logfloat-0.13.2: Log-domain floating point numbers

Data.Number.LogFloat

Description

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 Data.Number.LogFloat.Signed for doing signed log-domain calculations.

Synopsis

# `LogFloat` data type

data LogFloat Source

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 hence will underflow at that point. This behavior may change in the future.

Because `logFloat` performs the semantic conversion, we can use operators which say what we *mean* rather than saying what we're actually doing to the underlying representation. That is, equivalences like the following are true thanks to type-class overloading:

```logFloat (p + q) == logFloat p + logFloat q
logFloat (p * q) == logFloat p * logFloat q```

(Do note, however, that subtraction can and negation will throw errors: since `LogFloat` can only represent the positive half of `Double`. `Num` is the wrong abstraction to put at the bottom of the numeric type-class hierarchy; but alas, we're stuck with it.)

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. Also note that, for precision, if you're doing more than a few multiplications in the log-domain, you should use `product` rather than using '(*)' repeatedly.

Even more particularly, you should avoid addition whenever possible. Addition is provided because sometimes we need it, and the proper implementation is not immediately apparent. However, between two `LogFloat`s addition requires crossing the exp/log boundary twice; with a `LogFloat` and a `Double` 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 normal-domain operations first, do it!

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That is, true up-to underflow and floating point fuzziness. Which is, of course, the whole point of this module.

## Isomorphism to normal-domain

Constructor which does semantic conversion from normal-domain to log-domain. Throws errors on negative and NaN inputs. If `p` is non-negative, then following equivalence holds:

`logFloat p == logToLogFloat (log p)`

If `p` is NaN or negative, then the two sides differ only in which error is thrown.

Semantically convert our log-domain value back into the normal-domain. Beware of overflow/underflow. The following equivalence holds (without qualification):

`fromLogFloat == exp . logFromLogFloat`

## Isomorphism to log-domain

Constructor which assumes the argument is already in the log-domain. Throws errors on `notANumber` inputs.

Return the log-domain value itself without conversion.

sum :: [LogFloat] -> LogFloat Source

O(n). Compute the sum of a finite list of `LogFloat`s, being careful to avoid underflow issues. That is, the following equivalence holds (modulo underflow and all that):

`logFloat . sum == sum . map logFloat`

N.B., this function requires two passes over the input. Thus, it is not amenable to list fusion, and hence will use a lot of memory when summing long lists.

Since: 0.13

product :: [LogFloat] -> LogFloat Source

O(n). Compute the product of a finite list of `LogFloat`s, being careful to avoid numerical error due to loss of precision. That is, the following equivalence holds (modulo underflow and all that):

`logFloat . product == product . map logFloat`

Since: 0.13

pow :: LogFloat -> Double -> LogFloat infixr 8 Source

O(1). Compute powers in the log-domain; that is, the following equivalence holds (modulo underflow and all that):

`logFloat (p ** m) == logFloat p `pow` m`

Since: 0.13

# Accurate versions of logarithm/exponentiation

Definition: `log1p == log . (1+)`. Standard C libraries provide a special definition for `log1p` which is more accurate than doing the naive thing, especially for very small arguments. For example, the naive version underflows around `2 ** -53`, whereas the specialized version underflows around `2 ** -1074`. This function is used by (`+`) and (`-`) on `LogFloat`.

N.B. The `statistics:Statistics.Math` module provides a pure Haskell implementation of `log1p` for those who are interested. We do not copy it here because it relies on the `vector` package which is non-portable. If there is sufficient interest, a portable variant of that implementation could be made. Contact the maintainer if the FFI and naive implementations are insufficient for your needs.

This installation was compiled to use the FFI version.

Definition: `expm1 == subtract 1 . exp`. Standard C libraries provide a special definition for `expm1` which is more accurate than doing the naive thing, especially for very small arguments. This function isn't needed internally, but is provided for symmetry with `log1p`.

This installation was compiled to use the FFI version.