# Maximum Likelihood Estimation with Known Parameter Distribution

Consider i.i.d observation vector ${\bf x}$ from a distribution $F$ depending on vector of parameters $\boldsymbol{\theta}$ and single parameter $\alpha$.

We would like to estimate parameters [$\boldsymbol{\theta}, \alpha$] using maximum likelihood estimation but we also know that $\alpha$ is a realisation from a $N(\mu, \sigma^2)$.

If we didn't know the distribution of $\alpha$ we could just maximise the objective function $f({\bf x} | \boldsymbol{\theta}, \alpha)$ but knowing the distribution for $\alpha$, what does the objective function to maximise become?

• In what sense are you going to "estimate" the random variable $\alpha$, by maximizing some objective function that includes it? Commented Apr 18, 2014 at 15:50
• Yes, exactly. I was thinking something like $f({\bf x} | \boldsymbol{\theta}, \alpha) * \phi(\alpha)$ but this is only what my intuition is saying. Commented Apr 18, 2014 at 16:33
• The issue is not the form of the function. When parameters are treated as random variables, you are in Bayesian estimation framework, so no objective function and no maximization approach. Commented Apr 18, 2014 at 16:44
• Isn't that the definition for MAP estimation?
– Royi
Commented Apr 18, 2014 at 16:53
• Right, it does seem to be MAP estimation, in which case $f({\bf x} | \boldsymbol{\theta}, \alpha) * \phi(\alpha)$ is indeed the objective function to maximise and the non-random parameters can be thought of as coming from a probability distribution with all the probability at one point. Commented Apr 18, 2014 at 17:59