Phillip Stanley-Marbell

Foundations of Embedded Systems

Department of Engineering, University of Cambridge

http://physcomp.eng.cam.ac.uk

Topic 03: A Bayesian View of Measurements

(~30 minutes)

Version 0.2020

(Video)

25

Intended Learning Outcomes for This Topic

2

➌ Formulate measurements in an embedded system in terms of probability theory

By the end of this topic, you should be able to:

➋ Describe measurements in terms of events, probabilities, and distributions

➍ Propose ways to improve embedded systems using ideas from probability theory

➊ Recall the basic terminology for describing events and their probabilities

25

Three Key Components of Measurements

3

The measurement

instrument or sensor

The phenomenon being

measured: the measurand

The environment

(e.g., ambient vibrations, temperature, etc.)

The measurement is what

comes out of the

measurement instrument

Environment

➌

➊

➋

Measurement

Instrument

➌

➋

Recall from

Topic 02

25

A Bayesian View of Measurements

4

Our Objective (in what follows)

▶︎

Use information from (noisy) measurements and rules of probability

to get a better estimate of the measurand

Tools We Will Need…

▶︎

Will need to deﬁne the components of measurement processes (previous slide)

in a way that permits applying the standard tools of probability theory

(random variables, events, conditional probability, Bayes’s rule, etc.)

Our Approach

▶︎

Quantitatively combine ➊ measurements together with ➋ assumptions of the

probability distribution of noise in the measurement process (Topic 02)

and ➌ the assumptions of the probability distribution measurand

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A Bayesian View of Measurements: Terminology

5

Sample Quantize

θ

x

Signal / Measurand

Measurement

analog digital

Sensor

25

A Bayesian View of Measurements: Terminology

6

Sample Quantize

θ

x

Signal / Measurand

Measurement

analog digital

Parameter

(the thing we want to

estimate: the measurand)

Variable

(the evidence we are using to make

the estimate: the measurement)

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7

Terminology: Random variables, events, etc.

25

Probability Space:

8

(Ω, ℱ,℘)

The probability space deﬁnes the set Ω of simple events, sets of all possible

occurrences (subsets of Ω), and probabilities of occurrences

Sample space, :

Ω

▶ A set, consisting of all the basic events we are concerned with

Example: for a coin toss, Ω = {H, T}

Sigma algebra, :

ℱ

▶ A set, consisting of all possible occurrences we can build out of Ω

Example: for a coin toss, ℱ = {∅, Ω, {H}, {T}}

We will use the term event to refer to members of the set ℱ

Probability measure, :

℘

▶ A mapping from ℱ to [0, 1]

Assigns a probability, (a real number p ∈ [0, 1]) to event

One example of a probability measure is a probability mass function

25

Events and Random Variables

9

So far, we have talked about probability in terms of events and sets

Random Variables

▶ Rather than work with sets, we can map events to real values

Random Variables Deﬁne Events

▶ Event: The occurrence of a random variable taking on a speciﬁc value

▶ Deﬁnition: A random variable, X, on a sample space Ω, is a real-valued function on Ω; i.e., X : Ω→ℝ

▶ Let X be a random variable deﬁning the number of heads resulting from a coin toss

Example (coin toss)

▶ Sample space, Ω = {H, T}, sigma algebra of subsets of Ω, ℱ = {∅, Ω, {H}, {T}}

▶ Random variable, X: {∅, Ω, {H}, {T}} → [0, 1]

▶ Events: {X = 0}, {X = 1}

In general, an event deﬁned on a random variable X is of the form {s ∈ Ω | X(s) = x }

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Notation: Random Variables versus Instance Values

10

We will represent random variables with uppercase letters, late in alphabet

Random variables, instance values, and events:

Examples: X, Y, Z

I will sometimes use the abbreviation “rvar” for “random variable” (in slides and in speech)

▶ Events correspond to an rvar, say, X, (uppercase) taking on a speciﬁc instance value, say, x (lowercase)

▶ Probability of the event of an rvar X taking on the speciﬁc value x is written as Pr(X = x) or f

X

(x)

▶ Example (coin toss): Let X be an rvar representing number of heads; Pr(X = 0) = f

X

(0) = ½ (for a fair coin)

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Random Variables: Intuition

11

So far, we’ve presented a lot of notation; can we gain more intuition?

Imagine a phenomenon that can be represented with real values

Each time X takes on a speciﬁc value is an event

For die rolling example, with rvars X and Y: Pr(X = 1) = Pr(Y = 3), Pr(X = 4) = Pr(Y = 9), and so on

Example: the result of rolling a die

▶ Let X and Y be functions mapping the result of rolling die to a number

▶ If X = die result, then Ω→{1, 2, 3, 4, 5, 6}. Similarly, if Y = 2·(die result)+1, then Ω→{3, 5, 7, 9, 11, 13}

▶ X and Y are two different functions (random variables) deﬁned on the same set of events

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12

Terminology: Bayesian Statistics

25

Likelihoods, Priors, and Posteriors

13

Let and be events

✓

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x

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We deﬁne as the posterior probability of given

(or alternatively, as the likelihood of the event )

Pr(✓|x)

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x

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✓

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x

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x

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We deﬁne and as the prior probability of the event and the prior

probability of the event , respectively.

Pr(✓)

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Pr(x)

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✓

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To simplify notation, we will refer to the event of the random variable X taking on

instance value x, i.e., X = x, simply with x

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14

Putting It All Together: Random variables, events,

Bayesian inference, and measurements

25

Priors and Posterior Probability Distributions: Intuition

15

Prior Distributions

Posterior Distributions

▶︎ These distributions are deﬁned before we even perform a measurement

▶︎ These distributions are deﬁned after we perform a measurement

25

Bayes’s Rule: From Measurement to Measurand

1 of 2

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Pr(Hypothesis|D at a evidence) =

Pr(Data evidence|Hypothesis)Pr(Hypot he si s)

Pr(Data evidence)

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Bayes’s Rule: From Measurement to Measurand

2 of 2

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Pr(✓|x)=

Pr(x|✓)Pr(✓)

Pr(x)

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Marginals (essentially a normalization constant)

Posterior

Think of this as the distribution

of the signal given some set of

measurements, x = x

1

, …, x

n

Prior

Think of this as the

distribution of the signal

determined by physics

Likelihood

Think of this as the distribution of the measurement

determined by measurement process

Pr(Hypothesis|D at a evidence) =

Pr(Data evidence|Hypothesis)Pr(Hypot he si s)

Pr(Data evidence)

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Bayes’s Rule: From Measurement to Measurand

2 of 2

18

Posteri or =

(likelihood)(prior)

(marginal likel ih ood)

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Pr(✓|x)=

Pr(x|✓)Pr(✓)

Pr(x)

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Pr(Hypothesis|D at a evidence) =

Pr(Data evidence|Hypothesis)Pr(Hypot he si s)

Pr(Data evidence)

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25

Methods To Construct Prior,

19

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Signal A

dimensions

(e.g., LMT

-2

)

Signal N

dimensions

(e.g., L

2

)

➊

Dimensional Function Synthesis

Compile-time analysis

{ }

A ∝√(B/C),

A ∝√(D/E),

…

Run-time calibration

signal A, signal B, …, signal N

➋

Multi-modal sensor data values

A = 2𝜋√(B/C)

[LSM2018] J. Lim and P. Stanley-Marbell “Newton: A Language for Describing Physics", arXiv:1811.04626 [cs.PL], 2018.!

[TWSM 2020] V. Tsoutsouras, S. Willis, and P. Stanley-Marbell. Deriving Equations from Sensor Data using Dimensional Function Synthesis, to appear, Communications of the ACM, 2020.

More when we get to Topic 13 (How Physics Constrains Measurands)

25

20

Putting It All Together: A Brief End-to-End Example

25

End-to-End Example

1 of 2

21

Let be a measurand which is itself a function of another quantity

✓

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Measurand Prior:

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Let W be a random variable denoting noise in our measurement of

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Noise Model: If we assume noise is Gaussian with with mean µ and variance σ

2

Pr(W = w)=

1

p

2⇡

2

e



(wµ)

2

2

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(1)

<latexit sha1_base64="lAge9P5lp7nbya8F7ZDqYCbsXXQ=">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</latexit>

Adapted from an example in the article “Bayesian Inference in Applied Mathematics”, in Encyclopedia of Applied Mathematics.

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End-to-End Example

2 of 2

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Measurement Model:

If we assume the noise is additive on top of the measurand, we can substitute

measurand model for the mean in the noise model to get out measurement, X

Pr(X = x)=

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<latexit sha1_base64="SzNC9isIlB9qsJisFM8x/YQpcnw=">AAACqXicdVFda9swFJW9rzb7aLY97kUsjDmUBNmsa/JQKNvLXgYpLB8sSoOsyI6oLHvS9Wgw/nH7CXvcv5ncJHQd2wXB4dxz77m6Ny6UtEDIL8+/d//Bw0cHh63HT54+O2o/fzGxeWm4GPNc5WYWMyuU1GIMEpSYFUawLFZiGl99bPLT78JYmesvsCnEImOplonkDBy1bP+gsUilrpiSqa5bh3RkgtnZdRefYZoYxquwrqj9ZqCKaCGplWnGLqO6FpdVD+8kwTXuYapEAsFtTcqyjNX4+E5iy1JYC2AB6fZuhdTIdA3dpu2WwbDnus7Pue+tMa5bVOjVfuRlu0P60QkZng4w6Z+QcPgucoCQwfA9waEDTXTQLkbL9k+6ynmZCQ1cMWvnISlgUTEDkivhmpdWFIxfsVTMHdQsE3ZR3ay6xm8cs8JJbtzTgG/YPysqllm7yWKnzBis7d+5hvxXbl5CMlhUUhclCM23RkmpMOS4uRteSSM4qI0DjBvpZsV8zdxSwV3X7WD/Ufx/MIn6ocMXUef8w24bB+gVeo0CFKJTdI4+oREaI+699T57E2/qH/sX/sz/upX63q7mJboTPv8NZA7Qyg==</latexit>

Adapted from an example in the article “Bayesian Inference in Applied Mathematics”, in Encyclopedia of Applied Mathematics.

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