10
In a rather narrow sense, you are correct (1): if
$$\pi(\theta|x) \propto L(\theta|x)$$
then
$$\underbrace{\arg \max_\theta \pi(\theta|x)}_\text{MAP estimate} = \underbrace{\arg \max_\theta L(\theta|x)}_\text{ML estimate}$$
However, this coincidence is not of considerable interest as:
[Re. (1) and (3):] It is not invariant by repameterisation, i.e., the ...
4
When you have a binary outcome variable you typically use a link function to connect the probability of a positive response to the linear predictor that includes the random effects, i.e.,
\begin{equation}
\left \{ \begin{array}{l}
\log \displaystyle \frac{\pi_{it}}{1 - \pi_{it}} = x_{it}^\top \beta + z_{ij}^\top b_i,\\\\
\pi_{it} = \Pr(y_{it} = 1 \mid b_i),\\...
3
Estimating linear regression models, via OLS, and fitting distributions can both be accomplished using the same method, Maximum
Likelihood Estimation (MLE), and
Yes, you are correct on this. When using maximum likelihood, we are always fitting some kind of distribution to the data. The difference is however between the particular kinds of ...
2
The whole business of improper priors in Bayesian analysis arises because some users of Bayesian reasoning believe that proper priors tend to be derived from 'subjective' opinions and thus would taint the Bayesian inference with subjectivity. It turns out that ignoring the impropriety of the priors can nevertheless yield usable results.
There are other ...
2
You are quite confused. I see two major confusions: you are confusing the idea of the log-likelihood with that of the log-density and you do not understand the idea of an estimator.
The density function of a beta binomial is given by the formula you listed and can be evaluated in R by the function dbetabinom although note that dbetabinom actually uses a ...
2
The GEEs have been originally proposed for clustered/grouped categorical data. Namely, when we have clustered/grouped data, it is expected that measurements within the same clusters/groups are correlated. In the case of normal data, we can go from the univariate normal distribution to the multivariate one to account for these correlations using the variance-...
1
I can suggest a number of improvements that will make any minimization method more stable.
instead of
# Generates values from a normal distributed pdf
pdf_vals = norm_pdf(xvals, mu, sigma)
# Take log of normal distributed pdf values
ln_pdf_vals = np.log(pdf_vals)
you should analytically compute the log of the pdf. The reason is that the ...
1
Assume $X_1, ...,X_{15} \sim B(n,\pi)$. We know that $E(X) = n\pi$ and $V(X)=n\pi(1-\pi)$.
You can use moment method to get the estimate of the estimate of $n$ and $\pi$. At first get the sample mean $\bar X$ and sample variance $\hat V(X)$, then solve following you will get the estimate.
$$\bar X = \hat n \hat\pi$$
$$\hat V(X) = \hat n \hat\pi(1-\hat \pi)$...
1
Why use GEE (instead of Maximum Likelihood, ML)? Because the likelihood could be wrong. In fact, since we never know if it's right or not, GEEs safeguard against biases. We want results that require few assumptions. GEE's assumptions are some of the most general, hence the Generalized "G" of GEE.
... the problem with GEEs is I can't calculate the ...
1
Your likelihood is
$$
P(w_t\mid h) = \frac{\exp(s_\theta(w_t,h))}{\sum_{w'\in V} \exp(s_\theta (w', h))}.
$$
$w_t$ is one specific value, while $w'$ is the index of the sum, and it's taking on all values in $V$.
Then you take the log of this, then the derivative.
In the second part of your derivation, you're only looking at the second term, which came from ...
1
Yes, it is correct because we maximize the following data likelihood (say $x$ is the outcome of the experiment):
$$f(\theta)=p(D|\theta)=p(x|\theta)=\theta^x(1-\theta)^{1-x}$$
It won’t be completely accurate if we differentiate this (and it’s not well defined when $x=0,\theta=0$ and $x=1,\theta=1$) because the function won’t be differentiable in general at ...
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