Bayesian Regime Filter for Alpha Signal

- q$ if you were in regime 0.

Update step. Incorporate the new observation $A_t$ via Bayes' rule:

$\pi_t = \frac{\pi_{t|t-1} \cdot \phi(A_t; \mu_1, \sigma^2)}{\pi_{t|t-1} \cdot \phi(A_t; \mu_1, \sigma^2) + (1 - \pi_{t|t-1}) \cdot \phi(A_t; \mu_0, \sigma^2)}$

where $\phi(x; \mu, \sigma^2)$ is the Gaussian density. This can be simplified using the likelihood ratio $L_t = \phi(A_t; \mu_1, \sigma^2) / \phi(A_t; \mu_0, \sigma^2)$:

$\pi_t = \frac{\pi_{t|t-1} \cdot L_t}{\pi_{t|t-1} \cdot L_t + (1 - \pi_{t|t-1})}$

Since both densities are Gaussian with the same variance, the log-likelihood ratio has a clean form:

$\log L_t = \frac{(\mu_1 - \mu_0)}{\sigma^2} \left(A_t - \frac{\mu_1 + \mu_0}{2}\right)$

This is a linear function of the observation $A_t$, which is computationally convenient. Initialize with $\pi_0 = P(z_0 = 1)$, typically the ergodic probability $(1 - q) / (2 - p - q)$.

Part 2: De-Risking Rule and Threshold Selection

Key Insight: The threshold $\pi^{*}$ controls a trade-off between two costs: false de-risking (cutting exposure when the alpha regime is actually on, forfeiting returns) and false staying-in (keeping full exposure when alpha has disappeared, eating losses).

The Rule: Set position size proportional to the filtered probability, with a hard cutoff:

$w_t = \begin{cases} w_{\text{full}} \cdot \frac{\pi_t - \pi^{*}}{1 - \pi^{*}} & \text{if } \pi_t \geq \pi^{*} \\ 0 & \text{if } \pi_t < \pi^{*} \end{cases}$

This ramps down linearly as confidence in the alpha regime fades, and goes flat at zero below the threshold. A simpler binary version just toggles between full position ($\pi_t \geq \pi^{*}$) and no position ($\pi_t < \pi^{*}$).

**Choosing $\pi^{*}$:**

1. Simulation approach. Simulate the full Markov-switching model with known parameters. For each candidate threshold $\pi^{*}$, compute the false de-risking rate (fraction of time $\pi_t < \pi^{*}$ when $z_t = 1$) and the false exposure rate (fraction of time $\pi_t \geq \pi^{*}$ when $z_t = 0$). Plot the ROC curve and choose $\pi^{*}$ to hit a target false de-risking rate (e.g., 5%).

1. Cost-ratio approach. Let $c_{\text{miss}}$ be the cost of missing one day of alpha and $c_{\text{exposed}}$ be the cost of one day of false exposure. The optimal threshold satisfies:

$\pi^{*} = \frac{c_{\text{exposed}}}{c_{\text{miss}} + c_{\text{exposed}}}$

This is just the classical Bayes decision boundary. If missing alpha is much more costly than false exposure ($c_{\text{miss}} \gg c_{\text{exposed}}$), you set $\pi^{*}$ low and stay in longer. If drawdown risk dominates, you set $\pi^{*}$ high and cut early.

1. Practical calibration. In practice, you estimate $p$, $q$, $\mu_0$, $\mu_1$, $\sigma$ from historical data (often via EM algorithm), then simulate many paths to find the threshold that maximizes risk-adjusted returns (Sharpe ratio of the filtered strategy). Typical values of $\pi^{*}$ in equity regime models are in the range 0.3-0.5.

Answer: The forward filter is a predict-update recursion: predict via the Markov transition $\pi_{t|t-1} = p \pi_{t-1} + (1-q)(1-\pi_{t-1})$, then update via Bayes' rule using the Gaussian likelihood ratio. Position sizing is tied to $\pi_t$ with a threshold $\pi^{*}$ chosen by balancing the cost of false de-risking against the cost of false exposure, either through simulation-based ROC analysis or the Bayes-optimal decision boundary $\pi^{*} = c_{\text{exposed}} / (c_{\text{miss}} + c_{\text{exposed}})$.