Tag Archives: Vector space

The bias of the unbiased

A hilarious paper from Stanford shows the bias of the unbiased [ Proc. Natl. Acad. Sci. vol. 115 pp. E3635 – E3644 ’18 ].  No one wants to be considered biased or to use stereotypes, but this paper indicts all of us.  They use a technique called word embedding to look at a large body of printed material (Wikipedia, Google news articles etc. etc.) over the past 100 years, to look for word associations  -e.g. male trustworthy female submissive and the like. In word embedding models, each word in a given language is associated with a high dimensional vector (not clear to me how the dimensions are chosen) and the metric between the words is measured.  A metric is simply a mathematical device that takes two objects and associates a number with them.  The distance between cities is a good example.

 

The vector for France is close to vectors for Austria and Italy.  The difference between London and England (obtained by subtracting them) is parallel to the difference between to the difference between Paris and France.  This allows embeddings to capture analogy relationships such as London is to England as Paris is to France.

So word embeddings were used as a way to study gender and ethnic stereotypes in the 20th and 21st centuries in the USA.  Not only that but they plotted how the biases changed over time.

So in your mind the metric between bias == bad, stereotype == worse is clear

So just as women’s occupations have changed so have the descriptors of women.  Back in the day women, if they worked out of the home at all, were teachers or nurses.  A descendent of Jonathan Edwards was a grade school teacher in the town of my small rural high school.

As women moved into the wider workforce from them the descriptors of them changed.  The following is a pair of direct quotes from the article.”

“More importantly, these correlations are very similar over the decades, suggesting that the relationship between embedding bias score and “reality,” as measured by occupation participation, is consistent over time” ….”This consistency makes the interpretation of embedding bias more reliable; i.e., a given bias score corresponds to approximately the same percentage of the workforce in that occupation being women, regardless of the embedding decade.”

English translation:  As women’s percentage of workers in a given occupation changed the ‘bias score’ changed with it.

So what the authors describe and worse, define, as bias and stereotyping is actually an accurate perception of reality.  We’re all guilty.

The authors are following Humpty Dumpty in Alice in Wonderland  — ““When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.” “The question is,” said Humpty Dumpty, “which is to be master—that’s all.”

I find the paper hilarious and an example of the bias of the supposedly unbiased.

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How formal tensor mathematics and the postulates of quantum mechanics give rise to entanglement

Tensors continue to amaze. I never thought I’d get a simple mathematical explanation of entanglement, but here it is. Explanation is probably too strong a word, because it relies on the postulates of quantum mechanics, which are extremely simple but which lead to extremely bizarre consequences (such as entanglement). As Feynman famously said ‘no one understands quantum mechanics’. Despite that it’s never made a prediction not confirmed by experiments, so the theory is correct even if we don’t understand ‘how it can be like that’. 100 years of correct prediction of experimentation are not to be sneezed at.

If you’re a bit foggy on just what entanglement is — have a look at https://luysii.wordpress.com/2010/12/13/bells-inequality-entanglement-and-the-demise-of-local-reality-i/. Even better; read the book by Zeilinger referred to in the link (if you have the time).

Actually you don’t even need all the postulates for quantum mechanics (as given in the book “Quantum Computation and Quantum Information by Nielsen and Chuang). No differential equations. No Schrodinger equation. No operators. No eigenvalues. What could be nicer for those thirsting for knowledge? Such a deal ! ! ! Just 2 postulates and a little formal mathematics.

Postulate #1 “Associated to any isolated physical system, is a complex vector space with inner product (that is a Hilbert space) known as the state space of the system. The system is completely described by its state vector which is a unit vector in the system’s state space”. If this is unsatisfying, see an explication of this on p. 80 of Nielson and Chuang (where the postulate appears)

Because the linear algebra underlying quantum mechanics seemed to be largely ignored in the course I audited, I wrote a series of posts called Linear Algebra Survival Guide for Quantum Mechanics. The first should be all you need. https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/ but there are several more.

Even though I wrote a post on tensors, showing how they were a way of describing an object independently of the coordinates used to describe it, I did’t even discuss another aspect of tensors — multi linearity — which is crucial here. The post itself can be viewed at https://luysii.wordpress.com/2014/12/08/tensors/

Start by thinking of a simple tensor as a vector in a vector space. The tensor product is just a way of combining vectors in vector spaces to get another (and larger) vector space. So the tensor product isn’t a product in the sense that multiplication of two objects (real numbers, complex numbers, square matrices) produces another object of the exactly same kind.

So mathematicians use a special symbol for the tensor product — a circle with an x inside. I’m going to use something similar ‘®’ because I can’t figure out how to produce the actual symbol. So let V and W be the quantum mechanical state spaces of two systems.

Their tensor product is just V ® W. Mathematicians can define things any way they want. A crucial aspect of the tensor product is that is multilinear. So if v and v’ are elements of V, then v + v’ is also an element of V (because two vectors in a given vector space can always be added). Similarly w + w’ is an element of W if w an w’ are. Adding to the confusion trying to learn this stuff is the fact that all vectors are themselves tensors.

Multilinearity of the tensor product is what you’d think

(v + v’) ® (w + w’) = v ® (w + w’ ) + v’ ® (w + w’)

= v ® w + v ® w’ + v’ ® w + v’ ® w’

You get all 4 tensor products in this case.

This brings us to Postulate #2 (actually #4 on the book on p. 94 — we don’t need the other two — I told you this was fairly simple)

Postulate #2 “The state space of a composite physical system is the tensor product of the state spaces of the component physical systems.”

http://planetmath.org/simpletensor

Where does entanglement come in? Patience, we’re nearly done. One now must distinguish simple and non-simple tensors. Each of the 4 tensors products in the sum on the last line is simple being the tensor product of two vectors.

What about v ® w’ + v’ ® w ?? It isn’t simple because there is no way to get this by itself as simple_tensor1 ® simple_tensor2 So it’s called a compound tensor. (v + v’) ® (w + w’) is a simple tensor because v + v’ is just another single element of V (call it v”) and w + w’ is just another single element of W (call it w”).

So the tensor product of (v + v’) ® (w + w’) — the elements of the two state spaces can be understood as though V has state v” and W has state w”.

v ® w’ + v’ ® w can’t be understood this way. The full system can’t be understood by considering V and W in isolation, e.g. the two subsystems V and W are ENTANGLED.

Yup, that’s all there is to entanglement (mathematically at least). The paradoxes entanglement including Einstein’s ‘creepy action at a distance’ are left for you to explore — again Zeilinger’s book is a great source.

But how can it be like that you ask? Feynman said not to start thinking these thoughts, and if he didn’t know you expect a retired neurologist to tell you? Please.