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- On Monday, June 4, 2018
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As we saw, our target is to build a mixture model which does not require us to specify the number of k clusters/components from the beginning.
Using Dirichlet Processes allows us to have a mixture model with infinite components which can be thought as taking the limit of the finite model for k to infinity.
Where G is defined as and used as a short notation for which is a delta function that takes 1 if and 0 elsewhere.
Finally we can define a Density distribution which is our mixture distribution (countable infinite mixture) with mixing proportions and mixing components .
The G0 is the base distribution of DP and it is usually selected to be conjugate prior to our generative distribution F in order to make the computations easier and make use of the appealing mathematical properties.
The θi is a parameter vector which is drawn from the G distribution and contains the parameters of the cluster, F distribution is parameterized by θi and xi is the data point generated by the Generative Distribution F.
The model defined in the previous segment is mathematically solid, nevertheless it has a major drawback: for every new xi that we observe, we must sample a new θi taking into account the previous values of θ.
This way instead of using θi to denote both the cluster parameters and the cluster assignments, we use the latent variable zi to indicate the cluster id and then use this value to assign the cluster parameters.
As a result, we no longer need to sample a θ every time we get a new observation, but instead we get the cluster assignment by sampling zi from CRP.
Nevertheless this algorithms requires us to select a G0 which is a conjugate prior of F generative distribution in order to be able to solve analytically the equations and be able to sample directly from .
- On Thursday, June 27, 2019
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