3

You can simulate one of these lines like this: library(ggplot2) X<-rnorm(1000, 1, sqrt(4)) f<-function(x) exp(-x*x) df<-data.frame(sample_size=seq_len(1000), sample_means=cumsum(f(X))/seq_len(1000)) ggplot(df, aes(x=sample_size, y=sample_means)) + geom_line() + theme_minimal() + labs(x=NULL, y=NULL)


2

What is the meaning of distribution-wise asymmetric measure? The (forward) KL-divergence is distribution-wise asymmetric because if you calculate it as $$D_{KL} \langle P(X) \Vert P(Y) \rangle$$ where $P(X)$ and $P(Y)$ are two different probability distributions with the latter being the reference distribution, then $$D_{KL} \langle P(Y)\Vert P(X)\rangle \...


1

You correctly computed the gradients, but the optimization step is incorrect. Your gradient is computed at point $(\mu,\sigma)$, and, therefore, it should be used to update $\mu$ and $\sigma$ simultaneously. Correct update rule is $$ \begin{align} \mu_{t+1} &\longleftarrow \mu_t + \gamma \left(\frac{\overline{x}}{\sigma^2_t} - \frac{\mu_t}{\sigma^2_t}\...


1

The reason for the rules based approach is the simpson paradox. In short, the way a test group is sampled can easily overturn the conclusion of a study. It is very important to include sufficient patients from a particular risk group in the study. This is why rules are set. This approach however is the basis of causality theory. A data driven-approach would ...


1

Keeping it to discrete space, the joint probability of an event can be interpreted as the fraction of the size of this subset compared to the whole state space. In case of events, being sets themselves, the definition remains the same. $$P(X_i,Y_j) = \frac{|\{(x,y) \in (X_i,Y_j)|\}} {|\Omega|} $$ $|\Omega|$ is just the set of all combinations of single ...


1

Generalisation of Mcnemars is called Cochran–Mantel–Haenszel test There is an implementation in R, but I suppose porting to python should not be too hard. You can find the r version here


1

TL;DR While the data comes from the NYTimes and seems legit, the presentation is intentionally misleading and the subsequent assertions are baseless. I say "intentionally" because an unbiased and reputable analysis would not propagate such major allegations from the data they have presented. The data does not prove nor disprove voter fraud, so the ...


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