What is the score function of two parameters? According to this wikipedia article, score is the derivative of the log-likelihood function. However, I don't understand what if we have two parameters? For example, the logarithm of pdf has the following form:
$$ln f(x|a,b) = a\cdot g(x) + b \cdot h(x) + \phi(a,b)$$
I think that this is the sum of partial derivatives i.e.
$$s = g(x) + h(x) + \phi(a,b)_{a}' + \phi(a,b)_{b}'$$
but I'm not sure.
 A: The score for a multiple parameter problem (a vector parameter) is itself a vector. We need to take partial derivatives of the log likelihood with respect to each model parameter.
Let's consider an example. Find the score vector for $X_1, X_2, \ldots, X_n \sim N(\mu, \sigma^2)$ where the $X_i$ are iid $N(\mu, \sigma^2)$ samples. Let $\mathbf{x} = (x_1, x_2, \ldots, x_n)$ be a random sample of $(X_1, X_2, \ldots, X_n)$.
In this case, it can be shown that the log likelihood is
$$\ell(\mu, \sigma | \mathbf{x}) = -\frac{n}{2}\log (2 \pi) - n\log\sigma - \frac{1}{2\sigma^2} \sum_{i=1}^n (x_i - \mu)^2$$
So differentiation w.r.t. $\mu$ gives
$$ \frac{\partial \ell}{\partial \mu} = \frac{1}{\sigma^2} \sum_{i=1}^n (x_i - \mu)$$
differentiation w.r.t. $\sigma$ gives
$$ \frac{\partial \ell}{\partial \sigma} = -\frac{n}{\sigma} + \frac{1}{\sigma^3}\sum_{i=1}^n (x_i - \mu)^2.$$
The score is then the vector $S(\mu, \sigma) = \left( \frac{\partial \ell}{\partial \mu}, \frac{\partial \ell}{\partial \sigma} \right)^T$.
Concisely, we can write $S(\mathbf{\theta}) = \nabla_{\mathbf{\theta}} \ell(\mathbf{\theta}|\mathbf{x})$ for any vector parameter $\mathbf{\theta}$ where $\ell(\theta |\mathbf{x})$ is the log likelihood function.
