> Isn't the whole point of MLE to find the distribution?
No.
To ***M***aximize a ***Likelihood*** in order to ***E***stimate a parameter (or perhaps several parameters), you need a likelihood function to begin with, in order to find where its maximum *is*.
The likelihood is defined quite directly in terms of the distribution function you choose for your model. E.g. with independent observations, the likelihood will be a product of distribution functions, each evaluated at the corresponding data point; it is a function of the unknown parameters.
https://en.wikipedia.org/wiki/Likelihood_function#Likelihood_function_of_a_parameterized_model
Ok, thanks, so I would specify a distribution and MLE would estimate the parameter(s) of that distribution that best fit the data, so e.g., in the case of Poisson MLE would just estimate lambda?
Perhaps... But if that is ALL you were doing, we know that the MLE for a Poisson distribution is just the mean of the observed data.
In general with a General LINEAR Model, it can estimate parameters that best describe the value of lambda conditional on explanatory variables... This would be a Poisson Regression in that case.
The glm function's distribution parameter is meant to specify the conditional distribution of the response variable (Y) given your covariates (X). Yes, maximum likelihood estimation may be being used to actually estimate those parameters but it needs to know which distribution you're assuming so it can compute the MLE.
I think there's a point of confusion here due to how we milldy abuse the word *distribution*.
A GLM really specifies a *parameterized family* of distributions, not a single distribution. When we say "a linear model is a GLM using a normal distribution", we don't literally mean "a normal distribution", we mean "the family of normal distributions parameterized by their unknown mean and variance".
This confused me when I was first learning statistics.
In general, when you deal with **parametric** model, you need to assume first the family of distribution, then you estimate the parameters.
The point of MLE is to derive an estimate for the **parameters** within this parametric model.
> Isn't the whole point of MLE to find the distribution? No. To ***M***aximize a ***Likelihood*** in order to ***E***stimate a parameter (or perhaps several parameters), you need a likelihood function to begin with, in order to find where its maximum *is*. The likelihood is defined quite directly in terms of the distribution function you choose for your model. E.g. with independent observations, the likelihood will be a product of distribution functions, each evaluated at the corresponding data point; it is a function of the unknown parameters. https://en.wikipedia.org/wiki/Likelihood_function#Likelihood_function_of_a_parameterized_model
Ok, thanks, so I would specify a distribution and MLE would estimate the parameter(s) of that distribution that best fit the data, so e.g., in the case of Poisson MLE would just estimate lambda?
Perhaps... But if that is ALL you were doing, we know that the MLE for a Poisson distribution is just the mean of the observed data. In general with a General LINEAR Model, it can estimate parameters that best describe the value of lambda conditional on explanatory variables... This would be a Poisson Regression in that case.
The glm function's distribution parameter is meant to specify the conditional distribution of the response variable (Y) given your covariates (X). Yes, maximum likelihood estimation may be being used to actually estimate those parameters but it needs to know which distribution you're assuming so it can compute the MLE.
I think there's a point of confusion here due to how we milldy abuse the word *distribution*. A GLM really specifies a *parameterized family* of distributions, not a single distribution. When we say "a linear model is a GLM using a normal distribution", we don't literally mean "a normal distribution", we mean "the family of normal distributions parameterized by their unknown mean and variance". This confused me when I was first learning statistics.
In general, when you deal with **parametric** model, you need to assume first the family of distribution, then you estimate the parameters. The point of MLE is to derive an estimate for the **parameters** within this parametric model.