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I don't think complexity classes are the partition I'm after. The context in which this arose is that I'm writing a very dumb, limited-functionality Magic the Gathering simulator to estimate the early-game impact of different mana bases. Given a hand, I want an algorithm to play lands and...

In combinatorics, the twentyfold way (and better known predecessor the twelvefold way) classifies counting problems via a single mathematical framework, functions with certain properties mapping domains with certain properties to codomains with certain properties. While not exhaustive of all...

I think that pretty much answers my questions, thanks! The videos also connect with some of what you said, such as the final hidden layer not actually corresponding to probabilities of human-intelligible facets of the hand-written numbers (loops, straight lines, etc.). I just wasn't sure if it...

I think someone edited the question title using copy/paste and the LaTeX has come out wrong.

I am basing what I said on this four part video series on neural networks, initially mentioned in the first video at 10:14. Within the input layer, each node value has the "physical" interpretation as the grayscale-value of a particular pixel. The output layer seems have the interpretation of...

I am brand new to studying machine learning. Why are the nodes in a neural network made to have values ranging between 0 and 1? It seems like they should range between 0 and \frac{1}{n}, where n is the number of nodes in that same layer. Is it wrong to think of a node's value as a probability...

Problem: Find the Sprague-Grundy values of the graphs, and find a winning move, if any. Solution: The SG-value of the three-leaf clover is 2. The SG-value of the girl is 3. The SG-value of the dog is 2. And the SG-value of the tree is 5. So there exists a winning move on the tree that...

The only difference between the definitions on Wikipedia for cross-correlation and cross-covariance (for deterministic signals) is that cross-correlation shows definite integration, while cross-covariance shows indefinite integration. Is this really the difference between the two, or is this an...

Can someone explain this Wikipedia example? I get the idea that W is a weighted average of the movie's average rating and the global average rating, but not how the formula relates to the broader article. Which piece in this example is the loss function and which piece is the Bayes Estimator?

I have worked out a few successive derivations of Bayes' Theorem... \Pr(W) = \Pr(W) \Pr(W\ |\ C_1)\ =\ \frac{\Pr(C_1\ |\ W)\Pr(W)}{\Pr(C_1)} \Pr(W\ |\ (C_1\ \cap\ C_2))\ =\ \frac{\Pr(C_2\ |\ (W\ \cap\ C_1))\Pr(W\ |\ C_1)}{\Pr(C_2\ |\ C_1)} \Pr(W\ |\ (C_1\ \cap\ C_2\ \cap\ C_3)) =...

It won't help with the narrower problem, but in case you're in a Discrete Math or Combinatorics college course, I also found the subject rather unapproachable. I'm still not really good at these problems, but one thing I discovered that helped was the twelvefold way. What it offers is a...

According to Wikipedia... Suppose I want to relate \Pr(W\ |\ C), a degree of rational belief in W after evidence C becomes available, back to \Pr(W), a degree of rational belief in W before evidence C becomes available. Bayes' Theorem is just a double-application of the conditional...

According to this paper, "every submodular function can be represented as a maximum of additive valuations." It gives an algebraic description as well, but I'm having trouble seeing the high level idea. Is there a simple geometric intuition for this fact? Is there an obvious economic...

Sure, I understand the matrix. I guess the confusing part for me is that p-value, q-value, and power in hypothesis testing seem to mean something more "epistemic" than the more "ontological" conditional probabilities False Positive Rate, False Discovery Rate, Sensitivity, Positive Predictive...

According to the Wikipedia entry on q-values... The "positive" qualifier here seems to be a distraction, as the article later states... So the q-value controls the False Discovery Rate. Similarly, the p-value appears to control the False Positive Rate, and the power appears to control the...

In this video, the profession gives what he calls a simulation argument. He goes through it pretty quickly, seemingly assuming that students are already familiar with that type of argument. All I can find when searching for this are philosophical considerations of simulation theory, the idea...

Does a mechanical set of rules or algorithm exist for doing forward induction on a game tree, or is it an "every problem is too unique and requires its own reasoning" type of situation? I have seen a few examples of forward induction explained, but with pretty basic game trees, and I fear I may...

Suppose I am told the outcomes (x_1 through x_n) of n rolls (X_1 through X_n) of a fair, \theta-sided die, and want to find a point estimate of \theta. Let's state the problem without using the lexeme "likely" in order to eliminate one layer of confusion, since its technical and colloquial...

Thinking about this problem is leaving me more convinced of it being equivalent to a necktie paradox or two envelope paradox, of which I know some solutions but struggle to find them satisfying. I believe time can be more or less eliminated from the model. Bob (tracks expectation in USD and...

I agree that since Bob and Dave have identical portfolios, those portfolios should each have equal expected value regardless of how they acquired them, but that doesn't seem to explain the weirdness. Suppose the quoted transaction occurred by Bob directly trading $1 to Dave for £1. Who got the...