Fix problems with the MLE estimation

[smitdave]

We need to finalize the conclusions of the MLE. I had some concerns about the functional forms that we used after they failed to converge to the same point.

The MLE is now here:

https://github.com/dd-harp/memory.core/blob/main/mle.Rmd

[smitdave]

Note that the ordinary parameters describing a beta distribution are $\alpha$ and $\beta$ with mean

$$\mu = \frac{\alpha}{\alpha+\beta}$$

and variance

$$\sigma^2 = \frac{\alpha \beta}{(\alpha+\beta)^2 (\alpha+\beta+1)}$$

if we parameterize it directly, we can take any parameters $\mu$ and $\sigma$ where

$$\sigma^2 \leq \mu (1-\mu)$$

and we can use the ordinary R parameterization:

$$\beta = \alpha \left(\frac 1 \mu -1 \right)$$

$$\alpha = \left(\frac{1-\mu}{\sigma^2} - \frac 1 \mu \right)\mu^2$$

[smitdave]

...so we let $\mu = F_\mu(\alpha)$ where $0 < \mu < b(a)$ and we need a function giving the variance $$\sigma = F_\sigma(\mu)$$