PLSA

Probabilistic latent semantic analysis (PLSA), also known as probabilistic latent semantic indexing (PLSI, especially in information retrieval circles) is a statistical technique for the analysis of two-mode and co-occurrence data. In effect, one can derive a low-dimensional representation of the observed variables in terms of their affinity to certain hidden variables, just as in latent semantic analysis, from which PLSA evolved.

Compared to standard latent semantic analysis which stems from linear algebra and downsizes the occurrence tables (usually via a singular value decomposition), probabilistic latent semantic analysis is based on a mixture decomposition derived from a latent class model.

Considering observations in the form of co-occurrences ${\displaystyle (w,d)}$ of words and documents, PLSA models the probability of each co-occurrence as a mixture of conditionally independent multinomial distributions:

with 'c' being the words' topic. Note that the number of topics is a hyperparameter must be chosen in advance and is not estimated from the data. The first formulation is the symmetric formulation, where ${\displaystyle w}$ and ${\displaystyle d}$ are both generated from the latent class ${\displaystyle c}$ in similar ways (using the conditional probabilities ${\displaystyle P(d|c)}$ and ${\displaystyle P(w|c)}$), whereas the second formulation is the asymmetric formulation, where, for each document ${\displaystyle d}$, a latent class is chosen conditionally to the document according to ${\displaystyle P(c|d)}$, and a word is then generated from that class according to ${\displaystyle P(w|c)}$. Although we have used words and documents in this example, the co-occurrence of any couple of discrete variables may be modelled in exactly the same way.

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