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### Latent Tree Models

Nevin L. Zhang

Dept. of Computer Science & Engineering

The Hong Kong Univ. of Sci. & Tech.

http://www.cse.ust.hk/~lzhang

Latent Tree Models

- Part I: Non-Technical Overview (25 minutes)
- Part II: Definition and Properties (25 minutes)
- Part III: Learning Algorithms
(110 minutes, 30 minutes break half way)

- Part IV: Applications (50 minutes)

Part I: Non-Technical Overview

- Latent tree models
- What can LTMs be used for:
- Discovery of co-occurrence/correlation patterns
- Discovery of latent variable/structures
- Multidimensional clustering

- Examples
- Danish beer survey data
- Text data

Latent Tree Models (LTMs)

- Tree-structured probabilistic graphical models
- Leaves observed (manifest variables)
- Discrete or continuous

- Internal nodes latent (latent variables)
- Discrete

- Each edge is associated with a conditional distribution
- One node with marginal distribution
- Defines a joint distributions over all the variables
(Zhang, JMLR 2004)

- Leaves observed (manifest variables)

Latent Tree Analysis (LTA)

From data on observed variables, obtain latent tree model

Learning latent tree models: Determine

- Number of latent variables
- Numbers of possible states for latent variables
- Connections among nodes
- Probability distributions

LTA on Danish Beer Market Survey Data

- 463 consumers, 11 beer brands
- Questionnaire: For each brand:
- Never seen the brand before (s0);
- Seen before, but never tasted (s1);
- Tasted, but do not drink regularly (s2)
- Drink regularly (s3).

(Mourad et al. JAIR 2013)

Why variables grouped as such?

- Responses on brands in each group strongly correlated.
- GronTuborg and Carlsberg: Main mass-market beers
- TuborgClas and CarlSpec: Frequent beers, bit darker than the above
- CeresTop, CeresRoyal, Pokal, …: minor local beers

- In general, LTA partitions observed variables into groups such that
- Variables in each group are strongly correlated, and
- The correlations among each group can be properly be modeled using one single latent variable

Multidmensional Clustering

- Each Latent variable gives a partition of consumers.
- H1:
- Class 1: Likely to have tasted TuborgClas, Carlspecand Heineken , but do not drink regularly
- Class 2: Likely to have seen or tasted the beers, but did not drink regularly
- Class 3: Likely to drink TuborgClas and Carlspec regularly

- H1:
- H0 and H2 give two other partitions.
- In general, LTA is a technique for multiple clustering.
- In contrast, K-Means, mixture models give only one partition.

Unidimensional vs Multidimensional Clustering

- Grouping of objects intoclusters such that objects in the same cluster are similar while objects from different clusters are dissimilar.

- Result of clustering is often a partition of all the objects.

How to Cluster Those?

Style of picture

How to Cluster Those?

Type of object in picture

Multidimensional Clustering

- Complex data usually have multiple facets and can be meaningfully partitioned in multiple ways. Multidimensional clustering / Multi-Clustering
- LTA is a model-based method for multidimensional clustering.
- Other methods: http://www.siam.org/meetings/sdm11/clustering.pdf

Clustering of Variables and Objects

- LTA produces a partition of observed variables.
- For each cluster of variables, it produces a partition of objects.

Binary Text Data: WebKB

- 1041web pages collected from 4 CS departments in 1997
- 336 words

Why variables grouped as such?

- Words in each group tend to co-occur.
- On binary text data, LTA partitions word variables into groups such that
- Words in each group tend to co-occur and
- The correlations can be properly be explained using one single latent variable

LTA is a method for identifying co-occurrence relationships.

Multidimensional Clustering

LTA is an alternative approach to topic detection

- Y66=4: Object Oriented Programming (oop)
- Y66=2: Non-oop programming
- Y66=1: programming language
- Y66=3: Not on programming

More on this in Part IV

Summary

- Latent tree models:
- Tree-structured probabilistic graphical models
- Leaf nodes: observed variables
- Internal nodes: latent variable

- What can LTA be used for:
- Discovery of co-occurrence patterns in binary data
- Discovery of correlation patterns in general discrete data
- Discovery of latent variable/structures
- Multidimensional clustering
- Topic detection in text data
- Probabilistic modelling

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