Data.Number.LogFloat

This module presents a type for storing numbers in the log-domain. The main reason for doing this is to prevent underflow when multiplying many small probabilities as is done in Hidden Markov Models and other statistical models often used for natural language processing. The log-domain also helps prevent overflow when multiplying many large numbers. In rare cases it can speed up numerical computation (since addition is faster than multiplication, though logarithms are exceptionally slow), but the primary goal is to improve accuracy of results. A secondary goal has been to maximize efficiency since these computations are frequently done within a O(n^3) loop.

The LogFloat of this module is restricted to non-negative numbers for efficiency's sake, see the forthcoming Data.Number.LogFloat.Signed for doing signed log-domain calculations.

Basic functions

log :: (Floating a, Transfinite a) => a -> a

Since the normal log throws an error on zero, we have to redefine it in order for things to work right. Arguing from limits we can see that log 0 == negativeInfinity. Newer versions of GHC have this behavior already, but older versions and Hugs do not.

This function will raise an error when taking the log of negative numbers, rather than returning notANumber as the newer GHC implementation does. The reason being that typically this is a logical error, and notANumber allows the error to propegate silently.

In order to improve portability, the Transfinite class is required to indicate that the Floating type does in fact have a representation for negative infinity. Both native floating types (Double and Float) are supported. If you define your own instance of Transfinite, verify the above equation holds for your 0 and negativeInfinity. If it doesn't, then you should avoid importing our log and will probably want converters to handle the discrepancy when dealing with LogFloats.

toFractional :: (Real a, Fractional b) => a -> b

The most generic numeric converter I can come up with. All the built-in numeric types are Real, though Int and Integer aren't Fractional. Beware that converting transfinite values into Ratio types is error-prone and non-portable, as discussed in Data.Number.Transfinite.

LogFloat data type and conversion functions

data LogFloat

A LogFloat is just a Double with a special interpretation. The logFloat function is presented instead of the constructor, in order to ensure semantic conversion. At present the Show instance will convert back to the normal-domain, and so will underflow at that point. This behavior may change in the future.

Performing operations in the log-domain is cheap, prevents underflow, and is otherwise very nice for dealing with miniscule probabilities. However, crossing into and out of the log-domain is expensive and should be avoided as much as possible. In particular, if you're doing a series of multiplications as in lp * logFloat q * logFloat r it's faster to do lp * logFloat (q * r) if you're reasonably sure the normal-domain multiplication won't underflow, because that way you enter the log-domain only once, instead of twice.

Even more particularly, you should avoid addition whenever possible. Addition is provided because it's necessary at times and the proper implementation is not immediately transparent. However, between two LogFloats addition requires crossing the exp/log boundary twice; with a LogFloat and a regular number it's three times since the regular number needs to enter the log-domain first. This makes addition incredibly slow. Again, if you can parenthesize to do plain operations first, do it!

Instances

Eq LogFloat

Fractional LogFloat

Num LogFloat

Ord LogFloat

Real LogFloat

Show LogFloat

logFloat :: Real a => a -> LogFloat

A constructor which does semantic conversion from normal-domain to log-domain.

logToLogFloat :: Real a => a -> LogFloat

Constructor which assumes the argument is already in the log-domain.

fromLogFloat :: (Fractional a, Transfinite a) => LogFloat -> a

Return our log-domain value back into normal-domain. Beware of overflow/underflow.

logFromLogFloat :: (Fractional a, Transfinite a) => LogFloat -> a

Return the log-domain value itself without costly conversion

Exceptional numeric values

module Data.Number.Transfinite