# $2D$ Maximum Likelihood Fit

I have read a couple of places that it is possible to do a $$2D$$ (or $$3D$$) maximum likelihood fit, but I can't seem to understand how this would work. Suppose I'm considering a probability distribution function depending on $$2$$ observable variables, $$x$$ and $$y$$, given a multi-dimensional set of variable parameters, $$\alpha, \beta, \gamma$$, ..., PDF($$x_i,y_i$$|$$\alpha, \beta, \gamma, \ldots$$).

I would say that I have a good understanding of the ideas behind a maximum likelihood fit, but for some reason I cannot wrap my head around a multi-dimensional maximum likelihood fit.

How do I understand how a $$2D$$ maximum likelihood fit to determine the best estimate of the parameters $$\alpha, \beta, \gamma$$ works?

• Everything about it is the same as unidimensional case. What is unclear for you?
– Tim
Nov 28 '18 at 8:06

$$\hat{\theta} \in \{\arg\max_{\theta \in \Theta}\sum_{i=1}^n \ln f(x_i|\theta)\}$$

Notice that $$\Theta$$ need not be of just one dimension and it can be of multiple dimension.

The whole idea is to optimize a quantity and techniques from calculus or numerical optimization might be involved. For smooth cases, we typically begins by exploring the stationary point of the objective function.

• Likelihood is not mean but product. I guess you forgot log? The 1/n constant is not a part of the definition.
– Tim
Nov 28 '18 at 8:08
• I must admit, I still do not see where 2D fit comes in? I understand that 𝚯, the parameters we want to estimate, can be of several dimensions. But we would still do a fit around a single parameter at a time, right? And then possibly re-fit to the remaining parameters. However, what happens if f does not only depend on x, but also y?
– NKH
Nov 28 '18 at 14:01
• We solve them simultaneously rather than one parameter at a time. Nov 28 '18 at 14:04
• here is an example where the parameters are $\mu$ and $\sigma$. In that example, while the derivative with respect to $\mu$ can be solved independently, we should view the two first order constraints as a system of equations that we want to satisfy. Nov 28 '18 at 14:16
• here , a more complicated example where MLE for the parameters of beta distributed is staed in equation $(2.3)$ and $(2.4)$ and numerical method is used to find them. Nov 28 '18 at 14:25