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Bounding uncertainty
 On Being Approximate Fermi Questions

Part of approximate calculation is figuring out how uncertain you are.

 An estimating... How many lightbulbs are there in the US? Well, say there are 109 people in the US, and for each person there are 10 bulbs (more than 1, less than 100), thats about 1010 bulbs.
But how uncertain are we?
How close is the estimate?
 A bounding... How many lightbulbs are there in the US? Well, there are between 108 and 109 people in the US. For each person there is atleast 1 bulb, and definitely less than 103. So between 108 and 1012 bulbs.
Now we know our uncertainty.

Best-guess estimating gives us an estimate, but doesn't tell us how good the estimate is.
So we do bounds estimates too.

## Worst-case bounds What are the largest&smallest possible values.
Between what over&under-estimate are we sure the true value lies.
What are these bounds, this bounding box, which we are sure surrounds and contains our true value.

To to find this box, one does worst-case estimation, rather than best-guess estimation.
At each step, you do whatever makes for the worst estimate.
The overestimates are multipled together to give the worst possible overestimate. The underestimates are multipled to give worst underestimate.
And when dividing, the big overestimate is divided by the small underestimate to give the worst possible overestimate. And so on.

Worst-case boxes tend to get big quickly. Uncertainty at each step magnifies the surrounding uncertainty. There isn't the compensation one gets with a best-guess estimate, where a highish estimate here can be balanced with a lowish estimate there.
But you end up sure of where you stand.

## Likely bounds Another useful box is the likely box.
It is in between a best-guess estimate, and a worst-case estimate.

What are the largest&smallest likely values.
What box would you be surprised if the true value was outside of.

The estimation is a compromise between best-guess and worst-case.
One does the same kind of high-bound/low-bound estimating as with worst-case, but you use the surprise-bounds rather than the worst-case bounds. And if you would find it surprising for all your estimates to bee off in the same direction, you can fudge the box a bit smaller every few steps. Since there is less uncertainty at each step, and perhaps less uncertainty in the combination, the total uncertainty grows slower than the worst-case.

## Some pictures

 possibility likely best guess all together    not possible, too big Largest possible value possible, but unlikely Largest likely value likely Best guess likely Smallest likely value possible, but unlikely Smallest possible value not possible, too small

Some other ways to draw it... and [Tufte, Quant, 123-5]

 A View from the Back of the Envelope Comments encouraged. - Mitchell N Charity