# Finding maximum likelihood of a mixture normal distribution [duplicate]

I'm trying to understand how to solve the following problem:

I have to estimate each parameter of

$f(x)=pϕ(x│μ_1,σ_1^2 )+(1-p)ϕ(x|μ_2,σ_2^2)$

using maximum likelihood estimation. $ϕ(.|μ,σ^2)$ is the pdf of a normal distribution.

So, as I understand it, what I have to do is

$\mathcal{L}(\mu_1,\mu_2,\sigma_1,\sigma_2)= ∏_{i=1}^n\left(p\frac{1}{\sqrt{2π}\sigma_1i} e^{-\frac{(x_i-a_1i)^2 }{2\sigma_1i^2 }}+(1-p)\frac{1}{\sqrt{2π}\sigma_2i} e^{-\frac{(x_i-a_2i)^2}{2\sigma_2i^2 }}\right) = p^n(\frac{1}{\sqrt{2π}\sigma_1i})^n e^{-\frac{1}{2\sigma_1i^2}\sum(x_i-a_1i)^2}+(1-p)^n(\frac{1}{\sqrt{2π}\sigma_2i})^n e^{-\frac{1}{2\sigma_2i^2}\sum(x_i-a_2i)^2}$

Then $ln\mathcal{L}(\mu_1,\mu_2,\sigma_1,\sigma_2)= ln(p^n)ln(\frac{1}{\sqrt{2π}\sigma_1i})^n -\frac{1^2}{2\sigma_2i^2 }(x_i-a_1i)^2+ln((1-p)^n)ln(\frac{1}{\sqrt{2π}\sigma_2i})^n -\frac{1^2}{2\sigma_2i^2 }(x_i-a_2i)^2$

Is this ok so far? If so, could you show me how to proceed from here?

• The expansion of the logarithm is wrong. You may be interested in reading about the EM algorithm for mixture of Gaussians to solve this kind of problem. May 28, 2017 at 8:26
• Ok, I'll do that. But could you please tell me what I've done wrong? May 28, 2017 at 12:29
• Oh, sorry. It was before the logarithm. When expanding the product expression: (a+b)(c+d) = ac + ad + bc + bd. Actually you get \sum_{i} \log (...) May 28, 2017 at 12:55
• Is $p$ a known constant or an unknown parameter? In $ln\mathcal{L}(\mu_1,\mu_2,\sigma_1,\sigma_2)$, $p$ is not included. May 29, 2017 at 19:57