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. The sample complexity of learning Bayesian Networks Or Zuk*^, Shiri Margel* and Eytan Domany* *Dept. of Physics of Complex Systems Weizmann Inst. of Science ^Broad Inst. Of MIT and Harvard

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2 Let X 1,..,X n be binary random variables. A Bayesian Network is a pair B ≡. G – Directed Acyclic Graph (DAG). G =. V = {X 1,..,X n } the vertex set. Pa G (i) is the set of vertices X j s.t. (X j,X i ) in E. θ - Parameterization. Represent conditional probabilities: Together, they define a unique joint probability distribution P B over the n random variables. Introduction X2X2 0.80.21 0.050.950 10X1X1 X1X1 X2X2 X3X3 X5X5 X4X4 X 5 {X 1,X 4 } | {X 2,X 3 }

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3 Structure Learning We looked at a score based approach: For each graph G, one gives a score based on the data S(G) ≡ S N (G; D) (N is the sample size) Score is composed of two components: 1. Data fitting (log-likelihood) LL N (G;D) = max LL N (G,Ө;D) 2. Model complexity Ψ(N) |G| |G| = The Dimension. # parameters in (G,Ө). S N (G) = LL N (G;D) - Ψ(N) |G| This is known as the MDL (Minimum Description Length) score. Assumption : 1 << Ψ(N) << N. Score is consistent. Of special interest: Ψ(N) = ½log N. The BIC score (Bayesian Information Criteria) is asymptotically equivalent to the Bayesian score.

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4 Previous Work [Friedman&Yakhini 96] Unknown structure, no hidden variables. [Dasgupta 97] Known structure, Hidden variables. [Hoeffgen, 93] Unknown structure, no hidden variables. [Abbeel et al. 05] Factor graphs [Greiner et al. 97] classification error. Concentrated on approximating the generative distribution. Typical results: N > N 0 (ε,δ) D(P true || P learned ) 1- δ. D – some distance between distributions. Usually relative entropy (we use relative entropy from now on). We are interested in learning the correct structure. Intuition and practice A difficult problem (both computationally and statistically.) Empirical study: [Dai et al. IJCAI 97] New: [Wainwright et al. 06], [Bresler et al. 08] – undirected graphs

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5 Structure Learning Assume data is generated from B * =, with P B* generative distribution. Assume further that G* is minimal w. resp. to P B* : |G*| = min {|G|, P B* subset of M(G)) An error occurs when any ‘wrong’ graph G is preferred over G*. Many possible G’s. Complicated relations between them. Observation: Directed graphical models (with no hidden variables) are curved exponential families [Geiger et al. 01]. [Haughton 88] – The MDL score is consistent. [Haughton 89] – Bounds on the error probabilities: P (N) (under-fitting) ~ O(e -αN ) ; P (N) (over-fitting) ~ O(N -β ) Previously: Bounds only on β. Not on α, nor on the multiplicative constants.

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6 Structure Learning Simulations: 4-Nodes Networks. Totally 543 DAGs, in 185 equivalence classes. Draw at random a DAG G*. Draw all parameters θ uniformly from [0,1]. Generate 5,000 samples from P Gives scores S N (G) to all G’s and look at S N (G*)

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7 Structure Learning Relative entropy between the true and learned distributions: Fraction of Edge Learned Correctly Rank of the correct structure (equiv. class):

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8 All DAGs and Equivalence Classes for 3 Nodes

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9 Two Types of Error An error occurs when any ‘wrong’ graph G is preferred over G*. Many possible G’s. Study them one by one. Distinguish between two types of errors: 1. Graphs G which are not I-maps for P B* (‘under- fitting’). These graphs impose to many independency relations, some of which do not hold in P B*. 2. Graphs G which are I-maps for P B* (‘over-fitting’), yet they are over parameterized, |G| > |G*| Study each error separately.

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10 'Under-fitting' Errors 1. Graphs G which are not I-maps for P B* Intuitively, in order to get S N (G*) > S N (G), we need: a. P (N) to be closer to P B* than to any point Q in G b. The penalty difference Ψ(N) (|G| - |G*|) is small enough. (Only relevant for |G*| > |G|). For a., use concentration bounds (Sanov). For b., simple algebraic manipulations.

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11 Sanov's Theorem [Sanov 57]: Draw N sample from a probability distribution P. Let P (N) be the sample distribution. Then: Pr( D(P (N) || P) > ε) < N (n+1) 2 -εN Used in our case to show: (for some c>0) For |G| ≤ |G*|, we are able to bound c: 'Under-fitting' Errors 1. Graphs G which are not I-maps for P B*

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12 Upper-bound on decay exponent: c≤D(G||P B* )log 2. Could be very slow if G is close to P B* Lower-bound: Use Chernoff Bounds to bound the difference between the true and sample entropies. 'Under-fitting' Errors Two important parameters of the network: a. ‘Minimal probability’: b. ‘Minimal edge information’:

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13 Here errors are Moderate deviations events, as opposed to Large deviations events in the previous case. The probability of error does not decay exponentially with N, but is O(N -β ). By [Woodroofe 78], β=½(|G|-|G*|). Therefore, for large enough values of N, error is dominated by over-fitting. 'Over-fitting' Errors 2. Graphs G which are over-parameterized I- maps for P B*

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14 Perform simulations: Take a BN over 4 binary nodes. Look at two wrong models Example What happens for small values of N? X1X1 X2X2 X3X3 X4X4 G*G* X1X1 X2X2 X3X3 X4X4 G2G2 X1X1 X2X2 X3X3 X4X4 G1G1

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15 Example Errors become rare events. Simulate using importance sampling (30 iterations): [Zuk et al. UAI 06]

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16 Recent Results/Future Directions Want to minimize sum of errors (‘over-fitting’+’under- fitting’). Change penalty in the MDL score to Ψ(N) = ½log N – c log log N # variables n >> 1. Small Max. degree # parents ≤ d. Simulations for trees (computationally efficient: Chow-Liu) Hidden variables – Even more basic questions (e.g. identifiably, consistency) are unknown generally. Requiring exact model was maybe to strict – perhaps it is likely to learn wrong models which are close to the correct one. If we require only to learn 1-ε of the edges – how does this reduce sample complexity?

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