Understanding the Log-Sum-Exp Trick and Its Application in Hidden Markov Models (HMMs)

The log-sum-exp trick is a critical technique in numerical computations involving logarithms and exponentials. It is widely used in machine learning, especially in algorithms like the forward-backward procedure in Hidden Markov Models ( HMMs). In this post, we will cover:

1. What is the log-sum-exp trick?
2. Why do we need the log-sum-exp trick?
3. An example of using the log-sum-exp trick in solving HMMs.

1. What is the Log-Sum-Exp Trick?

The log-sum-exp trick is a numerically stable method for computing the logarithm of a sum of exponentials:

Direct computation of this term can result in numerical instability due to overflow or underflow when $x_i$ are large or small. To avoid this, we use the log-sum-exp trick:

Formulation

where:

Why This Works

• Subtracting $c$ ensures that the exponentials $e^{x_i - c}$ do not result in extremely large or small values.
• Adding $c$ back after the computation ensures the result remains correct.

2. Why Do We Need the Log-Sum-Exp Trick?

Numerical Instability in Logarithms and Exponentials

The sum of exponentials $ \sum_{i=1}^n e^{x_i} $ can:

• Overflow: When $x_i$ are very large, $e^{x_i}$ grows exponentially, leading to values that exceed the representable range of floating-point numbers.
• Underflow: When $x_i$ are very small, $e^{x_i}$ becomes extremely close to zero, which can result in numerical inaccuracies.

Example Scenario

In the context of HMMs, we often compute the forward probabilities $\alpha$ or backward probabilities $\beta$ using recursive equations. These involve sums of exponentials, which can become unstable without the log-sum-exp trick.

3. Example: Using the Log-Sum-Exp Trick in HMMs

(Please note that this is a simplified example for illustrative purposes. If you are looking for a more detailed tutorial on HMMs, please refer to my other posts.)

Problem: Forward Algorithm in HMMs

The forward probabilities $\alpha_t(i)$ are computed recursively as:

where:

• $\pi_{ji}$ is the transition probability from state $j$ to state $i$,
• $\psi_i(O_t)$ is the emission probability of observation $O_t$ given state $i$,
• $\alpha_{t-1}(j)$ is the forward probability for state $j$ at time $t-1$.

When implemented in log-space to prevent underflow, the recursion becomes:

Here, the log-sum-exp trick is used for the term:

Applying the Log-Sum-Exp Trick

Let:

Using the log-sum-exp trick:

1. Compute the maximum value:
   $$c = \max_j x_j.$$
2. Subtract $c$ and compute the scaled sum:
   $$\log\left(\sum_{j=1}^K e^{x_j}\right) = c + \log\left(\sum_{j=1}^K e^{x_j - c}\right).$$ Thus: $$\log \alpha_t(i) = \log \psi_i(O_t) + c + \log\left(\sum_{j=1}^K e^{\log \alpha_{t-1}(j) + \log \pi_{ji} - c}\right).$$

Full Forward Algorithm with Log-Sum-Exp

1. Initialization:
   $$\log \alpha_1(i) = \log \pi_i + \log \psi_i(O_1).$$
2. Recursion: For $t = 2, 3, \dots, T$:
   $$\log \alpha_t(i) = \log \psi_i(O_t) + \max_j (\log \alpha_{t-1}(j) + \log \pi_{ji}) \\ + \log \left(\sum_{j=1}^K e^{\log \alpha_{t-1}(j) + \log \pi_{ji} - \max_j (\log \alpha_{t-1}(j) + \log \pi_{ji})}\right).$$
3. Termination: Compute the final probability: $$ \log P(O) = \log \left(\sum_{i=1}^K e^{\log \alpha_T(i)}\right). $$

Key Advantages

• Numerical Stability: The subtraction of the maximum value $c$ prevents overflow and underflow.
• Efficiency: The algorithm avoids unnecessary computational errors without significant performance overhead.

Summary

The log-sum-exp trick is an essential tool for numerically stable computations in machine learning and statistics. In the context of HMMs, it ensures that the forward and backward algorithms are robust to extreme probabilities. By avoiding numerical instability, the log-sum-exp trick allows us to solve real-world problems accurately and efficiently.