This project implements a discrete Bayes Filter for probabilistic state estimation, developed as part of the RBE595 Robot Navigation course at Worcester Polytechnic Institute. The system estimates whether a door is open or closed by fusing noisy sensor measurements with a probabilistic action model through iterative predict-and-correct cycles.

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Algorithm

The Bayes Filter maintains a belief distribution over all possible states and updates it in two steps at each time instant:

Prediction Step

Given the robot’s action \(u_t\), the prior belief is propagated forward using the motion model:

\[\overline{bel}(x_t) = \sum_{x_{t-1}} p(x_t \mid u_t, x_{t-1}) \cdot bel(x_{t-1})\]

This marginalizes over all possible previous states, weighting each by its probability under the action model.

Correction Step

The predicted belief is then updated with the sensor measurement \(z_t\):

\[bel(x_t) = \eta \cdot p(z_t \mid x_t) \cdot \overline{bel}(x_t)\]

where \(\eta\) is a normalization constant ensuring the distribution sums to 1.

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State Estimation Scenarios

The filter was evaluated across three experimental scenarios:

Scenario	Actions	Observations	Convergence
Push + open measurement	Push door	Sense open	~4 iterations to 99.99%
Do nothing + closed measurement	No action	Sense closed	~6 iterations to 99.99%
Mixed actions and noisy sensing	Push / no-op	Mixed	~10 iterations

In all cases, the filter converges from a uniform prior to near-certain belief despite sensor noise, demonstrating the power of recursive Bayesian estimation.

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Key Properties

• State space: Binary (door open / door closed)

• Action model: Probabilistic push effectiveness (e.g., P(open

  push, closed) = 0.8)

• Sensor model: Noisy binary observation (e.g., P(sense open

  open) = 0.6)

• Initial belief: Uniform over all states
• Implementation: Pure NumPy in a Jupyter Notebook with inline visualizations

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