Forward Algorithm Calculator

Example Data Table

Component	Values	Explanation
Hidden States	Sunny, Rainy	These are the unobserved states in the model.
Observation Symbols	Happy, Sad	These are visible outputs emitted by hidden states.
Initial Probabilities	0.6, 0.4	Model starts in Sunny with 0.6 and Rainy with 0.4.
Transition Matrix	[0.7, 0.3] and [0.4, 0.6]	State switching probabilities between time steps.
Emission Matrix	[0.8, 0.2] and [0.3, 0.7]	Probability of each observation for each hidden state.
Observation Sequence	Happy, Sad, Happy	The visible symbols evaluated by the forward recursion.

Formula Used

The forward algorithm computes the probability of an observation sequence under a hidden Markov model by recursively summing all valid hidden-state paths.

Initialization: α₁(i) = π(i) × bᵢ(o₁)

Recursion: αₜ(j) = [Σ αₜ₋₁(i) × aᵢⱼ] × bⱼ(oₜ)

Termination: P(O|λ) = Σ α_T(i)

Where π(i) is the initial probability of state i, aᵢⱼ is the transition probability from state i to state j, and bⱼ(oₜ) is the emission probability of the observed symbol at time t.

When scaling is enabled, each time-step vector is normalized to limit floating-point underflow in long sequences.

How to Use This Calculator

Enter one hidden state per line in the hidden states box.
Enter one observation symbol per line in the observation symbols box.
Provide initial probabilities with the same count as hidden states.
Enter a square transition matrix where each row sums to 1.
Enter an emission matrix where rows map to states and columns map to observations.
Type the observation sequence using spaces or commas.
Enable scaling for long sequences or very small probabilities.
Press the calculate button to show results above the form.
Download the results as CSV or PDF if needed.

FAQs

1. What does the forward algorithm calculate?

It calculates the probability of an observed sequence under a hidden Markov model. It does this by summing probabilities across all hidden-state paths efficiently, without enumerating every path directly.

2. Why is scaling useful?

Scaling prevents very small probabilities from underflowing to zero during long recursions. It keeps the intermediate values numerically stable while preserving the log likelihood and overall interpretation.

3. What must the transition matrix satisfy?

Each row must represent a valid probability distribution. That means every value must be nonnegative, and each row must sum to 1 within a small numerical tolerance.

4. What must the emission matrix satisfy?

Each row corresponds to one hidden state, and each column corresponds to one observation symbol. Every row must sum to 1, and no probability may be negative.

5. Can I use words instead of numeric observation labels?

Yes. Observation symbols can be words such as Happy, Sad, Up, or Down. The sequence entries must exactly match the listed observation names, including spelling.

6. What is the difference between raw and scaled alpha values?

Raw alpha values are direct forward probabilities. Scaled alpha values are normalized at each step for stability. Scaled values are better for long sequences, while raw values are more literal.

7. Why might my model return a very small sequence probability?

Observation sequences often become extremely unlikely as their length grows, because probabilities are multiplied repeatedly. A tiny probability does not necessarily mean the model is wrong.

8. Where is this method used?

It is widely used in speech recognition, bioinformatics, activity detection, error correction, finance, and sequence modeling tasks involving hidden Markov models and uncertain hidden states.