Hidden Markov Models by Hand: States & Emissions, the Forward Algorithm, Viterbi Decoding, Smoothing & Simple Learning from Counts

What you'll build

What you will be able to do

Each lab ends with a concrete result that you build yourself and the checker confirms, cell by cell.

• Complete a two-state weather transition table, push today's distribution one day forward with SUMPRODUCT, compute a two-step probability and the probability of a whole four-day weather path.
• Add an emission table to the weather chain, multiply start, transition and emission probabilities for all 8 possible weather paths behind three logbook days, find the most likely path and add the joints up to the probability of the logbook itself.
• Fill the forward table α for five logbook days – start × emission on day 1, then yesterday's α times the transitions times today's emission – and check that three days reproduce the sum of all 8 paths from lesson 2.
• Replace the forward sum by a maximum, store for every state which previous state gave that maximum, read the best last state and walk the back-pointers backwards to the most likely weather path.
• Turn two five-day paths into log-probabilities with LN, compare them by the difference of logs, convert that back into a ratio with EXP and see a long product underflow to zero while its logarithm stays an ordinary number.
• Fill the backward table β from the last day towards the first, combine it with the forward table into posterior probabilities for every day and compare the posterior answer for day 2 with the Viterbi path.
• Count weather transitions with COUNTIFS on two shifted ranges, turn them into transition probabilities, estimate emission probabilities from labelled days and use add-one smoothing so that an outcome never seen in a state still gets a small chance.
• Run Viterbi with three hidden tags over the original sentence «the keeper lights lamps», score a toy tagger against gold tags and chart the posterior tag probabilities per word.