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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?

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    $\begingroup$ Everything about it is the same as unidimensional case. What is unclear for you? $\endgroup$ – Tim Nov 28 '18 at 8:06
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Maximum likelihood is about learning

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

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.

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  • $\begingroup$ Likelihood is not mean but product. I guess you forgot log? The 1/n constant is not a part of the definition. $\endgroup$ – Tim Nov 28 '18 at 8:08
  • $\begingroup$ 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? $\endgroup$ – NKH Nov 28 '18 at 14:01
  • $\begingroup$ We solve them simultaneously rather than one parameter at a time. $\endgroup$ – Siong Thye Goh Nov 28 '18 at 14:04
  • $\begingroup$ 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. $\endgroup$ – Siong Thye Goh Nov 28 '18 at 14:16
  • $\begingroup$ 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. $\endgroup$ – Siong Thye Goh Nov 28 '18 at 14:25

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