Contents
How is Em used in Gaussian mixture models?
In this note, we will introduce the expectation-maximization (EM) algorithm in the context of Gaussian mixture models. Let N(μ, σ2) denote the probability distribution function for a normal random variable.
When to stop a Gaussian mixture in Em?
If the log-likelihood has changed by less than some small ϵ, stop. Otherwise, go back to step 2. The EM algorithm is sensitive to the initial values of the parameters, so care must be taken in the first step. However, assuming the initial values are “valid,” one property of the EM algorithm is that the log-likelihood increases at every step.
How to calculate the marginal distribution of a Gaussian mixture?
Let N(μ, σ2) denote the probability distribution function for a normal random variable. In this scenario, we have that the conditional distribution Xi | Zi = k ∼ N(μk, σ2k) so that the marginal distribution of Xi is: P(Xi = x) = K ∑ k = 1P(Zi = k)P(Xi = x | Zi = k) = K ∑ k = 1πkN(x; μk, σ2k)
Which is the M-step of the EM model?
M-step: Estimate new parameters ^ μk, ^ σ2k and ^ πk with the current values of γZi(k) using equations (3), (4) and (5). Evaluate the log-likelihood with the new parameter estimates. If the log-likelihood has changed by less than some small ϵ, stop. Otherwise, go back to step 2.
How does the likelihood increase in the EM algorithm?
At each step of the iteration, the log likelihood increased, which shows that the fit of the Gaussian mixture model improved at each iteration. This is one of the features of the EM algorithm: the likelihood always increases on successive steps.
How to fit a multivariate Gaussian mixture model?
If x is a d-dimensional vector, you need to estimate τ i, μ i, and Σ i for each of the G groups, for a total of G* (1 + d + d* (d+1)/2) – 1 parameter estimates. The d* (d+1)/2 expression is the number of free parameters in the symmetric covariance matrix and the -1 term reflects the constraint Σ τ i = 1.
How to find the maximum likelihood of a mixture?
To find the maximum likelihood estimate for μ, we find the log-likelihood ℓ(μ), take the derivative with respect to μ, set it equal zero, and solve for μ: L(μ) = n ∏ i = 1 1 √2πσ2exp− (xi − μ)2 2σ2 ⇒ ℓ(μ) = n ∑ i = 1[log( 1 √2πσ2) − (xi − μ)2 2σ2] ⇒ d dμℓ(μ) = n ∑ i = 1xi − μ σ2
How to create a mixture model in Em?
Build site. Use external chunk to set knitr chunk options. Use session-info chunk. This document assumes basic familiarity with mixture models. In this note we introduced mixture models.