Tracking as Inference

1. Inference system

• Hidden State ($X$): True parameters we care about
• Measurement ($Y$): Noisy observation of underlying state

At each time step $t$, state changes (from $X_{t-1}$ to $X_t$), and we get a new observation $Y_t$

Our goal: recover most likely state $X_t$ given:

• All observation seen so far
• Knowledge about dynamics of state transitions

2. Steps of Tracking

2.1 Prediction

What is the next state of the object given past measurements? $$P(X_t | Y_0 = y_0, ..., Y_{t-1} = y_{t-1})$$

2.2 Correction

Compute an updated estimate of the state from prediction and measurements $$P(X_t | Y_0 = y_0, ..., Y_{t-1} = y_{t-1}, Y_t = y_t)$$

2.3 Tracking

The process of propagating this posterior distribution of state given measurements across time.

3. Simplifying Assumptions

3.1 Only the immediate past matters: (dynamics model)

$$P(X_t | X_0, ..., X_{t-1}) = P(X_t | X_{t-1})$$

3.2 Measurements depend only on the current state: (observation model)

$$P(Y_t | X_0, Y_0, ..., X_{t-1}, Y_{t-1}, X_t) = P(Y_t | X_t)$$

This is a Hidden Markov Model (HMM)