How to Calculate Viterbi Without Observed Sequence

The Viterbi algorithm is a dynamic programming technique used to find the most likely sequence of hidden states in a Markov model. When there's no observed sequence, the algorithm simplifies to finding the most probable path through the state space.

What is the Viterbi Algorithm?

The Viterbi algorithm is a method for finding the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events, especially in applications like speech recognition, DNA sequence analysis, and error correction.

In a hidden Markov model (HMM), the algorithm works by:

Initializing the probability of being in each state at time t=0
Iteratively calculating the probability of being in each state at time t+1
Tracking the most likely path to reach each state
Selecting the path with the highest probability at the final time step

Key Formula: The Viterbi algorithm uses dynamic programming to compute the maximum probability path through the state space.

Viterbi Without Observed Sequence

When there's no observed sequence, the Viterbi algorithm simplifies to finding the most probable path through the state space without considering emission probabilities. This is useful in scenarios where you only care about the sequence of states and not the observed outputs.

The simplified algorithm steps are:

Define the initial state probabilities
Define the transition probabilities between states
Use dynamic programming to find the path with maximum probability
Backtrack to find the most likely sequence

Note: Without observed data, the algorithm relies solely on the transition probabilities between states.

Step-by-Step Calculation

Define the states: Identify all possible hidden states in your model.
Set initial probabilities: Define the probability of starting in each state.
Define transition probabilities: Specify the probability of moving from one state to another.
Initialize the Viterbi table: Create a table to store the maximum probabilities and paths.
Fill the Viterbi table: For each time step, calculate the maximum probability of reaching each state.
Backtrack to find the path: Starting from the final state with maximum probability, trace back through the table to find the most likely sequence.

Viterbi Table Update: For each state s at time t, the probability is calculated as:

V(t, s) = max[V(t-1, s') * P(s|s')] for all possible previous states s'

Worked Example

Consider a simple weather model with two states: Sunny (S) and Rainy (R).

Initial probabilities: P(S) = 0.6, P(R) = 0.4

Transition probabilities:

P(S|S) = 0.7, P(R|S) = 0.3
P(S|R) = 0.4, P(R|R) = 0.6

We want to find the most probable sequence of weather over 3 days.

The calculation would proceed as follows:

Day 1: Choose the state with highest initial probability (Sunny)
Day 2: Choose the state with highest transition probability from Day 1 (Sunny)
Day 3: Choose the state with highest transition probability from Day 2

The most probable sequence would be Sunny → Sunny → Sunny with a probability of 0.6 * 0.7 * 0.7 = 0.294.

FAQ

What is the difference between Viterbi with and without observed sequence?

With observed sequences, the algorithm considers both transition and emission probabilities. Without observed sequences, it only considers transition probabilities to find the most likely state sequence.

When would I use Viterbi without observed sequence?

This approach is useful when you only care about the sequence of hidden states and don't have observed data to consider.

How does the Viterbi algorithm handle multiple states?

The algorithm evaluates all possible paths through the state space and selects the one with the highest probability at each step.