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Unformatted text preview: Model Learning and Clustering CPS170 Ron Parr material from: Lise Getoor, Andrew Moore, Tom [email protected], SebasDan Thrun, Rich Maclin Unsupervised Learning •  Supervised learning: Data <x1, x2, … xn, y> •  Unsupervised Learning: Data <x1, x2, … xn> •  So, what’s the big deal? •  Isn’t y just another feature? •  No explicit performance objecDve –  Bad news: Problem not necessarily well defined without further assumpDons –  Good news: Results can be useful for more than predicDng y 1 Model Learning •  Produce a global summary of the data •  Not an exact copy •  Consider space of models M and dataset D •  One approach: Maximize P(M|D) •  How to do this? Bayes Rule: P (M | D ) = P(D | M)P(M) P(D) € Example: Modeling Coin Flips •  Suppose we have observed: D=HTTHT •  Which is a [email protected] model? –  P(H=0.4) –  P(H=0.5) P (M | D ) = P(D | M)P(M) P(D) P(D | (P(H = 0.5)) = 0.5 5 = 0.312 2 3 P(D | (P(H = 0.4)) = 0.4 * 0.6 = 0.3456 € What about P(D) and P(M)??? € 2 Model Learning With Bayes Rule P (M | D ) = P(D | M)P(M) P(D) •  We call P(D|M) the likelihood •  We can ignore P(D)… Why? € •  What about P(M)? –  Call this a our prior probability on models –  If P(M) is uniform (all models equally likely) then maximizing P(D|M) is equivalent to maximizing P(M|D) (Call this the maximium likelihood approach.) Using Priors •  Suppose we have good reason to expect that the coin is fair •  Should we really conclude P(H)=0.4? •  Suppose we think P(P(H=0.5)) = 2 x P(P(H=0.4)) •  This means P(D|P(H=0.4)) must be 2X larger than P (D|P(H=0.5)) to compensate if P(H=0.4) is to maximize the posterior probability P (M | D ) = P(D | M)P(M) P(D) € 3 Data Can Overwhelm a Prior Specifying Priors •  In our coin example, we considered just two models P (H=0.4) and P(H=0.5) •  In general, we might want to specify a distribuDon over all possible coin probabiliDes •  This introduces complicaDons: –  P(M) is now a distribuDon over a conDnuous parameter –  Need to use calculus to find maximizer of P(D|M)P(M) 4 Conjugate Priors •  A likelihood and prior are said to be conjugate if their product has the same parametric form as the prior •  (This is outside the scope of the class, but we provide one nice example.) •  The beta distribution is conjugate to the binomial distribution –  Can think of the beta distribution as specifying a number of “imagined” heads and tails –  Maximum of the posterior adds together observed heads and tails with imagined heads and tails –  Examples: •  Prior of 100 heads and 100 tails is a strong prior towards fairness •  Prior of 1 head and 1 tail is a weak prior towards fairness Clustering as Modeling •  Clustering assigns points in a space to clusters •  Example: By examining x ­rays of cancer tumors, one might idenDfy different subtypes of cancer based upon growth [email protected] •  Each cluster has its own probabilisDc model describing how items of that cluster’s type behave 5 Examples of Clustering ApplicaDons •  MarkeDng: Help marketers discover disDnct groups in their customer bases, and then use this knowledge to develop targeted markeDng programs •  Land use: IdenDficaDon of areas of similar land use in an earth observaDon database •  Insurance: IdenDfying groups of motor insurance policy holders with similar claim cost •  City ­planning: IdenDfying groups of houses according to their house type, value, and geographical locaDon •  Earth ­quake studies: Observed earth quake epicenters should be clustered along conDnent faults Example of SubtleDes in Clustering •  Household Dataset: locaDon, income, number of children, rent/own, crime rate, number of cars •  Appropriate clustering may depend on use: –  Goal to minimize delivery Dme ⇒ cluster by locaDon –  Others? –  Clustering work ojen suffers from mismatch between the clustering objecDve funcDon and the performance criterion 6 Clustering Desiderata •  DecomposiDon or parDDon of data into groups so that –  Points in one group are similar to each other –  Are as different as possible from the points in other groups •  Measure of distance is fundamental •  Explicit representaDon: –  D(x(i),x(j)) for each x –  Only feasible for small domains •  Implicit representaDon by measurement: –  Distance computed from features –  Implement this as a funcDon Families of Clustering Algorithms •  ParDDon ­based methods –  e.g., K ­means •  Hierarchical clustering –  e.g., hierarchical agglomeraDve clustering •  ProbabilisDc model ­based clustering –  e.g., mixture models •  Graph ­based Methods –  e.g., spectral methods 7 K ­means •  Start with randomly chosen cluster centers •  Assign points to closest cluster •  Recompute cluster centers •  Reassign points •  Repeat unDl no changes K ­means example X(5) X(7) X(4) X(8) X(6) X(1) X(2) X(3) 8 K ­means example c3 X(5) X(7) c2 X(4) X(8) X(6) c1 X(1) X(2) X(3) K ­means example c3 X(5) X(7) c2 X(4) X(8) X(6) c1 X(1) X(2) X(3) 9 K ­means example c3 X(5) X(7) c2 X(4) X(8) X(6) c1 X(1) X(2) X(3) K ­means example c3 X(5) X(7) c2 X(4) X(8) X(6) c1 X(1) X(2) X(3) 10 K ­means example c3 X(4) X(5) X(7) c2 X(8) X(6) c1 X(1) X(2) X(3) K ­means example #2 X(5) X(7) X(4) X(8) X(6) X(1) X(2) X(3) 11 K ­means example #2 c3 X(5) X(7) X(4) X(8) X(6) c2 c1 X(1) X(2) X(3) K ­means example #2 c3 X(5) X(7) X(4) X(8) X(6) c2 X(1) X(2) c1 X(3) 12 Demo [email protected]://home.dei.polimi.it/[email protected]/Clustering/tutorial_html/AppletKM.html Complexity •  Does algorithm terminate? yes •  Does algorithm converge to opDmal clustering? Can only guarantee local opDmum •  Time complexity one iteraDon? nk 13 Understanding k ­Means •  Implicitly models data as coming from a Gaussian distribuDon centered at cluster centers •  log P(data) ~ sum of squared distances P( x i ∈ c j ) ∝ e − ( x i −c j ) 2 P(data) = ∏ P( x i ∈ cclustering( i ) ) i log(P(data)) = α ∑ ( x i − cclustering( i ) )2 i € Understanding k ­Means II •  Each step of k ­Means increases P(data) –  Reassigning points moves points to clusters for which their coordinates have higher probability –  RecompuDng means moves cluster centers to increase the average probability of points in the cluster •  Fixed number of assignments and monotonic score implies convergence 14 Understanding k ­Means III P (M | D ) = P(D | M)P(M) P(D) •  Can view k ­means as max likelihood method with a twist –  Unlike the coin toss example, there is a hidden variable with each € datum – the cluster membership –  k ­means iteraDvely improves its guesses about these hidden pieces of informaDon •  k ­means can be interpreted as an instance of a general approach to dealing with hidden variables called ExpectaDon MaximizaDon (EM) But How Do We Pick k? •  SomeDmes there will be an obvious choice given background knowledge or the intended use of the clustering output •  What if we just iterated over k? –  Picking k=n will always maximize P(D|M) –  We could introduce a prior over models using P(M) in Bayes rule •  Compare prior over models with regularizaDon: –  RegularizaDon in regression penalized overly complex soluDons –  We can assign models with a high number of clusters low probability to achieve a similar effect –  (In general, use of priors subsumes regularizaDon.) 15 Is Clustering Well Defined? •  Clustering is not clearly axiomaDzed •  Can we define an “opDmal” clustering w/o specifying an a priori preference (prior) on the cluster sizes or making addiDonal assumpDons? •  Kleinberg: Clustering is impossible under some plausible assumpDons (IOW, union of unstated assumpDons made by clustering algorithms is logically inconsistent) •  Recent efforts make progress putng clustering on more solid ground Model Learning Conclusion •  Ojen seek to find the most likely model given the data •  Can be viewed as maximizing the posterior P(M|D) using Bayes rule •  Model learning can be applied to: –  –  –  –  Coin flips Clustering Learning parameters of Bayes nets or HMMs etc. •  Some care must go into formulaDon of modeling assumpDons to avoid degenerate soluDons, e.g., assigning every point to its own cluster •  Priors can help avoid degenerate soluDons 16 ...
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