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What is a latent variable approach?
A latent variable is a variable that is inferred using models from observed data. Approaches to inferring latent variables from data include: using a single observed variable, multi-item scales, predictive models, dimension reduction techniques such as factor analysis, structural equation models, and mixture models.
Why do we need latent variables?
The use of latent variables can serve to reduce the dimensionality of data. Many observable variables can be aggregated in a model to represent an underlying concept, making it easier to understand the data. In this sense, they serve a function similar to that of scientific theories.
What is the role of latent variables in a model?
The primary role of the latent variables is to allow a complicated distribution over the observed variables to be represented in terms of a model constructed from simpler (typically exponential family) conditional distributions.
How to think of Z as a latent variable?
For the GMM context, we can think of z as a binary random variable with 1 of K representation (you can think of this as categorical representation), in which a particular element is equal to 1 and all other elements are 0. Exp 3: Marginal distribution over latent variable.
What is the joint probability distribution of LVM?
Generally a LVM p, is a probability distribution over 2 sets of variables x, z; p (x, z). x are observed variables at learning time in a data-set D and z are never observed. Joint probability distribution of the model can be written as p (x, z) = p (x|z) p (z).
Which is a valid expression for a latent variable?
The above expression is valid for a single data-point. For several data-points we will have a corresponding latent variable associated with each observed data-point. Rather than working with only the marginal distribution p (x), we can now work with the joint distribution p (x, z). Did we make things more complicated?