probabilities with small numbers

Navigating Probabilities with Extremely Small Numbers

Explore the challenges and solutions for accurately calculating and representing probabilities that are very close to zero, crucial in fields like statistics, machine learning, and scientific computing.

When dealing with probabilities, especially in complex systems or large datasets, it's common to encounter values that are extremely small. These numbers, often represented as 1e-100 or even smaller, can pose significant computational challenges. Standard floating-point arithmetic in most programming languages has limitations in precision, leading to underflow errors where these tiny probabilities are rounded down to zero. This article delves into why this happens and provides practical strategies, including the use of log-probabilities and arbitrary-precision arithmetic, to maintain accuracy.

The Problem of Floating-Point Underflow

Computers represent real numbers using floating-point formats (e.g., IEEE 754 standard for float and double). These formats have a finite range and precision. When a number becomes smaller than the smallest representable non-zero value, it's typically 'underflowed' to zero. For probabilities, this means that P(A) * P(B) * P(C) where P(A), P(B), and P(C) are all very small, can quickly become 0.0, even if the true probability is non-zero. This loss of information can severely impact the reliability of statistical models, especially in areas like Bayesian inference, hidden Markov models, or large-scale simulations where many small probabilities are multiplied together.

flowchart TD
    A[Start with small probabilities] --> B{Multiply P1 * P2 * ... * Pn}
    B --> C{Intermediate product becomes very small}
    C --> D{"Is product < min_representable_float?"}
    D -- Yes --> E[Underflow: Product becomes 0.0]
    D -- No --> F[Continue calculation]
    E --> G[Loss of information, incorrect results]
    F --> H[Accurate result]
    G --> I[End]
    H --> I[End]

Flowchart illustrating floating-point underflow with small probabilities.

Solution 1: Working with Log-Probabilities

The most common and effective solution to combat underflow is to work with log-probabilities instead of raw probabilities. The key mathematical property is that log(P1 * P2 * ... * Pn) = log(P1) + log(P2) + ... + log(Pn). By converting multiplications into additions, we avoid the issue of products becoming too small. Logarithms can handle a much wider range of values, preventing underflow. When the final probability is needed, you can simply exponentiate the sum of log-probabilities (exp(sum_of_log_probs)).

However, a challenge arises when you need to add probabilities, e.g., P(A) + P(B). In log-space, this becomes log(exp(log_P(A)) + exp(log_P(B))). A common numerical trick for this is the 'log-sum-exp' function: log(exp(a) + exp(b)) = a + log(1 + exp(b - a)) (assuming a >= b). This form helps prevent overflow when exp(a) or exp(b) are very large, and underflow when exp(b - a) is very small.

import math

def log_sum_exp(log_a, log_b):
    """Numerically stable computation of log(exp(log_a) + exp(log_b))."""
    if log_a == -math.inf:
        return log_b
    if log_b == -math.inf:
        return log_a
    
    # Use the log-sum-exp trick: log(exp(a) + exp(b)) = a + log(1 + exp(b - a))
    # assuming a >= b to prevent exp(b - a) from overflowing
    if log_a >= log_b:
        return log_a + math.log1p(math.exp(log_b - log_a))
    else:
        return log_b + math.log1p(math.exp(log_a - log_b))

# Example usage:
prob1 = 1e-200
prob2 = 1e-205

log_prob1 = math.log(prob1)
log_prob2 = math.log(prob2)

# Multiplying probabilities (adding log-probabilities)
log_product = log_prob1 + log_prob2
product = math.exp(log_product)
print(f"Product (raw): {prob1 * prob2}") # Will likely be 0.0
print(f"Product (log-space): {product}") # Accurate

# Adding probabilities (using log-sum-exp)
log_sum = log_sum_exp(log_prob1, log_prob2)
sum_probs = math.exp(log_sum)
print(f"Sum (raw): {prob1 + prob2}") # Will likely be prob1 (prob2 underflows)
print(f"Sum (log-space): {sum_probs}") # Accurate

Python example demonstrating log-sum-exp for stable probability addition and multiplication.

Solution 2: Arbitrary-Precision Arithmetic

For scenarios where log-probabilities are not suitable (e.g., direct comparison of very small numbers, or when the problem inherently requires maintaining the exact fractional representation), arbitrary-precision arithmetic libraries can be used. These libraries store numbers as sequences of digits, allowing for virtually unlimited precision, constrained only by available memory. While computationally more expensive than standard floating-point operations, they guarantee accuracy for extremely small or large numbers.

Libraries like Python's decimal module, Java's BigDecimal, or C++'s Boost.Multiprecision provide this capability. They are particularly useful in cryptographic applications, financial calculations, or scientific simulations where even minute inaccuracies can propagate into significant errors.

from decimal import Decimal, getcontext

# Set the precision for Decimal operations
getcontext().prec = 100 # 100 significant digits

prob1 = Decimal('1e-200')
prob2 = Decimal('1e-205')

# Multiplication
product = prob1 * prob2
print(f"Product with Decimal: {product}")

# Addition
sum_probs = prob1 + prob2
print(f"Sum with Decimal: {sum_probs}")

# A very small number
very_small_prob = Decimal('1e-500')
print(f"Very small prob with Decimal: {very_small_prob}")

# Compare with standard float (will underflow)
float_prob1 = 1e-200
float_prob2 = 1e-205
print(f"Float product: {float_prob1 * float_prob2}") # Output: 0.0
print(f"Float sum: {float_prob1 + float_prob2}") # Output: 1e-200 (prob2 underflows)

Python example using the decimal module for arbitrary-precision probability calculations.