How does expectation maximization relate to weighted least squares?

Question

For the past few days I have been trying to implement the EM-algorithm in order to segment stores into k-clusters. What I already did was derivation of the complete-log-likelihood and also performed the E-step. Where it goes wrong is the M-step.

I showed that I need to optimize the following (See derivation):

$-\frac{1}{2} \sum^{N T} P_{i s}\left(\log (S_{it})-\alpha^{[s]}-\beta^{[s]} \log \left(P_t\right)\right)^2$

When I take the derivative with respect to $\alpha^{[s]}$ and apply first order conditions, there shows up a $\beta^{[s]}$ in the expression and vice versa.

The main problem is my rusty linear algebra, because to solve this I should store $\alpha^{[s]}$ and $\beta^{[s]}$ in vector $\theta^{[s]}$ and optimize with respect to $\theta^{[s]}$, such that I get an expression that depends on the $P_t$ and $Log(S_{it})$

Additionally I need to show that this expression is basically the WLS solution

Q1: What is the derivation of the optimal $\theta^{[s]}$ Q2: How does this solution relation to WLS?

I added my derivations as an attachment to give extra context if someone is interested.

Context of the model:

Note: There are multiple stores and each store has time observations. Hence, I compute a density per store by taking all the time-observations of that store into account.

We consider a model for the sales of a particular product, measured at the store level at a weekly frequency. Denote the sales of the product in store $i=1, \ldots, N$ in week $t=1, \ldots, T$ by $S_{i t}$. The sales are explained by the price of the product at time t, that is, $p_t$. All stores have the same price in any given week. We specify the following heterogeneous model $$ \log S_{i t}=\alpha_i+\beta_i \log p_t+\varepsilon_{i t}, \quad \varepsilon_{i t} \sim N(0,1) . $$

The store-specific parameters are assumed to follow a mixture distribution with $K$ segments. ${ }^1$ We use $C_i \in\{1, \ldots, K\}$ to denote the (unobserved) cluster to which store $i$ belongs. Denote $\operatorname{Pr}\left[C_i=c\right]=\pi_c, c=1, \ldots, K$ with $\pi_c \geq 0$ and $\sum_{c=1}^K \pi_c=1$. For stores in segment $c$ the intercept and price elasticity equal $\alpha^{[c]}$ and $\beta^{[c]}$, respectively. Therefore $$ \alpha_i=\alpha^{[c]} \text { and } \beta_i=\beta^{[c]} \text { if } C_i=c . $$

Finally, denote $\theta=\left(\alpha^{[1]}, \beta^{[1]}, \ldots, \alpha^{[K]}, \beta^{[K]}\right)^{\prime}$ and $\pi=\left(\pi_1, \ldots, \pi_K^{\prime}\right)^{\prime}$.

Answer 1

While applying EM on a mixture of Gaussian (like your case), part of the steps do coincide with weighted least squares.

E step: Taking expectation w.r.t. $C$ on the loglikelihood results in a "weighted" log likelihood.

M step: Finding the parameter that maximize Gaussian log likelihoods is in essense a least square operation. Because Gaussian likelihood is quadratic exponential, natually the loglikelihood is quadratic.