Suppose the image was supposed to be a uniform distribution of a binary variable (ie red dot vs blue dot) along the x axis, with some probability of blue being "P". How do I calculate the probability that such a "clump/grouping" as that found in Node #1 could even come about under a uniform distribution? In other words how would I find the p-value that this is truly a uniform distribution given it has such a "cluster" at low values of the x axis?

Edit:: My apologies, asking if it's uniform is the incorrect question. I mean is the proportion (ratio) of red to blue consistent throughout the x axis. In which case standard logistic regression p value may be my best option.

For anyone who works with decision trees can you elaborate on the relationship between: minimizing gini impurity/ entropy accross leafs and minimizing the log loss of the entire model? Does minimizing one directly imply the other? If so I am trying to prove it rigorously. If not can you please give a counterexample showing these two metrics are NOT monotonic ( ie increasing one implies increase in the other ....)

What is the exact interpretation of the standard deviation of a Bernoulli variable (ie sqrt(p*(1-p)))? Furthermore if the standard deviation is so high for so many values can't we round the probability to .5 ? For example if p=.3 then the standard deviation is sqrt((.3)(1-.3)) ~ .45 <-- that seems like A LOT of uncertainty especially for small sample sizes so why not just round the .3 closer to .5 to reflect that uncertainty?? I know for large samples this is erroneous due to law of large numbers but what about small samples? Any clarification is appreciated.

Can anyone explain the last part of this YouTube video:-> https://youtu.be/kNPGXgzxoHw?si=kPeYaFSR7iHvk5gw. I understand up until the 5:00 minute mark where he mentions that each neuron creates a hyperplane? How exactly would this be the case? I'm not seeing how the activation function creates an entire partition of the feature space. Any clarification or further resources would be appreciated.

Sorry if this is more machine learning than actual statistics. If anyone here is familiar with how they work do you know if a neural network can be trained with coordinate descent instead of gradient descent?