Hidden Markov models

Idea: A state nobody publishes

A hidden state changes by a Markov rule. You see only observations that depend on it.

• Hidden: a deep or a thin market
• Seen: each day's spread
• Model: start, transition table, emission table

Formula: Filtering

$P(Z_t\mid Y_1,\dots ,Y_t)$

The chance of each hidden state given everything seen so far, updated every day.

Idea: Predict, then correct

Each day push the belief through the transitions, then weight it by the new data.

• Predict: the transition table, before any data
• Correct: Bayes with the emission table
• Today's output is tomorrow's input, no history stored

Example: A wide spread prints

Deep stays deep $0.9$, thin stays thin $0.8$. Wide spreads: $10\%$ of deep days, $60\%$ of thin. Belief in deep $0.8$, today prints wide. New belief?

1. Predict: $0.8\times 0.9+0.2\times 0.2=0.76$
2. Weight: $0.76\times 0.1=0.076$, $0.24\times 0.6=0.144$
3. Correct: $\frac{0.076}{0.22}=\frac{19}{55}$

Common mistake: Skipping the predict step

Bayes on last night's belief gives $\frac{0.08}{0.08+0.12}=\frac{2}{5}$ instead of $\frac{19}{55}$.

Instead The state can move overnight. Always predict first, then correct.

Formula: The long-run share

$\pi =\frac{b}{a+b}$

Share of a state: the other state's exit rate b over the sum of both exit rates.

Quick check: Share of deep days

Deep leaves with chance $0.1$ a day, thin with $0.2$. What is the long-run share of deep days?

Idea: Filter, smooth, decode

Keep the questions apart: only the filter can be traded.

• Filtering: data up to today
• Smoothing: also later data, so a backtest on it looks ahead
• Decoding: the single most likely hidden path

Idea: Say it out loud

Name the layers, then say "predict, then correct" and write the lines.

• The Markov rule is a claim about the hidden state
• Recursion: $2\times 2\times 100=400$ steps, not $2^{100}$ paths
• Tables fitted on calm data understate how fast liquidity vanishes