Question 1

What is a hidden Markov model, and what is "hidden" about it here?

Accepted Answer

A hidden Markov model (HMM) assumes a system moves through a sequence of unobservable (hidden) states according to a Markov chain -- the next state depends only on the current one, via a fixed transition probability matrix -- while at each time step it emits an observable symbol drawn from a state-specific emission distribution. In this simulation the hidden states are phonemes (SIL, K, AE, T, S) that a speaker is really passing through, and the observations are discretized acoustic feature symbols standing in for quantized spectral clusters extracted from an audio signal. You never see the phoneme sequence directly -- only the noisy acoustic symbols it produced -- which is exactly the estimation problem real automatic speech recognition (ASR) systems solve.

Question 2

What exactly does the Viterbi algorithm compute, and how does the trellis relate to it?

Accepted Answer

The Viterbi algorithm finds the single most probable hidden-state sequence given an observation sequence and a fixed HMM, using dynamic programming instead of checking every possible path (which would be exponential in the sequence length). The trellis is the DP table itself: one row per hidden state, one column per time step. Each cell delta[t][s] stores the log-probability of the best path that ends in state s at time t, computed as max over the previous state of (previous cell's score + log transition probability) plus the log emission probability of the current observation under state s. A backpointer records which previous state produced that maximum. After filling every column, backtracking through the backpointers from the highest-scoring final cell recovers the single best path in one linear pass.

Question 3

Why work in log-probabilities instead of raw probabilities?

Accepted Answer

Multiplying many probabilities together (one per time step, for a long utterance) shrinks the product toward zero so fast that ordinary floating-point numbers underflow to exactly 0.0 after only a few dozen frames, destroying all information about which path was better. Taking logarithms turns every multiplication into an addition, which is numerically stable over sequences of any length and, since log is monotonic, preserves the identity of the maximum-probability path exactly. This is why every practical Viterbi implementation, including this one, runs entirely in log-space and only exponentiates back to a probability if a human-readable number is needed.

Question 4

What does the noise slider actually change, and how does it affect decoding accuracy?

Accepted Answer

The noise slider sets the probability that, at a given time frame, the observation is replaced by a uniformly random acoustic symbol instead of one honestly sampled from the true hidden phoneme's emission distribution -- modeling acoustic mismatch such as background noise, microphone distortion or channel effects. At noise 0 the Viterbi decoder, which uses the exact same transition and emission model that generated the data, should recover the true phoneme path almost perfectly. As noise rises, an increasing fraction of frames carry acoustic evidence pointing to the wrong state, and the decoder must lean more heavily on the transition-probability structure (the fact that some phoneme sequences are simply more likely than others) to stay on the right path -- so decoding accuracy degrades gradually rather than catastrophically, which is exactly the graceful degradation real noisy-channel ASR systems exhibit.

Question 5

Why is the transition matrix left-to-right (Bakis) instead of allowing any state to follow any state?

Accepted Answer

Classical HMM-based speech recognizers model each word or phone as a left-to-right (Bakis-topology) chain: a state can loop on itself (letting a phoneme last for a variable number of frames, since people speak at different speeds), advance to the very next phoneme, or occasionally skip one ahead (modeling a phoneme that gets swallowed or coarticulated away) -- but time never runs backward within an utterance. This simulation's 5-state SIL-K-AE-T-S chain uses exactly that topology, with zero-probability transitions (stored as log-probability negative infinity) for any state pair that real speech physically never produces, which both matches genuine ASR practice and keeps the trellis visually interpretable as a forward sweep.

Question 6

How does this relate to how real voice assistants recognize speech?

Accepted Answer

Production ASR systems historically used exactly this HMM + Viterbi framework -- Gaussian-mixture or deep-neural-network acoustic models producing per-frame state likelihoods, phoneme and pronunciation HMMs strung together into word and sentence graphs, and Viterbi (or its beam-search approximation, used because real vocabularies make the full trellis too large) decoding the most likely word sequence -- and hybrid HMM/DNN systems remain in production use today alongside newer end-to-end neural approaches. This simulation strips the acoustic model down to a small hand-built discrete distribution and the vocabulary down to five phonemes so the entire dynamic program, backpointer trellis and decoding decision can be watched cell by cell, but the algorithmic core -- log-domain dynamic programming over a probabilistic state machine -- is identical to what shipped in real recognizers for decades.

Question 7

What is the difference between the log-likelihood of the decoded path and the decoding accuracy shown in the stats panel?

Accepted Answer

The log-likelihood of the decoded path is the Viterbi algorithm's own internal score: the sum of every log transition probability and log emission probability along the single most probable path it found, which is a property of the model and the observations alone and can be computed even without ever knowing the true generating path. Decoding accuracy, by contrast, compares the decoded path frame-by-frame against the true hidden path that was actually used to generate the synthetic utterance -- information a real recognizer never has access to, but which this simulation can show because it also plays the role of "ground truth generator." A high log-likelihood does not guarantee high accuracy (the model could be confidently wrong under heavy noise), which is exactly why both numbers are shown side by side.

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