holds iff H(Y,Z|X) = H(Y|X) + H(Z|X) Intuition: once X known, uncertainties in Y and Z are independent H(Y|X) = 0.5, H(Z|X) = 0.75, H(Y,Z|X) = 1.25 X Y Z h(Y,Z|X) x1 y1 z1 2.0 y2 z2 x2 y3 z3 1.0 z4 x3 z5 0.0 x4 z6 X Y h(Y|X) x1 y1 1.0 y2 x2 y3 0.0 x3 x4 X Z h(Z|X) x1 z1 1.0 z2 x2 z3 z4 x3 z5 0.0 x4 z6 = Information Theory for Data Management - Divesh & Suresh 106

Information Dependencies [DR00]
Result: Armstrong axioms for FDs derivable from InD inequalities Reflexivity: If Y  X, then X → Y H(Y|X) = 0 for Y  X Augmentation: X → Y  X,Z → Y,Z 0 ≤ H(Y,Z|X,Z) = H(Y|X,Z) ≤ H(Y|X) = 0 Transitivity: X → Y & Y → Z  X → Z 0 ≥ H(Y|X) + H(Z|Y) ≥ H(Z|X) ≥ 0 Information Theory for Data Management - Divesh & Suresh 107

Information Theory for Data Management - Divesh & Suresh
Database Normal Forms Goal: eliminate update anomalies by good database design Need to know the integrity constraints on all database instances Boyce-Codd normal form: Input: a set ∑ of functional dependencies For every (non-trivial) FD R.X → R.Y  ∑+, R.X is a key of R 4NF: Input: a set ∑ of functional and multi-valued dependencies For every (non-trivial) MVD R.X →→ R.Y  ∑+, R.X is a key of R Information Theory for Data Management - Divesh & Suresh 108

Information Theory for Data Management - Divesh & Suresh
Database Normal Forms Functional dependency: X → Y Which design is better? X Y Z x1 y1 z1 z2 x2 y2 z3 z4 x3 y3 z5 x4 y4 z6 X Y x1 y1 x2 y2 x3 y3 x4 y4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 = Information Theory for Data Management - Divesh & Suresh 109

Information Theory for Data Management - Divesh & Suresh
Database Normal Forms Functional dependency: X → Y Which design is better? Decomposition is in BCNF X Y Z x1 y1 z1 z2 x2 y2 z3 z4 x3 y3 z5 x4 y4 z6 X Y x1 y1 x2 y2 x3 y3 x4 y4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 = Information Theory for Data Management - Divesh & Suresh 110

Information Theory for Data Management - Divesh & Suresh
Database Normal Forms Multi-valued dependency: X →→ Y Which design is better? X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 X Y x1 y1 y2 x2 y3 x3 x4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 = Information Theory for Data Management - Divesh & Suresh 111

Information Theory for Data Management - Divesh & Suresh
Database Normal Forms Multi-valued dependency: X →→ Y Which design is better? Decomposition is in 4NF X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 X Y x1 y1 y2 x2 y3 x3 x4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 = Information Theory for Data Management - Divesh & Suresh 112

Well-Designed Databases [AL03]
Goal: use information theory to characterize “goodness” of a database design and reason about normalization algorithms Key idea: Information content measure of cell in a DB instance w.r.t. ICs Redundancy reduces information content measure of cells Results: Well-designed DB  each cell has information content > 0 Normalization algorithms never decrease information content Information Theory for Data Management - Divesh & Suresh 113

Well-Designed Databases [AL03]
Information content of cell c in database D satisfying FD X → Y Uniform distribution p(V) on values for c consistent with D\c and FD Information content of cell c is entropy H(V) H(V62) = 2.0 X Y Z x1 y1 z1 z2 x2 y2 z3 z4 x3 y3 z5 x4 y4 z6 V62 p(V62) h(V62) y1 0.25 2.0 y2 y3 y4 Information Theory for Data Management - Divesh & Suresh 114

Well-Designed Databases [AL03]
Information content of cell c in database D satisfying FD X → Y Uniform distribution p(V) on values for c consistent with D\c and FD Information content of cell c is entropy H(V) H(V22) = 0.0 X Y Z x1 y1 z1 z2 x2 y2 z3 z4 x3 y3 z5 x4 y4 z6 V22 p(V22) h(V22) y1 1.0 0.0 y2 y3 y4 Information Theory for Data Management - Divesh & Suresh 115

Well-Designed Databases [AL03]
Information content of cell c in database D satisfying FD X → Y Information content of cell c is entropy H(V) Schema S is in BCNF iff  D  S, H(V) > 0, for all cells c in D Technicalities w.r.t. size of active domain X Y Z x1 y1 z1 z2 x2 y2 z3 z4 x3 y3 z5 x4 y4 z6 c H(V) c12 0.0 c22 c32 c42 c52 2.0 c62 Information Theory for Data Management - Divesh & Suresh 116

Well-Designed Databases [AL03]
Information content of cell c in database D satisfying FD X → Y Information content of cell c is entropy H(V) H(V12) = 2.0, H(V42) = 2.0 X Y x1 y1 x2 y2 x3 y3 x4 y4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 V12 p(V12) h(V12) y1 0.25 2.0 y2 y3 y4 V42 p(V42) h(V42) y1 0.25 2.0 y2 y3 y4 Information Theory for Data Management - Divesh & Suresh 117

Well-Designed Databases [AL03]
Information content of cell c in database D satisfying FD X → Y Information content of cell c is entropy H(V) Schema S is in BCNF iff  D  S, H(V) > 0, for all cells c in D X Y x1 y1 x2 y2 x3 y3 x4 y4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 c H(V) c12 2.0 c22 c32 c42 Information Theory for Data Management - Divesh & Suresh 118

Well-Designed Databases [AL03]
Information content of cell c in DB D satisfying MVD X →→ Y Information content of cell c is entropy H(V) H(V52) = 0.0, H(V53) = 2.32 X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 V52 p(V52) h(V52) y3 1.0 0.0 V53 p(V53) h(V53) z1 0.2 2.32 z2 z3 z4 0.0 z5 z6 Information Theory for Data Management - Divesh & Suresh 119

Well-Designed Databases [AL03]
Information content of cell c in DB D satisfying MVD X →→ Y Information content of cell c is entropy H(V) Schema S is in 4NF iff  D  S, H(V) > 0, for all cells c in D X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 c H(V) c12 0.0 c22 c32 c42 c52 c62 c72 1.58 c82 c H(V) c13 0.0 c23 c33 c43 c53 2.32 c63 c73 2.58 c83 Information Theory for Data Management - Divesh & Suresh 120

Well-Designed Databases [AL03]
Information content of cell c in DB D satisfying MVD X →→ Y Information content of cell c is entropy H(V) H(V32) = 1.58, H(V34) = 2.32 X Y x1 y1 y2 x2 y3 x3 x4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 V32 p(V32) h(V32) y1 0.33 1.58 y2 y3 V34 p(V34) h(V34) z1 0.2 2.32 z2 z3 z4 0.0 z5 z6 Information Theory for Data Management - Divesh & Suresh 121

Well-Designed Databases [AL03]
Information content of cell c in DB D satisfying MVD X →→ Y Information content of cell c is entropy H(V) Schema S is in 4NF iff  D  S, H(V) > 0, for all cells c in D X Y x1 y1 y2 x2 y3 x3 x4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 c H(V) c12 1.0 c22 c32 1.58 c42 c52 c H(V) c14 2.32 c24 c34 c44 c54 2.58 c64 Information Theory for Data Management - Divesh & Suresh 122

Well-Designed Databases [AL03]
Normalization algorithms never decrease information content Information content of cell c is entropy H(V) X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 c H(V) c13 0.0 c23 c33 c43 c53 2.32 c63 c73 2.58 c83 Information Theory for Data Management - Divesh & Suresh 123

Well-Designed Databases [AL03]
Normalization algorithms never decrease information content Information content of cell c is entropy H(V) X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 X Y x1 y1 y2 x2 y3 x3 x4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 c H(V) c13 0.0 c23 c33 c43 c53 2.32 c63 c73 2.58 c83 c H(V) c14 2.32 c24 c34 c44 c54 2.58 c64 = Information Theory for Data Management - Divesh & Suresh 124

Well-Designed Databases [AL03]
Normalization algorithms never decrease information content Information content of cell c is entropy H(V) X Y Z x1 y1 z1 y2 z2 x2 y3 z3 z4 x3 z5 x4 z6 X Y x1 y1 y2 x2 y3 x3 x4 X Z x1 z1 z2 x2 z3 z4 x3 z5 x4 z6 c H(V) c13 0.0 c23 c33 c43 c53 2.32 c63 c73 2.58 c83 c H(V) c14 2.32 c24 c34 c44 c54 2.58 c64 = Information Theory for Data Management - Divesh & Suresh 125

Database Design: Summary
Good database design essential for preserving data integrity Information theoretic measures useful for integrity constraints FD X → Y holds iff InD measure H(Y|X) = 0 MVD X →→ Y holds iff H(Y,Z|X) = H(Y|X) + H(Z|X) Information theory to model correlations in specific database Information theoretic measures useful for normal forms Schema S is in BCNF/4NF iff  D  S, H(V) > 0, for all cells c in D Information theory to model distributions over possible databases Information Theory for Data Management - Divesh & Suresh 126

Information Theory for Data Management - Divesh & Suresh
Outline Part 1 Introduction to Information Theory Application: Data Anonymization Application: Data Integration Part 2 Review of Information Theory Basics Application: Database Design Computing Information Theoretic Primitives Open Problems Information Theory for Data Management - Divesh & Suresh 127 127

Domain size matters For random variable X, domain size = supp(X) = {xi | p(X = xi) > 0} Different solutions exist depending on whether domain size is “small” or “large” Probability vectors usually very sparse

Entropy: Case I - Small domain size
Suppose the #unique values for a random variable X is small (i.e fits in memory) Maximum likelihood estimator: p(x) = #times x is encountered/total number of items in set. 1 2 1 2 1 5 4 1 2 3 4 5

Entropy: Case I - Small domain size
HMLE = Sx p(x) log 1/p(x) This is a biased estimate: E[HMLE] < H Miller-Madow correction: H’ = HMLE + (m’ – 1)/2n m’ is an estimate of number of non-empty bins n = number of samples Bad news: ALL estimators for H are biased. Good news: we can quantify bias and variance of MLE: Bias <= log(1 + m/N) Var(HMLE) <= (log n)2/N

Entropy: Case II - Large domain size
|X| is too large to fit in main memory, so we can’t maintain explicit counts. Streaming algorithms for H(X): Long history of work on this problem Bottomline: (1+e)-relative-approximation for H(X) that allows for updates to frequencies, and requires “almost constant”, and optimal space [HNO08].

Streaming Entropy [CCM07]
High level idea: sample randomly from the stream, and track counts of elements picked [AMS] PROBLEM: skewed distribution prevents us from sampling lower-frequency elements (and entropy is small) Idea: estimate largest frequency, and distribution of what’s left (higher entropy)

Streaming Entropy [CCM07]
Maintain set of samples from original distribution and distribution without most frequent element. In parallel, maintain estimator for frequency of most frequent element normally this is hard but if frequency is very large, then simple estimator exists [MG81] (Google interview puzzle!) At the end, compute function of these two estimates Memory usage: roughly 1/e2 log(1/e) (e is the error)

Entropy and MI are related
I(X;Y) = H(X,Y) – H(X) – H(Y) Suppose we can c-approximate H(X) for any c > 0: Find H’(X) s.t |H(X) – H’(X)| <= c Then we can 3c-approximate I(X;Y): <= H’(X,Y)+c – (H’(X)-c) – (H’(Y)-c) <= H’(X,Y) – H’(X) – H’(Y) + 3c <= I’(X,Y) + 3c Similarly, we can 2c-approximate H(Y|X) = H(X,Y) – H(X) Estimating entropy allows us to estimate I(X;Y) and H(Y|X)

Computing KL-divergence: Small Domains
“easy algorithm”: maintain counts for each of p and q, normalize, and compute KL-divergence. PROBLEM ! Suppose qi = 0: pi log pi/qi is undefined ! General problem with ML estimators: all events not seen have probability zero !! Laplace correction: add one to counts for each seen element Slightly better: add 0.5 to counts for each seen element [KT81] Even better, more involved: use Good-Turing estimator [GT53] YIeld non-zero probability for “things not seen”.

Computing KL-divergence: Large Domains
Bad news: No good relative-approximations exist in small space. (Partial) good news: additive approximations in small space under certain technical conditions (no pi is too small). (Partial) good news: additive approximations for symmetric variant of KL-divergence, via sampling. For details, see [GMV08,GIM08]

Information-theoretic Clustering
Given a collection of random variables X, each “explained” by a random variable Y, we wish to find a (hard or soft) clustering T such that I(T,X) – bI(T, Y) is minimized. Features of solutions thus far: heuristic (general problem is NP-hard) address both small-domain and large-domain scenarios.

Agglomerative Clustering (aIB) [ST00]
Fix number of clusters k While number of clusters < k Determine two clusters whose merge loses the least information Combine these two clusters Output clustering Merge Criterion: merge the two clusters so that change in I(T;V) is minimized Note: no consideration of b (number of clusters is fixed)

Agglomerative Clustering (aIB) [S]
Elegant way of finding the two clusters to be merged: Let dJS(p,q) = (1/2)(dKL(p,m) + dKL(q,m)), m = (p+q)/2 dJS(p,q) is a symmetric distance between p, q (Jensen-Shannon distance) We merge clusters that have smallest dJS(p,q), (weighted by cluster mass) p m q

Iterative Information Bottleneck (iIB) [S]
aIB yields a hard clustering with k clusters. If you want a soft clustering, use iIB (variant of EM) Step 1: p(t|x) ← exp(-bdKL(p(V|x),p(V|t)) assign elements to clusters in proportion (exponentially) to distance from cluster center ! Step 2: Compute new cluster centers by computing weighted centroids: p(t) = Sx p(t|x) p(x) p(V|t) = Sx p(V|t) p(t|x) p(x)/p(t) Choose b according to [DKOSV06]

Dealing with massive data sets
Clustering on massive data sets is a problem Two main heuristics: Sampling [DKOSV06]: pick a small sample of the data, cluster it, and (if necessary) assign remaining points to clusters using soft assignment. How many points to sample to get good bounds ? Streaming: Scan the data in one pass, performing clustering on the fly How much memory needed to get reasonable quality solution ?

LIMBO (for aIB) [ATMS04] BIRCH-like idea:
Maintain (sparse) summary for each cluster (p(t), p(V|t)) As data streams in, build clusters on groups of objects Build next-level clusters on cluster summaries from lower level

Information Theory for Data Management - Divesh & Suresh
Outline Part 1 Introduction to Information Theory Application: Data Anonymization Application: Data Integration Part 2 Review of Information Theory Basics Application: Database Design Computing Information Theoretic Primitives Open Problems Information Theory for Data Management - Divesh & Suresh 143 143

THANK YOU ! Open Problems
Data exploration and mining – information theory as first-pass filter Relation to nonparametric generative models in machine learning (LDA, PPCA, ...) Engineering and stability: finding right knobs to make systems reliable and scalable Other information-theoretic concepts ? (rate distortion, higher-order entropy, ...) THANK YOU !

References: Information Theory
[CT] Tom Cover and Joy Thomas: Information Theory. [BMDG05] Arindam Banerjee, Srujana Merugu, Inderjit Dhillon, Joydeep Ghosh. Learning with Bregman Divergences, JMLR 2005. [TPB98] Naftali Tishby, Fernando Pereira, William Bialek. The Information Bottleneck Method. Proc. 37th Annual Allerton Conference, 1998 Information Theory for Data Management - Divesh & Suresh 145

References: Data Anonymization
[AA01] Dakshi Agrawal, Charu C. Aggarwal: On the design and quantification of privacy preserving data mining algorithms. PODS 2001. [AS00] Rakesh Agrawal, Ramakrishnan Srikant: Privacy preserving data mining. SIGMOD 2000. [EGS03] Alexandre Evfimievski, Johannes Gehrke, Ramakrishnan Srikant: Limiting privacy breaches in privacy preserving data mining. PODS 2003. Information Theory for Data Management - Divesh & Suresh 146 146 146

References: Data Integration
[AMT04] Periklis Andritsos, Renee J. Miller, Panayiotis Tsaparas: Information-theoretic tools for mining database structure from large data sets. SIGMOD 2004. [DKOSV06] Bing Tian Dai, Nick Koudas, Beng Chin Ooi, Divesh Srivastava, Suresh Venkatasubramanian: Rapid identification of column heterogeneity. ICDM 2006. [DKSTV08] Bing Tian Dai, Nick Koudas, Divesh Srivastava, Anthony K. H. Tung, Suresh Venkatasubramanian: Validating multi-column schema matchings by type. ICDE 2008. [KN03] Jaewoo Kang, Jeffrey F. Naughton: On schema matching with opaque column names and data values. SIGMOD 2003. [PPH05] Patrick Pantel, Andrew Philpot, Eduard Hovy: An information theoretic model for database alignment. SSDBM 2005. Information Theory for Data Management - Divesh & Suresh 147

References: Database Design
[AL03] Marcelo Arenas, Leonid Libkin: An information theoretic approach to normal forms for relational and XML data. PODS 2003. [AL05] Marcelo Arenas, Leonid Libkin: An information theoretic approach to normal forms for relational and XML data. JACM 52(2), , 2005. [DR00] Mehmet M. Dalkilic, Edward L. Robertson: Information dependencies. PODS 2000. [KL06] Solmaz Kolahi, Leonid Libkin: On redundancy vs dependency preservation in normalization: an information-theoretic study of XML. PODS 2006. Information Theory for Data Management - Divesh & Suresh 148

References: Computing IT quantities
[P03] Liam Panninski. Estimation of entropy and mutual information. Neural Computation 15: [GT53] I. J. Good. Turing’s anticipation of Empirical Bayes in connection with the cryptanalysis of the Naval Enigma. Journal of Statistical Computation and Simulation, 66(2), 2000. [KT81] R. E. Krichevsky and V. K. Trofimov. The performance of universal encoding. IEEE Trans. Inform. Th. 27 (1981), [CCM07] Amit Chakrabarti, Graham Cormode and Andrew McGregor. A near-optimal algorithm for computing the entropy of a stream. Proc. SODA 2007. [HNO] Nich Harvey, Jelani Nelson, Krzysztof Onak. Sketching and Streaming Entropy via Approximation Theory. FOCS 2008 [ATMS04] Periklis Andritsos, Panayiotis Tsaparas, Renée J. Miller and Kenneth C. Sevcik. LIMBO: Scalable Clustering of Categorical Data. EDBT 2004 Information Theory for Data Management - Divesh & Suresh 149 149 149

References: Computing IT quantities
[S] Noam Slonim. The Information Bottleneck: theory and applications. Ph.D Thesis. Hebrew University, 2000. [GMV08] Sudipto Guha, Andrew McGregor, Suresh Venkatasubramanian. Streaming and sublinear approximations for information distances. ACM Trans Alg. 2008 [GIM08] Sudipto Guha, Piotr Indyk, Andrew McGregor. Sketching Information Distances. JMLR, 2008. Information Theory for Data Management - Divesh & Suresh 150 150 150