*(NB: Actually a weighted sum. But if you just normalise it (divide by the overall total) you’ll get a weighted average.)*

*The Economist*'s Which MBA? website scores MBA programmes on:

faculty quality, student quality, student diversity, percentage who found jobs through the careers service, student assessment of career service, percentage in work three months after graduation, increase in salary, potential to network, internationalism of alumni, student rating of alumni effectiveness, and a few other metrics

— and lets you adjust how important each of these factors are to you, determining*your*ranking of MBA programmes (using their data=methodology),

rather than pretending there’s a universal or objective weighting of importance of factors (as the US News & World Report ranking of US undergrad schools does).- My friend made a spreadsheet of all the factors that determined what city she wants to move to.

She scored each city on various factors, then assigned each of those factors an importance, added and timesed and got a total score for each city. (I don’t think the result is meaningful, because I don’t think the space is linear. But the exercise itself was fun and gave her a reason to do the research.) - In my car radio I have knobs for “treble" and "bass", which weight particular functional forms more heavily than others.
- When you do a Gaussian Blur in photoshop

.

or smooth a time series against a Gaussian kernel,

you’re (basically) covectoring against a Normal curve. In other words you weight the neighbours with heights of`2**−distance² = 1/2, 1/16, 1/512, ...`

.

(I actually think of the Gaussian now as an optimal smoother, primarily, instead of as Bell Curve religion. But that’s a story for another time.) - The standard “regression beta"—the OLS squares minimisation problem—is to adjust a covector—the tilts of the various data columns

=properties you’ve observed and quantified (plus a column of ones) to match

a straight-line fit up against whatever you’ve chosen as`y`

. - An artist in a coffee shop once told me he had found some great numerical parameters for the particular visual (like a Winamp style one) he was creating. He was clearly thinking about the parameter space as such, but the maximisation procedure he was following was probably not a mechanical one.
- If polynomials are sequences where, instead of being limited to a largest digit of 9 in the hundreds digit, we’re not limited to positive, negative, fraction, whatever, in the
`xx=x²`

constant, then the constants you line up — whether they have some well-known name or pattern like combinatorial sequence, Sheffer sequence, Schur polynomial, Taylor series, or have no name — are the covector. (This overstretches my simplification that covectors are averages. Here they really need to be sums.) - A client wanted customers to be able to browse his wares easier in his online store. This boils down to bubbling up to the top what they want to see and sorting down what they don’t want to see. One idea he had was to give the customer a number of “sliders” and let them choose which aspects were important to them. So instead of sorting first by price, then sorting within that sort by alphabetical, you would catalogue various properties of the stuff in your ecommerce storefront, multiply those by a fixed number chosen by the customer, add those subscores together to get a total score, and then sort on that total score. That way the list can be mixed. (The customer wants to penalise high prices and non-red dresses, but doesn’t want to see only $2 purse accessories that somehow got parsed by the computer as “dress”.)

Another way to say this is he wanted to let customers define their own “scoring metric” and sort results based on that.

All of these are **covectors**.

In order to not get confused about the meaning of “parameter" versus "variable" — let me just use the concrete examples above. The weighting scheme on the MBA programme is the covector and the observed properties of each MBA programme are the vector. Multiply the vector for a particular school and the **covector (weighting scheme)** you’ve chosen, and you get “your score” (a single number). Do this for each school and you can then sort the results to get “your ranking”.

If you changed the weighting scheme, you change the covector, i.e. you change the parameters. This is “moving in the dual space” and it outputs a different “your ranking”.

So the next time someone says to you "Canonically identify a vector space with its dual via `g↦∫fg"`

, that’s basically what they mean.

(By the way, this duality is also used in the reproducing kernel Hilbert space, a key part of machine learning.)