Consider an observed random sample of size n, w1, . . . , wn, from a normal distribution N(µ, σ2 ). To the 75 observations in the dataset Data-A1a.csv apply the EM algorithm to fit via maximum likelihood the two-component normal mixture density with common variances, f(w; Ψ) = X 2 i=1 πi φ(w; µi , σ2 ), where φ(w; µ, σ2 ) = (2πσ2 ) −1/2 exp{−1 2 (w − µ) 2 /σ2 } and Ψ = (π1, µ1, µ2, σ2 ) T . To this end, (i) [1/2 mark] Specify the EM framework
Question
Consider an observed random sample of size n, w1, . . . , wn, from a normal distribution N(µ, σ2 ). To the 75 observations in the dataset Data-A1a.csv apply the EM algorithm to fit via maximum likelihood the two-component normal mixture density with common variances, f(w; Ψ) = X 2 i=1 πi φ(w; µi , σ2 ), where φ(w; µ, σ2 ) = (2πσ2 ) −1/2 exp{−1 2 (w − µ) 2 /σ2 } and Ψ = (π1, µ1, µ2, σ2 ) T . To this end, (i) [1/2 mark] Specify the EM framework
Solution
The EM (Expectation-Maximization) framework is a two-step iterative process used for finding maximum likelihood estimates of parameters in statistical models, especially for models which depend on unobserved latent variables. It is particularly useful for problems where the direct computation of the maximum likelihood estimate is difficult.
Here is the EM framework for the given problem:
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Initialization: Choose initial values for the parameters Ψ = (π1, µ1, µ2, σ2). These can be chosen randomly, or based on some heuristic.
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E-Step (Expectation Step): Calculate the expected value of the log-likelihood function, given the observed data and the current estimate of the parameters. In the context of a mixture model, this involves calculating the posterior probabilities of the latent variable, which indicate the probability that each observation belongs to each component of the mixture.
For a two-component normal mixture model, the posterior probability that observation w belongs to component i is given by:
γi(w) = πi * φ(w; µi, σ2) / Σj πj * φ(w; µj, σ2)
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M-Step (Maximization Step): Find the parameters that maximize the expected log-likelihood found on the E-step. This typically involves setting the derivative of the expected log-likelihood with respect to the parameters to zero and solving for the parameters.
For a two-component normal mixture model, the updates for the parameters are given by:
πi = Σw γi(w) / n
µi = Σw γi(w) * w / Σw γi(w)
σ2 = Σw γi(w) * (w - µi)^2 / Σw γi(w)
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Check for Convergence: If the parameters have not changed significantly, or the increase in the log-likelihood is below a certain threshold, stop the algorithm. Otherwise, return to the E-step with the updated parameters.
This process is repeated until the parameters converge to a stable set of values. The final estimates of the parameters are then used to fit the two-component normal mixture density to the data.
Similar Questions
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