### Trang Le

#math graduate. Postdoc fellow with Jason Moore.

Trang Le, PhD

2021-04-05

Guest lecture

Foundations of AI

@trang1618

- Entropy
- Microstate
- Macrostate
- Information gain
- Mutual information
- KL divergence
- Epistasis

*and some applications...*

You toss a fair coin.

- If heads, + $1.
- If tails, \(-\) $1.

What's the expected value of your winning?

Expected winning = \((+\$1)\times 0.5 + (-\$1)\times 0.5= \$0.\)

\[p_{heads}\]

\[p_{tails}\]

\[E[X] = \sum_{i=1}^k x_ip_i = x_1p_1 + x_2p_2 + \cdots + x_kp_k\]

If you play this game \(n\) times, as \(n\) grows, the average will almost surely converge to this expected value.

What if \(p_{heads} = 0.8, p_{tails} = 0.2\) ?

Expected winning = \((+1)\times 0.8 + (-1)\times 0.2 = 0.6.\)

- disorder?
- uncertainty?
- surprise?
- information?

A **microstate** specifies the position and velocity of all the atoms in the coffee.

A **macrostate** specifies temperature, pressure on the side of the cup, total energy, volume, etc.

- themodynamics/statistical mechanics

a

Boltzmann constant

number of configurations

a

\[S = - k_B\sum_i p_i\log p_i = k_B\log \Omega\]

*(assume each microstate is equally probable)*

\[S = - k_B\sum_i p_i\log p_i\]

expected value of

log(probability that a microstate \(i\) is occupied)

[J/K]

## Entropy always increases.

Why is that?

Because it's overwhelmingly more likely that it will.

- information theory: expected value of self-information

a

Shannon's entropy

probability of the

possible outcome \(x_i\)

a

\[H(X) = -\sum_{i = 1}^kp(x_i)\log_2p(x_i)\]

[bits]

*Source coding theorem.*

The Shannon entropy of a model ~ the number of bits of **information gain** by doing an *experiment/sampling* on a system your model describes.

\[p_{heads} = 0.5, p_{tails} =0.5\]

\[H = ?\]

\[ H = -p_{heads}log_2(p_{heads})-p_{tails}log_2(p_{tails})\]

\[ = -0.5*log_2(0.5)-0.5*log_2(0.5)\]

\[ = -0.5*(-1)-0.5*(-1) = 1\]

*Source coding theorem.*

The Shannon entropy of a model ~ the number of bits of **information gain **by doing an experiment on a system your model describes.

\[H(X) = -\sum_{i = 1}^kp(x_i)\log_2p(x_i)\]

*Source coding theorem.*

The Shannon entropy of a model ~ the number of bits of **information gain **by doing an experiment on a system your model describes.

\[p_{heads} = 0.5, p_{tails} =0.5\]

\[H = 1\]

What if \(p_{heads} = 1, p_{tails} =0\)?

\[H = 0\]

What if \(p_{heads} = 0, p_{tails} =1\)?

\[H = 0\]

\[p_{heads}\]

\[H(X) = -\sum_{i = 1}^kp(x_i)\log_2p(x_i)\]

\[H(X|Y) = -\sum_{x \in X, y \in Y}p(x, y)\log_2\frac{p(x, y)}{p(y)}= H(X,Y) - H(Y)\]

\[H(X) = -\sum_{x \in X}p(x)\log_2p(x)\]

\[H(X, Y) = -\sum_{x \in X}\sum_{y \in Y}p(x, y)\log_2p(x, y)\]

Mutual information

\[I(X;Y)= H(X) - H(X|Y)\]

\[= H(X) + H(Y) - H(X,Y)\]

joint entropy

\[I(X;Y)= D_{KL}(P_{X,Y}|P_X \times P_Y)\]

Kullback-Leibler divergence

a

Discrete probability distributions \(P\) and \(Q\) defined on probability space \(X\)

\[D_{KL}(p|q) = \sum_{x \in X} p(x) \log\frac{p(x)}{q(x)}\]

Cross entropy

\[H_q(p) = \sum_{x\in X} p(x) \log\frac{1}{q(x)}\]

of information theory

The gain in phenotype/class information obtained by considering A and B jointly *beyond* the class information that would be gained by considering variables A and B *independently*.

\[I(A,B; C) = I(AB;C) - I(A;C) - I(B;C)\]

a

information gained about the phenotype *C* when locus *A* is known

information gained about the phenotype C when locus B is known

a

joint attribute constructed from attributes A and B

a

Let S be a sample of training examples

- \(p_+\) is the proportion of
**positive**examples in \(S\) -
\(p_-\) is the proportion of
**negative**examples in \(S\)

Entropy measures the impurity of \(S\).

\[ Entropy(S) = -p_+log_2(p_+)-p_-log_2(p_-)\]

Information gain = Entropy decrease

\[ Gain(S,model) = Entropy(S) - \sum_{v \in Values(model)} \frac{|S_v|}{|S|}Entropy(S_v)\]

\(S: [9+, 5-]\)

\(p_+ = 9/14, p_- = 5/14\)

\[ Entropy(S) = -p_+log_2(p_+)-p_-log_2(p_-)\]

\[ = -\frac{9}{14}log_2\frac{9}{14}-\frac{5}{14}log_2\frac{5}{14}\]

\[ \approx 0.94\]

**+**

**+**

**+**

**+**

**+**

**+**

**+**

**+**

**+**

**+**

**\(-\)**

**\(-\)**

**\(-\)**

**\(-\)**

**\(-\)**

\(S\)

\[ Gain(S,model) = Entropy(S) - \sum_{v \in Values(model)} \frac{|S_v|}{|S|}Entropy(S_v)\]

\(Gain(S, Humidity)\)

\(= 0.94-\frac{7}{14}0.985- \frac{7}{14}0.592\)

\(= 0.152\)

\(Gain(S, Wind)\)

\(= 0.94-\frac{8}{14}0.811- \frac{6}{14}(1)\)

\(= 0.048\)

Wikipedia

*“Only entropy comes easy.”
-Anton Chekhov*

By Trang Le

Guest lecture for Penn BMIN 520-401 course, Spring 2020

- 454

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