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This problem would seem to have a simple solution, or a number of simple solutions. The unconstrained MLEs of the $\mu_i$ are simply $\hat\mu_1=\bar x$, $\hat\mu_2=\bar y$ and $\hat\mu_3=\bar z$. You …

I assume that you are referring to the total residual deviance that is computed when you fit a generalized linear model. Your question alludes to a widespread misconception. Regardless of what you mig …

Parametric link functions The "softplus" model you have proposed is called a parametric link function in generalized linear model theory. Technically, the link function is the inverse of your softplu …

Yes, certainly it exists and there is no problem with its implementation. Fisher scoring is one of the most commonly-used algorithms in statistics and is it usually implemented exactly as defined in y …

You can compute MLE for the gamma distribution using the dglm package, which is available from the CRAN repository. Here is an example run. Note that the two parameters being estimated in this example …

Bayes formula assumes that there is a distribution over every quantity, so there is no need to rewrite it. Bayes formula applies to both discrete and continuous random quantities. If the quantities ar …

The MLE of $p$ is available in closed form. There is no need to run an algorithm to maximize the likelihood. The negative binomial is an example of an linear exponential family and, for all linear exp …

In this regression, then are only two observations and two fitted values, which are $\hat p_m=17/91$ for males and $\hat p_f=32/109$ for females. Converting to the logit linear predictor scale gives: …