Date: January 15, 2024

Topic: Formal Definitions

Recall

Bayes filter is a recursive method that takes past beliefs and updates them to produce a future belief.

It is used in a variety of applications like Histogram, Kalman and particle filters

Notes

Bayes Filter

$bel(x_t)=\eta p(z_t|x_t) \int p(x_t|u_t,x_{t-1})bel(x_{t-1})dx_{t-1}$

A framework for recursive state estimate

• Belief: Estimate of the state
• Observations: What we perceive (measurement)
• Motions: Shift in state (movement)

The above will produce an updated belief in a future timestep

Update Loop:

1. At $t_0$:
   1. original_belief + F(measurement, movement) = updated_belief_t0
2. At $t_1$:
   1. original_belief = updated_belief_t0
   2. original_belief + F(measurement, movement) = updated_belief_t1

• Repeat the above process for as many time-steps as needed

<aside> 💡 Movement: Law of Total Probability

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<aside> 💡 Measurement: Bayes Rule

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Realizations of Bayes Filter:

• Histogram Filter
• Kalman Filter
• Particle Filter

Conditional Probability:

Probability of $Y$ happening given that $X$ has occurred

$$ P(Y|X) = \frac{P(Y\cap X)}{P(X)} $$

<aside> 📌 SUMMARY: Bayes filter is what many filter applications are built on. The two terms in the filter are a result of movement (Total probability) and measurement (Histogram filter)

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Date: January 16, 2024

Topic: Theorem of Total Probability

Recall

The probability of a future time-step is dependent on the summation of the probabilities of the previous locations

Notes

Law of Total Probability (Movement)

The total probability of an outcome can be realized via several distinct events

$P(X_i^t) = \sum P(X_j^{t-1})\cdot P(X_i|X_j)$

• The current probability is dependent on the summation of the probability of $X_j$ at a previous time-step multiplied by the probability that $X_j$ moves to $X_i$.
• Hence these probabilities account for overshoot, undershoot and desired movements.

<aside> 💡 Total probability is applied for movement, and is shown in the Bayes filter above (within integral)

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Law of total probably is the sum of the probability of $A$ occurring given that $B$ occurs, multiplied by the probability that $B$ occurs.

<aside> 📌 SUMMARY: Total probability is used for estimating future positions based on prior movements

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Date: January 20, 2024

Topic: Bayes Rule