1. Probabilistic Latent Preference Analysis
Nathan Liu, Min Zhao, Qiang Yang
Presented by Zachary

2. Outline Motivation Probabilistic Latent Semantic Analysis
Bradley-Terry Model
Probabilistic Latent Preference Analysis
Experiments and evaluation

3. Motivation
Ultimate goal of recommender system is to produce a list of items that a specific user would prefer.
Rating V.S. Ranking
Item
True rating
Predicted rating 1 4
4.5
2.8 2 5
4.3
4.1 3
2.1
2.6
MAE: 0.4
MAE: 0.9

4. Probabilistic Latent Semantic Analysis
A model-based approach to CF
Also known as Aspect Model

5. Probabilistic Latent Semantic Analysis
The rating a user give to an item is decomposed into a sum of products.
z is the latent class
Probability that class z (can be seen as community
in CF) would assign score r to item i.
Mixing proportion

6. Probabilistic Latent Semantic Analysis
Model as a Gaussian distribution
Mixing proportion can be modeled as a categorical distribution

7. Probabilistic Latent Semantic Analysis
To make predictions, we compute the expected rating

8. Probabilistic Latent Semantic Analysis
Model parameters can be learnt by maximizing the following log likelihood of observed data
This can be readily solved using EM algorithm

9. Bradley-Terry Model
A probability distribution defined for ranking of a set of n items
Consider a pair of items i and j, we define
Indicator that item i is preferred to item j
Model parameter , can be
interpreted as the utility associated
with an item.

10. Bradley-Terry Model
Bradley-Terry model defined for a pair of items can be extended to define a probability distribution over rankings
A ranking
Normalization constant
All possible ranking

11. Bradley-Terry Model Previous formula can be simplified to
Rank of i in ranking π
Starting from 1
Normalization constant not depending on π

12. Bradley-Terry Model There is a single most probable ranking
Can be obtained by sorting in decreasing order

13. Probabilistic Latent Preference Analysis
A model similar to pLSA
Latent class mixture model
With Bradley-Terry model incorporated

14. Probabilistic Latent Preference Analysis
Define to be 1 if and 0 otherwise
is unknown if either or is unknown or
Define each observed pairwise preference as

15. Probabilistic Latent Preference Analysis
is modeled as Bradley-Terry Model with parameter

16. Probabilistic Latent Preference Analysis
Once we have defined , we can compute the log likelihood of all observed preference as:
Denote the set (u,i,j) triples for which is observed

17. Probabilistic Latent Preference Analysis
Parameters can, again, be learnt using EM algorithm
We have closed form solution for
However, no closed form solution for Bradley-Terry model exists
There is an efficient numerical algorithm (refer to paper)

18. Probabilistic Latent Preference Analysis
Ranking prediction
Combinatorial search
Time complexity is too high

19. Probabilistic Latent Preference Analysis
Ranking prediction
Approximate solution
Sort to get a ranking

20. Experiments and evaluation
Metric used for evaluation
NDCG
Rate assigned by u to item at position p
Set of users included in test data
Normalization term

21. Experiments and evaluation
Data set used for evaluation
Eachmovie (random picked partial data)
Netflix (random picked partial data)
Pick 10,600 users randomly
10,000 users for training
100 users for parameter tuning (k)
500 active users, 80% training, 20% testing

22. Experiments and evaluation
Impact of k (number of latent classes)
Eachmovie
Netflix

23. Experiments and evaluation