A simple example

Belief: A probability distribution that models uncertainty over possible states X of the world

Suppose I roll a fair die, but don’t tell you the result X

Q1: What should your belief be about X?

A1: Prior: p(X=x)=⅙ for all x in {1, …, 6}

Q2: Suppose that now I look at X, and tell you that its value is even. What should your belief be now?

A2: Posterior (conditional): p(X=x|X in {2,4,6}) = $\frac{p (X = x)}{p ({2, 4, 6})} = 1 / 3$

Key point: In this example, it is not the world that is changing, but rather our information about the world!

Remember: Beliefs model our state of knowledge of the world

A blue dice cup tipped over next to two dice

1

Try it: update a belief

I roll a hidden die. Ask questions and watch your belief change. The die never changes; only what you know about it does.

Ask about the roll

2

Example: measure weight with imperfect scales

Day 0Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8Day 9Day 10

Initialization!

Question: What are Day 2’s Estimate & Prediction for Day 3?

Question: What are Day 4’s Estimate & Prediction for Day 5?

Day	Measurement	Prediction	Estimate	Truth
0	-	-	-160.0	160.0
1	158.0	161.0	159.8	161.0
2	164.2	160.8	?¹162.16	162.0
3	160.3	?²163.16	162.02	163.0
4	159.9	163.02	?161.77	164.0
5	162.1	?162.77	162.50	165.0
6	164.6	163.50	163.94	166.0
7	169.6	164.94	166.80	167.0
8	167.4	167.80	167.64	168.0
9	166.4	168.64	167.75	169.0
10	171.0	168.75	169.65	170.0

Predict:

prediction = estimate + gain_rate * time_step

Sensor reading:

measurement

Update:

estimate = prediction + 410(measurement − prediction)

Giving:

gain_rate = 1.0 lb/daytime_step = 1.0 day

3

Try it: tune the filter

Same predict and update loop. Change how much you trust the scale and how fast you think the weight grows.

Filter settings

Update gain Assumed gain_rate

Measurements

4

2D Example

Tracking the constant velocity of an aircraft

A radar dish tracking an aircraft at two times, t1 and t2, along a straight line

\dot{x} = v = \frac{d x}{d t}

n	1	2	3	4	5	6	7	8	9	10
$z_{n}$	30171	30353	30756	30799	31018	31278	31276	31379	31748	32175
${\hat{x}}_{n, n}$	30194.2	30383.64	30612.73	30818.93	31025.7	31242.3	31418.8	31566.3	31739.4	31964.1
${\hat{\dot{x}}}_{n, n}$	39.42	38.65	42.2	41.7	41.55	42.44	38.9	34.2	34.4	39.67
${\hat{x}}_{n + 1, n}$	30391.3	30576.9	30823.9	31027.6	31233.4	31454.5	31613.15	31737.24	31911.4	32162.45
${\hat{\dot{x}}}_{n + 1, n}$	39.42	38.65	42.2	41.7	41.55	42.44	38.9	34.2	34.4	39.67

?

Δ t = 5 s

{\hat{x}}_{0, 0} = 30,000 m

Predict:

{\hat{x}}_{n, n - 1} = {\hat{x}}_{n - 1, n - 1} + Δ t {\hat{\dot{x}}}_{n - 1, n - 1}

{\hat{\dot{x}}}_{n, n - 1} = {\hat{\dot{x}}}_{n - 1, n - 1} = 40 m/s

State Update Equation

α = 0.2

β = 0.1

{\hat{x}}_{n, n} = {\hat{x}}_{n, n - 1} + α (z_{n} - {\hat{x}}_{n, n - 1})

{\hat{\dot{x}}}_{n, n} = {\hat{\dot{x}}}_{n, n - 1} + β (\frac{z_{n} - {\hat{x}}_{n, n - 1}}{Δ t})

z_{n}

Measurement at time n

{\hat{x}}_{n, n}

Estimation of x at time n

{\hat{x}}_{n + 1, n}

Prediction of x at time n

{\hat{x}}_{1, 0} = 30,200 m

{\hat{\dot{x}}}_{1, 0} = 40 m/s

5

Try it: tune α and β

α sets how far a residual moves the position estimate; β sets how much it changes the velocity.

n	1	2	3	4	5	6	7	8	9	10
$z_{n}$
${\hat{x}}_{n, n}$
${\hat{\dot{x}}}_{n, n}$
${\hat{x}}_{n + 1, n}$
${\hat{\dot{x}}}_{n + 1, n}$

α (position) β (velocity)

Random flights speed up after n = 5

6