# ML Wiki

## Probabilistic LSA

This is a probabilistic extension to Latent Semantic Analysis

### Problem

Notation and problem:

• $D_1, \ ... \ , D_n$ are documents
• $T_1, \ ... \ , T_k$ are topics (sort of "clusters")
• each document may belong to several topics - so these "clusters" are Fuzzy
• probability of $D_i$ belonging to $T_j$ is $P(T_j \mid D_i)$
• but cluster membership is secondary in this problem
• the main problem is to find latent topics that generated documents - which is why it's called Topic Modeling
• let $t_1, \ ... \ , t_d$ be $d$ terms from the lexicon
• then the probability that $t_l$ occurs in $T_j$ is $P(t_l \mid T_j)$

### Learning

Thus, we need to estimate the following probabilities:

• $P(T_j \mid D_i)$ and $P(t_l \mid T_j)$
• usually parameters are learned via maximum likelihood methods like Expectation Maximization

We need to learn $P(T_j \mid D_i)$ and $P(t_l \mid T_j)$

• $P(t_l \mid D_i)$ can be expressed via them:
• $P(t_l \mid D_i) = \sum\limits_{j=1}^k p(t_l \mid T_i) \, P(T_j \mid D_i)$
• thus, for each $t_l$ and $D_i$ we can generate $n \times d$ matrix of probabilities
• these probabilities are learned from term-document matrix $X$: $X_{il}$ is # of times $t_l$ occurred in $D_i$
• so we can use Maximum Likelihood Estimator to maximize the product of probabilities of terms we observed

Optimization:

• we will optimize the log likelihood $\sum_{i,l} X_{il} \cdot \log P(t_l, D_i)$
• s.t. $\sum_l P(t_l \mid T_j) = 1$ for all $T_j$ and $\sum_j P(T_j \mid D_i) = 1$ for all $D_i$
• can use Lagrange Multipliers for this

## Latent Dirichlet Allocation

is an extension of Probabilistic LSA

• model term-topic probabilities and topic-document probabilities with Dirichlet Distribution
• so LDA is a Bayesian version of PLSA
• but LSA overfits less than PLSA because it has less parameters to fit

## References

• Hofmann, Thomas. "Probabilistic latent semantic analysis." 1999. [1]

## Sources

• Aggarwal, Charu C., and ChengXiang Zhai. "A survey of text clustering algorithms." Mining Text Data. Springer US, 2012. [2]