

Why interval arithmetic is so useful (2007) [pdf] - maho
http://subs.emis.de/LNI/Seminar/Seminar07/148.pdf

======
imh
I think this kind of thing is great, but it runs into some of the same
considerations that make probabilistic graphical models hard.

If I have two variables defined as

    
    
        x = [0,1];
        y = [0,1];
        z = x - y;
    

Then we see that z should be the interval [-1,1]. But for a similar problem,

    
    
        a = [0,1];
        b = a - a;
        x = a;
        y = a;
        z = x - y;
    

b should clearly be the interval [0,0]. z is computed the same way as the
first example, as the difference between two [0,1] intervals. But this time we
want it to be [0,0]. That means we want to keep around some notion similar to
the joint distribution of x and y. That makes arithmetic immensely harder than
simple rules.

Not to say it's not worth it, just that it's hard.

~~~
anon4
You get the same behaviour with regular floats:

    
    
        >>> a=10.0**10
        >>> a
        10000000000.0
        >>> b = 0.1
        >>> b+a
        10000000000.1
        >>> c=b+a
        >>> d=c-a
        >>> d
        0.10000038146972656
        >>> d == b
        False
        >>> a + b - b == b
        False
        >>> 
    

You could build a variable-tracking system on top of the numbers, or you could
just use a symbolic computation package, then simplify the final equations so
each variable figures only once.

------
paperwork
Could such intervals be represented by random variables of uniform
distributions?

~~~
Someone
No. You cannot do that because adding/subtracting/multiplying/... two
uniformly distributed independent variables does not produce a uniformly
distributed variable (see for example
[http://en.m.wikipedia.org/wiki/Irwin–Hall_distribution](http://en.m.wikipedia.org/wiki/Irwin–Hall_distribution))

Thats true for other distributions, too. You could use the obvious choice of a
normal distribution, if all you did is add and subtract, but that's (about?)
as good as it gets.

