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, Write down the expressions for the E- and M-steps. on the (k + 1) th iteration of the EMalgorithm.
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, Write down the expressions for the E- and M-steps. on the (k + 1) th iteration of the EMalgorithm.
Solution
The EM algorithm is a two-step process that includes the E-step (Expectation) and the M-step (Maximization). It is used to find the maximum likelihood estimates of parameters in statistical models, where the model depends on unobserved latent variables.
In the context of a two-component normal mixture model, the EM algorithm works as follows:
- E-Step (Expectation):
In this step, we calculate the expected value of the log-likelihood function, with respect to the conditional distribution of the data given the current parameter estimates.
For a two-component normal mixture model, the E-step involves calculating the posterior probabilities, which represent the probability that a given observation comes from a particular component of the mixture.
The posterior probability for the kth iteration, Z_ik, that observation i comes from component k is given by:
Z_ik = π_k * N(w_i | µ_k, σ^2) / Σ_k'[ π_k' * N(w_i | µ_k', σ^2) ]
where π_k is the mixing proportion for component k, N(w_i | µ_k, σ^2) is the normal density function for component k, and the denominator is a sum over all components.
- M-Step (Maximization):
In this step, we maximize the expected log-likelihood found in the E-step with respect to the parameters to update our estimates of the parameters.
For a two-component normal mixture model, the M-step involves updating the estimates for the mixing proportions, means, and variances.
The updated mixing proportion for component k, π_k, is given by:
π_k = Σ_i Z_ik / n
The updated mean for component k, µ_k, is given by:
µ_k = Σ_i Z_ik * w_i / Σ_i Z_ik
The updated variance, σ^2, is given by:
σ^2 = Σ_i,k Z_ik * (w_i - µ_k)^2 / Σ_i,k Z_ik
where the sums are over all observations.
These steps are repeated until the estimates of the parameters converge.
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