07. Calculate Localization Posterior

To continue developing our intuition for this filter and prepare for later coding exercises, let's walk through calculations for determining posterior probabilities at several pseudo positions x, for a single time step. We will start with a time step after the filter has already been initialized and run a few times. We will cover initialization of the filter in an upcoming concept.

pseudo_position (x) P(location) P(observation∣location) Raw P(location∣observation) Normalized P(location∣observation)
1 1.67E-02 0.00E+00 0.00E+00 0.00E+00
2 3.86E-02 6.99E-03 ? 2.59E-02
3 4.90E-02 8.52E-02 4.18E-03 4.01E-01
4 3.86E-02 ? 5.42E-03 5.21E-01
5 1.69E-02 3.13E-02 5.31E-04 5.10E-02
6 6.51E-03 9.46E-04 6.16E-06 ?
7 ? 3.87E-06 6.55E-08 6.29E-06
8 3.86E-02 0.00E+00 0.00E+00 0.00E+00

Normalized P(location_observation) vs. Raw P(location|observation): The Raw P(location|observation) is the result prior to dividing by the total probability of P(observation), the P(b) term (denominator) of the generalized Bayes`rule. The normalized P(location|observation) is the result of after dividing by P(observation).

Remember the general form for Bayes' Rule:

P(a|b) = \frac{P(b|a) \, P(a)}{P(b)}

With respect to localization, these terms are:

  1. P(location|observation) : This is P(a|b), the normalized probability of a position given an observation (posterior)
  • P(observation|location) : This is P(b|a), the probability of an observation given a position (likelihood)
  • P(location) : This is P(a), the prior probability of a position
  • P(observation) : This is P(b), the total probability of an observation

P(observation∣location)

QUESTION:

What is P(observation∣location) for x = 4? Write the answer in scientific notation with an accuracy of two decimal places, for example 3.14E-15

SOLUTION:

NOTE: The solutions are expressed in RegEx pattern. Udacity uses these patterns to check the given answer

P(Posterior)

QUESTION:

What is the raw posterior probability P(location|observation) for x = 2? Be sure to put the answer in scientific notation with an accuracy of two decimal places, for example 3.14E-15.

SOLUTION:

NOTE: The solutions are expressed in RegEx pattern. Udacity uses these patterns to check the given answer

Normalized Posterior Probability

QUESTION:

What is the normalized posterior probability for x = 6? Write the answer in scientific notation with an accuracy of two decimal places, for example 3.14E-15.

SOLUTION:

NOTE: The solutions are expressed in RegEx pattern. Udacity uses these patterns to check the given answer

P(Position)

QUESTION:

What is the position probability for x = 7? Write your answer in scientific notation with an accuracy of two decimal places, for example 3.14E-15

SOLUTION:

NOTE: The solutions are expressed in RegEx pattern. Udacity uses these patterns to check the given answer

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