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
Formulate measurements in an embedded system in terms of probability theory
By the end of this topic, you should be able to:
Describe measurements in terms of events, probabilities, and distributions
Propose ways to improve embedded systems using ideas from probability theory
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
Measurement
Instrument
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 measurements together with assumptions of the
probability distribution of noise in the measurement process (Topic 02)
and the assumptions of the probability distribution measurand
25
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
25
7
Terminology: Random variables, events, etc.
25
Probability Space:
8
(Ω, ,)
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, :
A set, consisting of all possible occurrences we can build out of Ω
Example: for a coin toss, = {, Ω, {H}, {T}}
We will use the term event to refer to members of the set
Probability measure, :
A mapping from 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
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, = {, 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 }
25
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)
25
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
25
12
Terminology: Bayesian Statistics
25
Likelihoods, Priors, and Posteriors
13
Let and be events
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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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<latexit sha1_base64="ldTChBJmjtlLCtT4XaCHdmQrWdk=">AAAB7nicdZDLSgMxFIYz9VbrrerSTbAVXJXMYG27K7pxWcFeoB1KJs20oZkLyRmhDH0JF25U3Po6Ln0bM20FFf0h8PGfc8g5vxdLoYGQDyu3tr6xuZXfLuzs7u0fFA+POjpKFONtFslI9TyquRQhb4MAyXux4jTwJO960+us3r3nSosovINZzN2AjkPhC0bBWL3yACYcaHlYLJGKUyWNWh2TSpXYjQvHACH1xiXBtoFMJbRSa1h8H4wilgQ8BCap1n2bxOCmVIFgks8Lg0TzmLIpHfO+wZAGXLvpYt85PjPOCPuRMi8EvHC/T6Q00HoWeKYzoDDRv2uZ+Vetn4Bfd1MRxgnwkC0/8hOJIcLZ8XgkFGcgZwYoU8LsitmEKsrARGQy+DoU/w8dp2IbvnVKzatVGnl0gk7RObJRDTXRDWqhNmJIogf0hJ6t2Hq0XqzXZWvOWs0cox+y3j4BRRuPvQ==</latexit>
<latexit sha1_base64="ldTChBJmjtlLCtT4XaCHdmQrWdk=">AAAB7nicdZDLSgMxFIYz9VbrrerSTbAVXJXMYG27K7pxWcFeoB1KJs20oZkLyRmhDH0JF25U3Po6Ln0bM20FFf0h8PGfc8g5vxdLoYGQDyu3tr6xuZXfLuzs7u0fFA+POjpKFONtFslI9TyquRQhb4MAyXux4jTwJO960+us3r3nSosovINZzN2AjkPhC0bBWL3yACYcaHlYLJGKUyWNWh2TSpXYjQvHACH1xiXBtoFMJbRSa1h8H4wilgQ8BCap1n2bxOCmVIFgks8Lg0TzmLIpHfO+wZAGXLvpYt85PjPOCPuRMi8EvHC/T6Q00HoWeKYzoDDRv2uZ+Vetn4Bfd1MRxgnwkC0/8hOJIcLZ8XgkFGcgZwYoU8LsitmEKsrARGQy+DoU/w8dp2IbvnVKzatVGnl0gk7RObJRDTXRDWqhNmJIogf0hJ6t2Hq0XqzXZWvOWs0cox+y3j4BRRuPvQ==</latexit>
<latexit sha1_base64="ldTChBJmjtlLCtT4XaCHdmQrWdk=">AAAB7nicdZDLSgMxFIYz9VbrrerSTbAVXJXMYG27K7pxWcFeoB1KJs20oZkLyRmhDH0JF25U3Po6Ln0bM20FFf0h8PGfc8g5vxdLoYGQDyu3tr6xuZXfLuzs7u0fFA+POjpKFONtFslI9TyquRQhb4MAyXux4jTwJO960+us3r3nSosovINZzN2AjkPhC0bBWL3yACYcaHlYLJGKUyWNWh2TSpXYjQvHACH1xiXBtoFMJbRSa1h8H4wilgQ8BCap1n2bxOCmVIFgks8Lg0TzmLIpHfO+wZAGXLvpYt85PjPOCPuRMi8EvHC/T6Q00HoWeKYzoDDRv2uZ+Vetn4Bfd1MRxgnwkC0/8hOJIcLZ8XgkFGcgZwYoU8LsitmEKsrARGQy+DoU/w8dp2IbvnVKzatVGnl0gk7RObJRDTXRDWqhNmJIogf0hJ6t2Hq0XqzXZWvOWs0cox+y3j4BRRuPvQ==</latexit>
x
<latexit sha1_base64="6zfLjapaAqmlCwdMvpksV/zSof0=">AAAB6XicdZDLTgIxFIbP4A3xhrp00wgmrkiHiMCO6MYlRrkkMCGd0oGGziVtx0gmPIILN2rc+kQufRs7gIka/ZMmX/5zTnrO70aCK43xh5VZWV1b38hu5ra2d3b38vsHbRXGkrIWDUUouy5RTPCAtTTXgnUjyYjvCtZxJ5dpvXPHpOJhcKunEXN8Mgq4xynRxrop3hcH+QIulSu4Xq0hXKpgu35WNoBxrX6OkW0gVQGWag7y7/1hSGOfBZoKolTPxpF2EiI1p4LNcv1YsYjQCRmxnsGA+Ew5yXzVGToxzhB5oTQv0Gjufp9IiK/U1HdNp0/0WP2upeZftV6svZqT8CCKNQvo4iMvFkiHKL0bDblkVIupAUIlN7siOiaSUG3SMRl8HYr+h3a5ZBu+LhcaF8s0snAEx3AKNlShAVfQhBZQGMEDPMGzNbEerRfrddGasZYzh/BD1tsnhpqNkQ==</latexit>
<latexit sha1_base64="6zfLjapaAqmlCwdMvpksV/zSof0=">AAAB6XicdZDLTgIxFIbP4A3xhrp00wgmrkiHiMCO6MYlRrkkMCGd0oGGziVtx0gmPIILN2rc+kQufRs7gIka/ZMmX/5zTnrO70aCK43xh5VZWV1b38hu5ra2d3b38vsHbRXGkrIWDUUouy5RTPCAtTTXgnUjyYjvCtZxJ5dpvXPHpOJhcKunEXN8Mgq4xynRxrop3hcH+QIulSu4Xq0hXKpgu35WNoBxrX6OkW0gVQGWag7y7/1hSGOfBZoKolTPxpF2EiI1p4LNcv1YsYjQCRmxnsGA+Ew5yXzVGToxzhB5oTQv0Gjufp9IiK/U1HdNp0/0WP2upeZftV6svZqT8CCKNQvo4iMvFkiHKL0bDblkVIupAUIlN7siOiaSUG3SMRl8HYr+h3a5ZBu+LhcaF8s0snAEx3AKNlShAVfQhBZQGMEDPMGzNbEerRfrddGasZYzh/BD1tsnhpqNkQ==</latexit>
<latexit sha1_base64="6zfLjapaAqmlCwdMvpksV/zSof0=">AAAB6XicdZDLTgIxFIbP4A3xhrp00wgmrkiHiMCO6MYlRrkkMCGd0oGGziVtx0gmPIILN2rc+kQufRs7gIka/ZMmX/5zTnrO70aCK43xh5VZWV1b38hu5ra2d3b38vsHbRXGkrIWDUUouy5RTPCAtTTXgnUjyYjvCtZxJ5dpvXPHpOJhcKunEXN8Mgq4xynRxrop3hcH+QIulSu4Xq0hXKpgu35WNoBxrX6OkW0gVQGWag7y7/1hSGOfBZoKolTPxpF2EiI1p4LNcv1YsYjQCRmxnsGA+Ew5yXzVGToxzhB5oTQv0Gjufp9IiK/U1HdNp0/0WP2upeZftV6svZqT8CCKNQvo4iMvFkiHKL0bDblkVIupAUIlN7siOiaSUG3SMRl8HYr+h3a5ZBu+LhcaF8s0snAEx3AKNlShAVfQhBZQGMEDPMGzNbEerRfrddGasZYzh/BD1tsnhpqNkQ==</latexit>
<latexit sha1_base64="6zfLjapaAqmlCwdMvpksV/zSof0=">AAAB6XicdZDLTgIxFIbP4A3xhrp00wgmrkiHiMCO6MYlRrkkMCGd0oGGziVtx0gmPIILN2rc+kQufRs7gIka/ZMmX/5zTnrO70aCK43xh5VZWV1b38hu5ra2d3b38vsHbRXGkrIWDUUouy5RTPCAtTTXgnUjyYjvCtZxJ5dpvXPHpOJhcKunEXN8Mgq4xynRxrop3hcH+QIulSu4Xq0hXKpgu35WNoBxrX6OkW0gVQGWag7y7/1hSGOfBZoKolTPxpF2EiI1p4LNcv1YsYjQCRmxnsGA+Ew5yXzVGToxzhB5oTQv0Gjufp9IiK/U1HdNp0/0WP2upeZftV6svZqT8CCKNQvo4iMvFkiHKL0bDblkVIupAUIlN7siOiaSUG3SMRl8HYr+h3a5ZBu+LhcaF8s0snAEx3AKNlShAVfQhBZQGMEDPMGzNbEerRfrddGasZYzh/BD1tsnhpqNkQ==</latexit>
x
<latexit sha1_base64="6zfLjapaAqmlCwdMvpksV/zSof0=">AAAB6XicdZDLTgIxFIbP4A3xhrp00wgmrkiHiMCO6MYlRrkkMCGd0oGGziVtx0gmPIILN2rc+kQufRs7gIka/ZMmX/5zTnrO70aCK43xh5VZWV1b38hu5ra2d3b38vsHbRXGkrIWDUUouy5RTPCAtTTXgnUjyYjvCtZxJ5dpvXPHpOJhcKunEXN8Mgq4xynRxrop3hcH+QIulSu4Xq0hXKpgu35WNoBxrX6OkW0gVQGWag7y7/1hSGOfBZoKolTPxpF2EiI1p4LNcv1YsYjQCRmxnsGA+Ew5yXzVGToxzhB5oTQv0Gjufp9IiK/U1HdNp0/0WP2upeZftV6svZqT8CCKNQvo4iMvFkiHKL0bDblkVIupAUIlN7siOiaSUG3SMRl8HYr+h3a5ZBu+LhcaF8s0snAEx3AKNlShAVfQhBZQGMEDPMGzNbEerRfrddGasZYzh/BD1tsnhpqNkQ==</latexit>
<latexit sha1_base64="6zfLjapaAqmlCwdMvpksV/zSof0=">AAAB6XicdZDLTgIxFIbP4A3xhrp00wgmrkiHiMCO6MYlRrkkMCGd0oGGziVtx0gmPIILN2rc+kQufRs7gIka/ZMmX/5zTnrO70aCK43xh5VZWV1b38hu5ra2d3b38vsHbRXGkrIWDUUouy5RTPCAtTTXgnUjyYjvCtZxJ5dpvXPHpOJhcKunEXN8Mgq4xynRxrop3hcH+QIulSu4Xq0hXKpgu35WNoBxrX6OkW0gVQGWag7y7/1hSGOfBZoKolTPxpF2EiI1p4LNcv1YsYjQCRmxnsGA+Ew5yXzVGToxzhB5oTQv0Gjufp9IiK/U1HdNp0/0WP2upeZftV6svZqT8CCKNQvo4iMvFkiHKL0bDblkVIupAUIlN7siOiaSUG3SMRl8HYr+h3a5ZBu+LhcaF8s0snAEx3AKNlShAVfQhBZQGMEDPMGzNbEerRfrddGasZYzh/BD1tsnhpqNkQ==</latexit>
<latexit sha1_base64="6zfLjapaAqmlCwdMvpksV/zSof0=">AAAB6XicdZDLTgIxFIbP4A3xhrp00wgmrkiHiMCO6MYlRrkkMCGd0oGGziVtx0gmPIILN2rc+kQufRs7gIka/ZMmX/5zTnrO70aCK43xh5VZWV1b38hu5ra2d3b38vsHbRXGkrIWDUUouy5RTPCAtTTXgnUjyYjvCtZxJ5dpvXPHpOJhcKunEXN8Mgq4xynRxrop3hcH+QIulSu4Xq0hXKpgu35WNoBxrX6OkW0gVQGWag7y7/1hSGOfBZoKolTPxpF2EiI1p4LNcv1YsYjQCRmxnsGA+Ew5yXzVGToxzhB5oTQv0Gjufp9IiK/U1HdNp0/0WP2upeZftV6svZqT8CCKNQvo4iMvFkiHKL0bDblkVIupAUIlN7siOiaSUG3SMRl8HYr+h3a5ZBu+LhcaF8s0snAEx3AKNlShAVfQhBZQGMEDPMGzNbEerRfrddGasZYzh/BD1tsnhpqNkQ==</latexit>
<latexit sha1_base64="6zfLjapaAqmlCwdMvpksV/zSof0=">AAAB6XicdZDLTgIxFIbP4A3xhrp00wgmrkiHiMCO6MYlRrkkMCGd0oGGziVtx0gmPIILN2rc+kQufRs7gIka/ZMmX/5zTnrO70aCK43xh5VZWV1b38hu5ra2d3b38vsHbRXGkrIWDUUouy5RTPCAtTTXgnUjyYjvCtZxJ5dpvXPHpOJhcKunEXN8Mgq4xynRxrop3hcH+QIulSu4Xq0hXKpgu35WNoBxrX6OkW0gVQGWag7y7/1hSGOfBZoKolTPxpF2EiI1p4LNcv1YsYjQCRmxnsGA+Ew5yXzVGToxzhB5oTQv0Gjufp9IiK/U1HdNp0/0WP2upeZftV6svZqT8CCKNQvo4iMvFkiHKL0bDblkVIupAUIlN7siOiaSUG3SMRl8HYr+h3a5ZBu+LhcaF8s0snAEx3AKNlShAVfQhBZQGMEDPMGzNbEerRfrddGasZYzh/BD1tsnhpqNkQ==</latexit>
We define and as the prior probability of the event and the prior
probability of the event , respectively.
Pr()
<latexit sha1_base64="nUh8bp4XV9BHVTOWmouXKSA5aes=">AAAB83icdZDLSgMxFIYzXmu9VV26CbZC3ZTMYG27K7pxWcFeoDOUTJq2oZmLyRmhDH0OF25U3PowLn0b04ugoj8EPv5zDufk92MpNBDyYa2srq1vbGa2sts7u3v7uYPDlo4SxXiTRTJSHZ9qLkXImyBA8k6sOA18ydv++GpWb99zpUUU3sIk5l5Ah6EYCEbBWF7BbaiiCyMO9KzQy+VJySmTWqWKSalM7Nq5Y4CQau2CYNvATHm0VKOXe3f7EUsCHgKTVOuuTWLwUqpAMMmnWTfRPKZsTIe8azCkAddeOj96ik+N08eDSJkXAp673ydSGmg9CXzTGVAY6d+1mflXrZvAoOqlIowT4CFbLBokEkOEZwngvlCcgZwYoEwJcytmI6ooA5OTyeDro/h/aDkl2/CNk69fLtPIoGN0gorIRhVUR9eogZqIoTv0gJ7Qs5VYj9aL9bpoXbGWM0foh6y3TziDkV4=</latexit>
<latexit sha1_base64="nUh8bp4XV9BHVTOWmouXKSA5aes=">AAAB83icdZDLSgMxFIYzXmu9VV26CbZC3ZTMYG27K7pxWcFeoDOUTJq2oZmLyRmhDH0OF25U3PowLn0b04ugoj8EPv5zDufk92MpNBDyYa2srq1vbGa2sts7u3v7uYPDlo4SxXiTRTJSHZ9qLkXImyBA8k6sOA18ydv++GpWb99zpUUU3sIk5l5Ah6EYCEbBWF7BbaiiCyMO9KzQy+VJySmTWqWKSalM7Nq5Y4CQau2CYNvATHm0VKOXe3f7EUsCHgKTVOuuTWLwUqpAMMmnWTfRPKZsTIe8azCkAddeOj96ik+N08eDSJkXAp673ydSGmg9CXzTGVAY6d+1mflXrZvAoOqlIowT4CFbLBokEkOEZwngvlCcgZwYoEwJcytmI6ooA5OTyeDro/h/aDkl2/CNk69fLtPIoGN0gorIRhVUR9eogZqIoTv0gJ7Qs5VYj9aL9bpoXbGWM0foh6y3TziDkV4=</latexit>
<latexit sha1_base64="nUh8bp4XV9BHVTOWmouXKSA5aes=">AAAB83icdZDLSgMxFIYzXmu9VV26CbZC3ZTMYG27K7pxWcFeoDOUTJq2oZmLyRmhDH0OF25U3PowLn0b04ugoj8EPv5zDufk92MpNBDyYa2srq1vbGa2sts7u3v7uYPDlo4SxXiTRTJSHZ9qLkXImyBA8k6sOA18ydv++GpWb99zpUUU3sIk5l5Ah6EYCEbBWF7BbaiiCyMO9KzQy+VJySmTWqWKSalM7Nq5Y4CQau2CYNvATHm0VKOXe3f7EUsCHgKTVOuuTWLwUqpAMMmnWTfRPKZsTIe8azCkAddeOj96ik+N08eDSJkXAp673ydSGmg9CXzTGVAY6d+1mflXrZvAoOqlIowT4CFbLBokEkOEZwngvlCcgZwYoEwJcytmI6ooA5OTyeDro/h/aDkl2/CNk69fLtPIoGN0gorIRhVUR9eogZqIoTv0gJ7Qs5VYj9aL9bpoXbGWM0foh6y3TziDkV4=</latexit>
<latexit sha1_base64="nUh8bp4XV9BHVTOWmouXKSA5aes=">AAAB83icdZDLSgMxFIYzXmu9VV26CbZC3ZTMYG27K7pxWcFeoDOUTJq2oZmLyRmhDH0OF25U3PowLn0b04ugoj8EPv5zDufk92MpNBDyYa2srq1vbGa2sts7u3v7uYPDlo4SxXiTRTJSHZ9qLkXImyBA8k6sOA18ydv++GpWb99zpUUU3sIk5l5Ah6EYCEbBWF7BbaiiCyMO9KzQy+VJySmTWqWKSalM7Nq5Y4CQau2CYNvATHm0VKOXe3f7EUsCHgKTVOuuTWLwUqpAMMmnWTfRPKZsTIe8azCkAddeOj96ik+N08eDSJkXAp673ydSGmg9CXzTGVAY6d+1mflXrZvAoOqlIowT4CFbLBokEkOEZwngvlCcgZwYoEwJcytmI6ooA5OTyeDro/h/aDkl2/CNk69fLtPIoGN0gorIRhVUR9eogZqIoTv0gJ7Qs5VYj9aL9bpoXbGWM0foh6y3TziDkV4=</latexit>
Pr(x)
<latexit sha1_base64="o+AJzGX0lQVrzOU9OhX8TwPclKU=">AAAB7nicdZDLSgMxFIbPeK31VnXpJtgKdVMyg7XtrujGZQV7gXYomTTThmYuJBmxlL6ECzcqbn0dl76NmbaCiv4Q+PjPOeSc34sFVxrjD2tldW19YzOzld3e2d3bzx0ctlSUSMqaNBKR7HhEMcFD1tRcC9aJJSOBJ1jbG1+l9fYdk4pH4a2exMwNyDDkPqdEG6tT6DVk8f6s0M/lcckp41qlinCpjO3auWMA42rtAiPbQKo8LNXo5957g4gmAQs1FUSpro1j7U6J1JwKNsv2EsViQsdkyLoGQxIw5U7n+87QqXEGyI+keaFGc/f7xJQESk0Cz3QGRI/U71pq/lXrJtqvulMexolmIV185CcC6Qilx6MBl4xqMTFAqORmV0RHRBKqTUQmg69D0f/Qckq24RsnX79cppGBYziBIthQgTpcQwOaQEHAAzzBsxVbj9aL9bpoXbGWM0fwQ9bbJ3F4jzI=</latexit>
<latexit sha1_base64="o+AJzGX0lQVrzOU9OhX8TwPclKU=">AAAB7nicdZDLSgMxFIbPeK31VnXpJtgKdVMyg7XtrujGZQV7gXYomTTThmYuJBmxlL6ECzcqbn0dl76NmbaCiv4Q+PjPOeSc34sFVxrjD2tldW19YzOzld3e2d3bzx0ctlSUSMqaNBKR7HhEMcFD1tRcC9aJJSOBJ1jbG1+l9fYdk4pH4a2exMwNyDDkPqdEG6tT6DVk8f6s0M/lcckp41qlinCpjO3auWMA42rtAiPbQKo8LNXo5957g4gmAQs1FUSpro1j7U6J1JwKNsv2EsViQsdkyLoGQxIw5U7n+87QqXEGyI+keaFGc/f7xJQESk0Cz3QGRI/U71pq/lXrJtqvulMexolmIV185CcC6Qilx6MBl4xqMTFAqORmV0RHRBKqTUQmg69D0f/Qckq24RsnX79cppGBYziBIthQgTpcQwOaQEHAAzzBsxVbj9aL9bpoXbGWM0fwQ9bbJ3F4jzI=</latexit>
<latexit sha1_base64="o+AJzGX0lQVrzOU9OhX8TwPclKU=">AAAB7nicdZDLSgMxFIbPeK31VnXpJtgKdVMyg7XtrujGZQV7gXYomTTThmYuJBmxlL6ECzcqbn0dl76NmbaCiv4Q+PjPOeSc34sFVxrjD2tldW19YzOzld3e2d3bzx0ctlSUSMqaNBKR7HhEMcFD1tRcC9aJJSOBJ1jbG1+l9fYdk4pH4a2exMwNyDDkPqdEG6tT6DVk8f6s0M/lcckp41qlinCpjO3auWMA42rtAiPbQKo8LNXo5957g4gmAQs1FUSpro1j7U6J1JwKNsv2EsViQsdkyLoGQxIw5U7n+87QqXEGyI+keaFGc/f7xJQESk0Cz3QGRI/U71pq/lXrJtqvulMexolmIV185CcC6Qilx6MBl4xqMTFAqORmV0RHRBKqTUQmg69D0f/Qckq24RsnX79cppGBYziBIthQgTpcQwOaQEHAAzzBsxVbj9aL9bpoXbGWM0fwQ9bbJ3F4jzI=</latexit>
<latexit sha1_base64="o+AJzGX0lQVrzOU9OhX8TwPclKU=">AAAB7nicdZDLSgMxFIbPeK31VnXpJtgKdVMyg7XtrujGZQV7gXYomTTThmYuJBmxlL6ECzcqbn0dl76NmbaCiv4Q+PjPOeSc34sFVxrjD2tldW19YzOzld3e2d3bzx0ctlSUSMqaNBKR7HhEMcFD1tRcC9aJJSOBJ1jbG1+l9fYdk4pH4a2exMwNyDDkPqdEG6tT6DVk8f6s0M/lcckp41qlinCpjO3auWMA42rtAiPbQKo8LNXo5957g4gmAQs1FUSpro1j7U6J1JwKNsv2EsViQsdkyLoGQxIw5U7n+87QqXEGyI+keaFGc/f7xJQESk0Cz3QGRI/U71pq/lXrJtqvulMexolmIV185CcC6Qilx6MBl4xqMTFAqORmV0RHRBKqTUQmg69D0f/Qckq24RsnX79cppGBYziBIthQgTpcQwOaQEHAAzzBsxVbj9aL9bpoXbGWM0fwQ9bbJ3F4jzI=</latexit>
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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
25
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
16
Pr(Hypothesis|D at a evidence) =
Pr(Data evidence|Hypothesis)Pr(Hypot he si s)
Pr(Data evidence)
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<latexit sha1_base64="K74TVzu3fWbNXK2Bik7ekQnl+Lk=">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</latexit>
25
Bayes’s Rule: From Measurement to Measurand
2 of 2
17
Pr(|x)=
Pr(x|)Pr()
Pr(x)
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<latexit sha1_base64="nJHFsAkS6JOZOt0w8FLOetUMkHE=">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</latexit>
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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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<latexit sha1_base64="K74TVzu3fWbNXK2Bik7ekQnl+Lk=">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</latexit>
25
Bayes’s Rule: From Measurement to Measurand
2 of 2
18
Posteri or =
(likelihood)(prior)
(marginal likel ih ood)
<latexit sha1_base64="ZIcqQoe7Ln6XwjFOENWT2F/fvr8=">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</latexit>
<latexit sha1_base64="ZIcqQoe7Ln6XwjFOENWT2F/fvr8=">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</latexit>
<latexit sha1_base64="ZIcqQoe7Ln6XwjFOENWT2F/fvr8=">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</latexit>
<latexit sha1_base64="ZIcqQoe7Ln6XwjFOENWT2F/fvr8=">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</latexit>
Pr(|x)=
Pr(x|)Pr()
Pr(x)
<latexit sha1_base64="nJHFsAkS6JOZOt0w8FLOetUMkHE=">AAACLXicdVDLSgMxFM3UV31XXboJtkK7KZnio10IRRFcVrBa6JSSSTM2mHmQ3BHL0C9y4bd04cIHbv0NM23FB3ogcDjnXG7ucSMpNBDyZGVmZufmF7KLS8srq2vruY3NSx3GivEmC2WoWi7VXIqAN0GA5K1Iceq7kl+5Nyepf3XLlRZhcAGDiHd8eh0ITzAKRurmTgtOQxUd6HOg2DFJwHclfIQdT1GWpN7dVJ5kSl/x0nDil4aFbi5PypV9UjusYlLeJ3Ztr2IIIdXaAcG2ISnyaIpGNzdyeiGLfR4Ak1Trtk0i6CRUgWCSD5ecWPOIsht6zduGBtTnupOMzx3iXaP0sBcq8wLAY/X7REJ9rQe+a5I+hb7+7aXiX147Bq/aSUQQxcADNlnkxRJDiNPucE8ozkAODKFMCfNXzPrUFAWmYdPB56H4f3JZKduGn1fy9eNpG1m0jXZQEdnoENXRGWqgJmLoHo3QM3qxHqxH69V6m0Qz1nRmC/2A9f4BtWandw==</latexit>
<latexit sha1_base64="nJHFsAkS6JOZOt0w8FLOetUMkHE=">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</latexit>
<latexit sha1_base64="nJHFsAkS6JOZOt0w8FLOetUMkHE=">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</latexit>
<latexit sha1_base64="nJHFsAkS6JOZOt0w8FLOetUMkHE=">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</latexit>
Pr(Hypothesis|D at a evidence) =
Pr(Data evidence|Hypothesis)Pr(Hypot he si s)
Pr(Data evidence)
<latexit sha1_base64="K74TVzu3fWbNXK2Bik7ekQnl+Lk=">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</latexit>
<latexit sha1_base64="K74TVzu3fWbNXK2Bik7ekQnl+Lk=">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</latexit>
<latexit sha1_base64="K74TVzu3fWbNXK2Bik7ekQnl+Lk=">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</latexit>
<latexit sha1_base64="K74TVzu3fWbNXK2Bik7ekQnl+Lk=">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</latexit>
25
Methods To Construct Prior,
19
f
(| x )=
f
X
(x|)f
()
P
2D()
f
X
(x|)f
()
. (1)
<latexit sha1_base64="nzT5PcWN5SsNgq66hQnqLV6joNQ=">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</latexit>
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Signal A
dimensions
(e.g., LMT
-2
)
Signal N
dimensions
(e.g., L
2
)
Dimensional Function Synthesis
Compile-time analysis
{ }
A (B/C),
A (D/E),
Run-time calibration
signal A, signal B, …, signal N
signal A, signal B, …, signal N
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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( t)=
1
+
✓(0) 1
e
t
(1)
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Measurand Prior:
(t )=
1
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✓(0) 1
e
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<latexit sha1_base64="yCQbGEgUHq1YuF1tqgf32eH4tpU=">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</latexit>
Let W be a random variable denoting noise in our measurement of
<latexit sha1_base64="ldTChBJmjtlLCtT4XaCHdmQrWdk=">AAAB7nicdZDLSgMxFIYz9VbrrerSTbAVXJXMYG27K7pxWcFeoB1KJs20oZkLyRmhDH0JF25U3Po6Ln0bM20FFf0h8PGfc8g5vxdLoYGQDyu3tr6xuZXfLuzs7u0fFA+POjpKFONtFslI9TyquRQhb4MAyXux4jTwJO960+us3r3nSosovINZzN2AjkPhC0bBWL3yACYcaHlYLJGKUyWNWh2TSpXYjQvHACH1xiXBtoFMJbRSa1h8H4wilgQ8BCap1n2bxOCmVIFgks8Lg0TzmLIpHfO+wZAGXLvpYt85PjPOCPuRMi8EvHC/T6Q00HoWeKYzoDDRv2uZ+Vetn4Bfd1MRxgnwkC0/8hOJIcLZ8XgkFGcgZwYoU8LsitmEKsrARGQy+DoU/w8dp2IbvnVKzatVGnl0gk7RObJRDTXRDWqhNmJIogf0hJ6t2Hq0XqzXZWvOWs0cox+y3j4BRRuPvQ==</latexit>
<latexit sha1_base64="ldTChBJmjtlLCtT4XaCHdmQrWdk=">AAAB7nicdZDLSgMxFIYz9VbrrerSTbAVXJXMYG27K7pxWcFeoB1KJs20oZkLyRmhDH0JF25U3Po6Ln0bM20FFf0h8PGfc8g5vxdLoYGQDyu3tr6xuZXfLuzs7u0fFA+POjpKFONtFslI9TyquRQhb4MAyXux4jTwJO960+us3r3nSosovINZzN2AjkPhC0bBWL3yACYcaHlYLJGKUyWNWh2TSpXYjQvHACH1xiXBtoFMJbRSa1h8H4wilgQ8BCap1n2bxOCmVIFgks8Lg0TzmLIpHfO+wZAGXLvpYt85PjPOCPuRMi8EvHC/T6Q00HoWeKYzoDDRv2uZ+Vetn4Bfd1MRxgnwkC0/8hOJIcLZ8XgkFGcgZwYoU8LsitmEKsrARGQy+DoU/w8dp2IbvnVKzatVGnl0gk7RObJRDTXRDWqhNmJIogf0hJ6t2Hq0XqzXZWvOWs0cox+y3j4BRRuPvQ==</latexit>
<latexit sha1_base64="ldTChBJmjtlLCtT4XaCHdmQrWdk=">AAAB7nicdZDLSgMxFIYz9VbrrerSTbAVXJXMYG27K7pxWcFeoB1KJs20oZkLyRmhDH0JF25U3Po6Ln0bM20FFf0h8PGfc8g5vxdLoYGQDyu3tr6xuZXfLuzs7u0fFA+POjpKFONtFslI9TyquRQhb4MAyXux4jTwJO960+us3r3nSosovINZzN2AjkPhC0bBWL3yACYcaHlYLJGKUyWNWh2TSpXYjQvHACH1xiXBtoFMJbRSa1h8H4wilgQ8BCap1n2bxOCmVIFgks8Lg0TzmLIpHfO+wZAGXLvpYt85PjPOCPuRMi8EvHC/T6Q00HoWeKYzoDDRv2uZ+Vetn4Bfd1MRxgnwkC0/8hOJIcLZ8XgkFGcgZwYoU8LsitmEKsrARGQy+DoU/w8dp2IbvnVKzatVGnl0gk7RObJRDTXRDWqhNmJIogf0hJ6t2Hq0XqzXZWvOWs0cox+y3j4BRRuPvQ==</latexit>
<latexit sha1_base64="ldTChBJmjtlLCtT4XaCHdmQrWdk=">AAAB7nicdZDLSgMxFIYz9VbrrerSTbAVXJXMYG27K7pxWcFeoB1KJs20oZkLyRmhDH0JF25U3Po6Ln0bM20FFf0h8PGfc8g5vxdLoYGQDyu3tr6xuZXfLuzs7u0fFA+POjpKFONtFslI9TyquRQhb4MAyXux4jTwJO960+us3r3nSosovINZzN2AjkPhC0bBWL3yACYcaHlYLJGKUyWNWh2TSpXYjQvHACH1xiXBtoFMJbRSa1h8H4wilgQ8BCap1n2bxOCmVIFgks8Lg0TzmLIpHfO+wZAGXLvpYt85PjPOCPuRMi8EvHC/T6Q00HoWeKYzoDDRv2uZ+Vetn4Bfd1MRxgnwkC0/8hOJIcLZ8XgkFGcgZwYoU8LsitmEKsrARGQy+DoU/w8dp2IbvnVKzatVGnl0gk7RObJRDTXRDWqhNmJIogf0hJ6t2Hq0XqzXZWvOWs0cox+y3j4BRRuPvQ==</latexit>
Noise Model: If we assume noise is Gaussian with with mean µ and variance σ
2
Pr(W = w)=
1
p
2⇡
2
e
(wµ)
2
2
2
(1)
<latexit sha1_base64="lAge9P5lp7nbya8F7ZDqYCbsXXQ=">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</latexit>
<latexit sha1_base64="lAge9P5lp7nbya8F7ZDqYCbsXXQ=">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</latexit>
<latexit sha1_base64="lAge9P5lp7nbya8F7ZDqYCbsXXQ=">AAACUnicdVLBThsxFPQG2kJK25Qee7GIKoUDkXdVSnJAQu2FY5AIQYpD5HXeLi62d2t7iyJr/7GVuPAjPXApDiQSrdonWRrNzHu2x05LKawj5DZqrK0/e/5iY7P5cuvV6zett9tntqgMhyEvZGHOU2ZBCg1DJ5yE89IAU6mEUXr1ZaGPvoOxotCnbl7CRLFci0xw5gI1bX2lKeRCeyZFruvmJh2YzujwehcfYpoZxn1ce2q/GecTWgpqRa7YRVLXcOH38NLSucZ7mKpqNwjBtvJgXDcp6Nlq9rTVJt1kn/QPeph090nc/5gEQEiv/4ngOIBFtdGyBtPWTzoreKVAOy6ZteOYlG7imXGCSwjDKwsl41csh3GAmimwE/+QSY0/BGaGs8KEpR1+YJ92eKasnas0OBVzl/ZvbUH+SxtXLutNvNBl5UDzx42ySmJX4EXAeCYMcCfnATBuRDgr5pcs5OTCM4QMVhfF/wdnSTcO+CRpH31eprGB3qMd1EExOkBH6BgN0BBx9AP9Qr8jFN1Ed43wSx6tjWjZ8w79UY2te2zuskE=</latexit>
<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.
25
End-to-End Example
2 of 2
22
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)=
1
p
2⇡
2
e
(x
(
1
+
(
✓(0)1
)
e
t
)
)
2
2
2
(1)
<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>
<latexit sha1_base64="SzNC9isIlB9qsJisFM8x/YQpcnw=">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</latexit>
<latexit sha1_base64="SzNC9isIlB9qsJisFM8x/YQpcnw=">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</latexit>
<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.
25
Further Reading
23
▶︎ Complete these online self-assessments on https://f-of-e.org/
▶︎ Like learning to swim, you can’t learn all you need from a textbook
Best next step: Get some practice and test your understanding
https://f-of-e.org/chapter-03/#exercises
You will have the opportunity to apply these ideas as part of Coursework #4
25
Things to Do
24
Complete a “muddiest point” two-question survey using this link