Answered: Show that the M step for ML estimation…

Show that the M step for ML estimation of a mixture of Gaussians is given by: ΣΤΑΧ Tk Σ; rix(x; - μ.)(x; – μ.)" _Σraxx – rμμ Tk T'k με ΣΗ = =

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter8: Further Techniques And Applications Of Integration

Section8.2: Integration By Parts

Problem 41E

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Question

I really don't understand this concepts within the attached.

Show that the M step for ML estimation of a mixture of Gaussians is given by:
ΣTAX
Th
Σ;rk(x
flk =
ΣΕ
=
- μπ.)(x - μ.)^ _Erixx – rμμ
-
Tk
Th
=
T

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Step 1: Given Data:

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Step 2: Derivation of the M-step updates for maximum likelihood estimation (MLE) of a mixture of Gaussians:

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Step 3: Continuing the derivation:

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The log-likelihood can be expressed as:

There are two Sigmas, are they read a the Sum of sum or is it the Sum multiplied by the sum? (see attached)

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