Any further reading suggestion for question 7?
Here are my opinions for the exs:
I still not sure about ex.1, ex.5.E, ex.5.F, ex.6
ex.1
But, what does “match” means in question B?
ex.2
A. If I use binary code for the three class, like 00, 01, 11, then the distance between 00 and 01 is smaller than that between 00 and 11, that is oppose to the fact that the three class is of equal probability.
B. I think I should use one-hot coding mentioned in this chapter, because for any independent observation(which I think is a class), there contains no distance information between any pair of them.
ex.3
Two ternaries can have 9 different representation, so my answer is 2.
This ternary is suitable for electronics because in a physical wire, there will be three distinctive condition: positive voltage, negative voltage, zero voltage.
ex.4
A. Bradley-Terry model is like
When there are only two classes, softmax just fit this.
B. No matter how many classes there will be, if I put a higher score for class A compared to class B, the the B-T model will still let me chose class A, and after 3 times of comparing, I will chose the class with the highest score, that still holds true for the softmax.
ex.5
ex.6
This is my procedure for question A, but I can’t prove that the second derivative is just the variance.
As for the rest of the questions, I don’t even understand the question.
ex.7
A. Because of exp(−𝐸/𝑘𝑇), if I double T, alpha will go to 1/2, and if I halve it, alpha will go to 2, so T and alpha goes in opposite direction.
B. If T converge to 0, the possibility for any class will converge to 0, and the proportion between two class i and j exp( -(Ei - Ej) /kT) will also converge to 0. Like a frozen object of which all molecules is static.
C. If T converge to ∞, the proportion between two class i and j will converge to 1, which means every class has the same possibility to show up.
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Exercise 6 Show that g(x) is translation invariant, i.e., g(x+b) = g(x)
I don’t see how this can be true for b different from 0.
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Ex1.
Ex6 (issue: (1) translational invariant or equivariant? (Softmax is invariant, but log-sum-exp should be equivariant); (2) b or negative b? Adding maximum can make overflow problem worse).
The same result… I think we may say the log-partition function is translational equivariant rather than invariant. See also this page.
To ex.1, maybe we can take softmax distribution as Bernoulli distribution with a probability of $p = softmax(o)$, so the variance is:
$$Var[X] = E[X^2] - E[X]^2 = \text{softmax}(o)(1 - \text{softmax}(o))$$
I don’t know whether this suppose is right
and my solutions to the exs: 4.1
Does this mean that each coordinate of each y label vector is independent of each other? Also, shouldn’t the y_j of the last 2 equations also have a “i” superscript?
I read the URL given, but it doesn’t clarify too much for this specific case.