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Problem

Difficulty of Maximum Likelihood Estimation with Latent Variables

If we have complete data {X,Z}\left\{X, Z\right\}, it is easy to maximize the likelihood p(X,Zθ)p(X, Z | \theta). Unfortunately, with incomplete data (XX only), we must marginalize over the latent variables ZZ, so:

ln(p(Xθ))=ln(Zp(X,Zθ))\ln(p(X|\theta)) = \ln\left(\sum_Z p(X,Z|\theta)\right)

The parameters θ\theta inside the logarithm make it difficult to compute the maximum likelihood directly.

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Updated 2026-06-16

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Data Science