variance (also known as second central moment) is a way of measuring spread:

\begin{align} Var(X) &= E[(X-E(X))^{2}] \\ &= E[X^{2}] - (E[X])^{2} \\ &= \qty(\sum_{x}^{} x^{2} p\qty(X=x)) - (E[X])^{2} \end{align}

“on average, how far is the probability of \(X\) from its expectation”

The expression(s) are derived below. Recall that standard deviation is a square root of the variance.

computing variance:

\begin{align} Var(X) &= E[(X - \mu)^{2}] \\ &= \sum_{x}^{} (x-\mu)^{2} p(X) \end{align}

based on the law of the Unconscious statistician. And then, we do algebra:

1. Secrets of Silicon Valley - Horowitz
   1. Looking for people who have feel for the problem: people need to believe in the problem
   2. Team: can people come with execution? people that are good at startups which are usually not good at later stage stuff
      1. Buy a startup and kick out the founders
      2. This is very typical
   3. Team and idea are easy to decouple
2. Vetting problems
   1. Lack of market
   2. Technically insatiability
   3. “Unbelievable stupidity”: calcium is so cheap
   4. Idea goes through many morphs; getting the credit back
3. People wiling to have a meeting?
4. Decoupling value proposition

=> iStudio as a service

A vector is an element of a vector space. They are also called a point.

vector semantics is a sense encoding method.

“a meaning of the word should be tied to how they are used”

we measure similarity between word vectors with cosine similarity. see also vector-space model.

motivation

idea 1

neighboring words can help infer semantic meaning of new words: “we can define a word based on its distribution in language use”

idea 2

meaning should be in a point in space, just like affective meaning (i.e. a score in each dimension).

A vector space is an object between a field and a group; it has two ops—addition and scalar multiplication. Its not quite a field and its more than a group.

constituents

• A set \(V\)
• An addition on \(V\)
• An scalar multiplication on \(V\)

such that…

requirements

• commutativity in add.: \(u+v=v+u\)
• associativity in add. and mult.: \((u+v)+w=u+(v+w)\); \((ab)v=a(bv)\): \(\forall u,v,w \in V\) and \(a,b \in \mathbb{F}\)
• distributivity: goes both ways \(a(u+v) = au+av\) AND!! \((a+b)v=av+bv\): \(\forall a,b \in \mathbb{F}\) and \(u,v \in V\)
• additive identity: \(\exists 0 \in V: v+0=v \forall v \in V\)
• additive inverse: \(\forall v \in V, \exists w \in V: v+w=0\)
• multiplicative identity: \(1v=v \forall v \in V\)

additional information

• Elements of a vector space are called vectors or points.

vector space “over” fields

Scalar multiplication is not in the set \(V\); instead, “scalars” \(\lambda\) come from this magic faraway land called \(\mathbb{F}\). The choice of \(\mathbb{F}\) for each vector space makes it different; so, when precision is needed, we can say that a vector space is “over” some \(\mathbb{F}\) which contributes its scalars.