intro to probability - machinesintro to probability andrei barbu andrei barbu (mit) probability...
TRANSCRIPT
![Page 1: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/1.jpg)
Intro to Probability
Andrei Barbu
Andrei Barbu (MIT) Probability August 2018
![Page 2: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/2.jpg)
Some problemsSome problems
A means to capture uncertainty
You have data from two sources, are they different?
How can you generate data or fill in missing data?
What explains the observed data?
Andrei Barbu (MIT) Probability August 2018
![Page 3: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/3.jpg)
Some problemsSome problems
A means to capture uncertainty
You have data from two sources, are they different?
How can you generate data or fill in missing data?
What explains the observed data?
Andrei Barbu (MIT) Probability August 2018
![Page 4: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/4.jpg)
Some problemsSome problems
A means to capture uncertainty
You have data from two sources, are they different?
How can you generate data or fill in missing data?
What explains the observed data?
Andrei Barbu (MIT) Probability August 2018
![Page 5: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/5.jpg)
Some problemsSome problems
A means to capture uncertainty
You have data from two sources, are they different?
How can you generate data or fill in missing data?
What explains the observed data?
Andrei Barbu (MIT) Probability August 2018
![Page 6: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/6.jpg)
Some problemsSome problems
A means to capture uncertainty
You have data from two sources, are they different?
How can you generate data or fill in missing data?
What explains the observed data?
Andrei Barbu (MIT) Probability August 2018
![Page 7: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/7.jpg)
UncertaintyUncertainty
Every event has a probability of occurring, some event always occurs, andcombining separate events adds their probabilities.
Why these axioms? Many other choices are possible:Possibility theory, probability intervals
Belief functions, upper and lower probabilities
Andrei Barbu (MIT) Probability August 2018
![Page 8: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/8.jpg)
UncertaintyUncertainty
Every event has a probability of occurring, some event always occurs, andcombining separate events adds their probabilities.
Why these axioms? Many other choices are possible:Possibility theory, probability intervals
Belief functions, upper and lower probabilities
Andrei Barbu (MIT) Probability August 2018
![Page 9: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/9.jpg)
UncertaintyUncertainty
Every event has a probability of occurring, some event always occurs, andcombining separate events adds their probabilities.
Why these axioms? Many other choices are possible:
Possibility theory, probability intervalsBelief functions, upper and lower probabilities
Andrei Barbu (MIT) Probability August 2018
![Page 10: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/10.jpg)
UncertaintyUncertainty
Every event has a probability of occurring, some event always occurs, andcombining separate events adds their probabilities.
Why these axioms? Many other choices are possible:Possibility theory, probability intervals
Belief functions, upper and lower probabilities
Andrei Barbu (MIT) Probability August 2018
![Page 11: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/11.jpg)
UncertaintyUncertainty
Every event has a probability of occurring, some event always occurs, andcombining separate events adds their probabilities.
Why these axioms? Many other choices are possible:Possibility theory, probability intervals
Belief functions, upper and lower probabilities
Andrei Barbu (MIT) Probability August 2018
![Page 12: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/12.jpg)
UncertaintyUncertainty
Every event has a probability of occurring, some event always occurs, andcombining separate events adds their probabilities.
Why these axioms? Many other choices are possible:Possibility theory, probability intervals
Belief functions, upper and lower probabilities
Andrei Barbu (MIT) Probability August 2018
![Page 13: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/13.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative p
I’m offering to pay R for −pR dollars.
Probabilities don’t sum to one
Take out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sum
If the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 14: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/14.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative p
I’m offering to pay R for −pR dollars.
Probabilities don’t sum to one
Take out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sum
If the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 15: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/15.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative p
I’m offering to pay R for −pR dollars.
Probabilities don’t sum to one
Take out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sum
If the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 16: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/16.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative pI’m offering to pay R for −pR dollars.
Probabilities don’t sum to one
Take out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sum
If the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 17: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/17.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative pI’m offering to pay R for −pR dollars.
Probabilities don’t sum to one
Take out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sum
If the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 18: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/18.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative pI’m offering to pay R for −pR dollars.
Probabilities don’t sum to oneTake out a bet that always pays off.
If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sum
If the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 19: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/19.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative pI’m offering to pay R for −pR dollars.
Probabilities don’t sum to oneTake out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.
If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sum
If the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 20: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/20.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative pI’m offering to pay R for −pR dollars.
Probabilities don’t sum to oneTake out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sum
If the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 21: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/21.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative pI’m offering to pay R for −pR dollars.
Probabilities don’t sum to oneTake out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sum
If the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 22: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/22.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative pI’m offering to pay R for −pR dollars.
Probabilities don’t sum to oneTake out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sumIf the value is bigger, I still pay out more.
If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 23: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/23.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative pI’m offering to pay R for −pR dollars.
Probabilities don’t sum to oneTake out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sumIf the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 24: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/24.jpg)
Probability is special: Dutch booksProbability is special: Dutch books
I offer that if X happens I pay out R, otherwise I keep your money.
Why should I always value my bet at pR, where p is the probability of X?
Negative pI’m offering to pay R for −pR dollars.
Probabilities don’t sum to oneTake out a bet that always pays off.If the sum is below 1, I pay R for less than R dollars.If the sum is above 1, buy the bet and sell it to me for more.
When X and Y are incompatible the value isn’t the sumIf the value is bigger, I still pay out more.If the value is smaller, sell me my own bets.
Decisions under probability are “rational”.
Andrei Barbu (MIT) Probability August 2018
![Page 25: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/25.jpg)
Experiments, theory, and fundingExperiments, theory, and funding
Andrei Barbu (MIT) Probability August 2018
![Page 26: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/26.jpg)
Experiments, theory, and fundingExperiments, theory, and funding
Andrei Barbu (MIT) Probability August 2018
![Page 27: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/27.jpg)
DataData
Mean µX = E[X] =∑
xxp(x)
Variance σ2X = var(X) = E[(X − µ)2]
Covariance cov(X,Y) = E[(X − µx)(Y − µY)]
Correlation ρX,Y =cov(X,Y)σXσY
Andrei Barbu (MIT) Probability August 2018
![Page 28: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/28.jpg)
DataData
Mean µX = E[X] =∑
xxp(x)
Variance σ2X = var(X) = E[(X − µ)2]
Covariance cov(X,Y) = E[(X − µx)(Y − µY)]
Correlation ρX,Y =cov(X,Y)σXσY
Andrei Barbu (MIT) Probability August 2018
![Page 29: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/29.jpg)
DataData
Mean µX = E[X] =∑
xxp(x)
Variance σ2X = var(X) = E[(X − µ)2]
Covariance cov(X,Y) = E[(X − µx)(Y − µY)]
Correlation ρX,Y =cov(X,Y)σXσY
Andrei Barbu (MIT) Probability August 2018
![Page 30: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/30.jpg)
DataData
Mean µX = E[X] =∑
xxp(x)
Variance σ2X = var(X) = E[(X − µ)2]
Covariance cov(X,Y) = E[(X − µx)(Y − µY)]
Correlation ρX,Y =cov(X,Y)σXσY
Andrei Barbu (MIT) Probability August 2018
![Page 31: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/31.jpg)
DataData
Mean µX = E[X] =∑
xxp(x)
Variance σ2X = var(X) = E[(X − µ)2]
Covariance cov(X,Y) = E[(X − µx)(Y − µY)]
Correlation ρX,Y =cov(X,Y)σXσY
Andrei Barbu (MIT) Probability August 2018
![Page 32: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/32.jpg)
Mean and varianceMean and variance
Often implicitly assume that our data comes from a normaldistribution.
That our samples are i.i.d. (independent indentically distributed).
Generally these don’t capture enough about the underlying data.
Uncorrelated does not mean independent!
Andrei Barbu (MIT) Probability August 2018
![Page 33: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/33.jpg)
Mean and varianceMean and variance
Often implicitly assume that our data comes from a normaldistribution.
That our samples are i.i.d. (independent indentically distributed).
Generally these don’t capture enough about the underlying data.
Uncorrelated does not mean independent!
Andrei Barbu (MIT) Probability August 2018
![Page 34: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/34.jpg)
Mean and varianceMean and variance
Often implicitly assume that our data comes from a normaldistribution.
That our samples are i.i.d. (independent indentically distributed).
Generally these don’t capture enough about the underlying data.
Uncorrelated does not mean independent!
Andrei Barbu (MIT) Probability August 2018
![Page 35: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/35.jpg)
Mean and varianceMean and variance
Often implicitly assume that our data comes from a normaldistribution.
That our samples are i.i.d. (independent indentically distributed).
Generally these don’t capture enough about the underlying data.
Uncorrelated does not mean independent!
Andrei Barbu (MIT) Probability August 2018
![Page 36: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/36.jpg)
Correlation vs independenceCorrelation vs independence
V = N(0, 1), X = sin(V), Y = cos(V)
Correlation only measures linear relationships.
Andrei Barbu (MIT) Probability August 2018
![Page 37: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/37.jpg)
Correlation vs independenceCorrelation vs independence
V = N(0, 1), X = sin(V), Y = cos(V)
Correlation only measures linear relationships.
Andrei Barbu (MIT) Probability August 2018
![Page 38: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/38.jpg)
Correlation vs independenceCorrelation vs independence
V = N(0, 1), X = sin(V), Y = cos(V)
Correlation only measures linear relationships.
Andrei Barbu (MIT) Probability August 2018
![Page 39: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/39.jpg)
Data dinosaursData dinosaurs
Andrei Barbu (MIT) Probability August 2018
![Page 40: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/40.jpg)
Data dinosaursData dinosaurs
Andrei Barbu (MIT) Probability August 2018
![Page 41: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/41.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 42: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/42.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 43: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/43.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 44: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/44.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 45: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/45.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 46: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/46.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 47: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/47.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 48: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/48.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 49: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/49.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 50: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/50.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 51: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/51.jpg)
DataData
Mean
Variance
Covariance
Correlation
Are two players the same?
How do you know and how certain are you?
What about two players is different?
How do you quantify which differences matter?
Here’s a player, how good will they be?
What is the best information to ask for?
What is the best test to run?
If I change the size of the board, how mightthe results change?
. . .
Andrei Barbu (MIT) Probability August 2018
![Page 52: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/52.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)p(¬x) = 1− p(x)Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 53: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/53.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)p(¬x) = 1− p(x)Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 54: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/54.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)
Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)p(¬x) = 1− p(x)Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 55: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/55.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, A
Random variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)p(¬x) = 1− p(x)Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 56: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/56.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)p(¬x) = 1− p(x)Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 57: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/57.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)
The probability of either event p(A ∩ B)p(¬x) = 1− p(x)Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 58: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/58.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)
p(¬x) = 1− p(x)Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 59: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/59.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)p(¬x) = 1− p(x)
Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 60: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/60.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)p(¬x) = 1− p(x)Joint probabilities P(x, y)
Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 61: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/61.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)p(¬x) = 1− p(x)Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 62: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/62.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)p(¬x) = 1− p(x)Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 63: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/63.jpg)
Probability as an experimentProbability as an experiment
A machine enters a random state, its current state is an event.
Events, x, have probabilities associated, p(X = x) (p(x) shorthand)Sets of events, ARandom variables, X, are a function of the event
The probability of two events p(A ∪ B)The probability of either event p(A ∩ B)p(¬x) = 1− p(x)Joint probabilities P(x, y)Independence P(x, y) = P(x)P(y)
Conditional probabilities P(x|y) = P(x,y)p(y)
Law of total probability∑
A a = 1 when events A are a disjoint cover
Andrei Barbu (MIT) Probability August 2018
![Page 64: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/64.jpg)
Beating the lotteryBeating the lottery
Andrei Barbu (MIT) Probability August 2018
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Beating the lotteryBeating the lottery
Andrei Barbu (MIT) Probability August 2018
![Page 66: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/66.jpg)
Analyzing a testAnalyzing a test
You want to play the lottery, and have a method to win.
0.5% of tickets are winners, and you have a test to verify this.
You are 85% accurate (5% false positives, 10% false negatives)
Is this test useful? How useful? Should you be betting?
Andrei Barbu (MIT) Probability August 2018
![Page 67: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/67.jpg)
Analyzing a testAnalyzing a test
You want to play the lottery, and have a method to win.
0.5% of tickets are winners, and you have a test to verify this.
You are 85% accurate (5% false positives, 10% false negatives)
Is this test useful? How useful? Should you be betting?
Andrei Barbu (MIT) Probability August 2018
![Page 68: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/68.jpg)
Analyzing a testAnalyzing a test
You want to play the lottery, and have a method to win.
0.5% of tickets are winners, and you have a test to verify this.
You are 85% accurate (5% false positives, 10% false negatives)
Is this test useful? How useful? Should you be betting?
Andrei Barbu (MIT) Probability August 2018
![Page 69: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/69.jpg)
Analyzing a testAnalyzing a test
You want to play the lottery, and have a method to win.
0.5% of tickets are winners, and you have a test to verify this.
You are 85% accurate (5% false positives, 10% false negatives)
Is this test useful? How useful? Should you be betting?
Andrei Barbu (MIT) Probability August 2018
![Page 70: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/70.jpg)
Analyzing a testAnalyzing a test
You want to play the lottery, and have a method to win.
0.5% of tickets are winners, and you have a test to verify this.
You are 85% accurate (5% false positives, 10% false negatives)
Is this test useful? How useful? Should you be betting?
Andrei Barbu (MIT) Probability August 2018
![Page 71: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/71.jpg)
Analyzing a testAnalyzing a test
You want to play the lottery, and have a method to win.
0.5% of tickets are winners, and you have a test to verify this.
You are 85% accurate (5% false positives, 10% false negatives)
Is this test useful? How useful? Should you be betting?
Andrei Barbu (MIT) Probability August 2018
![Page 72: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/72.jpg)
Analyzing a testAnalyzing a test
You want to play the lottery, and have a method to win.
0.5% of tickets are winners, and you have a test to verify this.
You are 85% accurate (5% false positives, 10% false negatives)
Is this test useful? How useful? Should you be betting?
Andrei Barbu (MIT) Probability August 2018
![Page 73: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/73.jpg)
Is this test useful?Is this test useful?
(D−,T+) (D−,T−) (D+,T+) (D+,T−)What percent of the time when my test comes up true am I winner?D+ ∩T+
T+
= 0.9×0.0050.9×0.005+0.995×0.05 = 8.3% =
P(T + |D+)P(D+)P(T+)
P(A|B) =P(B|A)P(A)
P(B)posterior =
likelihood× priorprobability of data
Andrei Barbu (MIT) Probability August 2018
![Page 74: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/74.jpg)
Is this test useful?Is this test useful?
(D−,T+) (D−,T−) (D+,T+) (D+,T−)
What percent of the time when my test comes up true am I winner?D+ ∩T+
T+
= 0.9×0.0050.9×0.005+0.995×0.05 = 8.3% =
P(T + |D+)P(D+)P(T+)
P(A|B) =P(B|A)P(A)
P(B)posterior =
likelihood× priorprobability of data
Andrei Barbu (MIT) Probability August 2018
![Page 75: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/75.jpg)
Is this test useful?Is this test useful?
(D−,T+) (D−,T−) (D+,T+) (D+,T−)What percent of the time when my test comes up true am I winner?
D+ ∩T+T+
= 0.9×0.0050.9×0.005+0.995×0.05 = 8.3% =
P(T + |D+)P(D+)P(T+)
P(A|B) =P(B|A)P(A)
P(B)posterior =
likelihood× priorprobability of data
Andrei Barbu (MIT) Probability August 2018
![Page 76: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/76.jpg)
Is this test useful?Is this test useful?
(D−,T+) (D−,T−) (D+,T+) (D+,T−)What percent of the time when my test comes up true am I winner?D+ ∩T+
T+
= 0.9×0.0050.9×0.005+0.995×0.05 = 8.3% =
P(T + |D+)P(D+)P(T+)
P(A|B) =P(B|A)P(A)
P(B)posterior =
likelihood× priorprobability of data
Andrei Barbu (MIT) Probability August 2018
![Page 77: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/77.jpg)
Is this test useful?Is this test useful?
(D−,T+) (D−,T−) (D+,T+) (D+,T−)What percent of the time when my test comes up true am I winner?D+ ∩T+
T+= 0.9×0.005
0.9×0.005+0.995×0.05
= 8.3% =P(T + |D+)P(D+)
P(T+)
P(A|B) =P(B|A)P(A)
P(B)posterior =
likelihood× priorprobability of data
Andrei Barbu (MIT) Probability August 2018
![Page 78: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/78.jpg)
Is this test useful?Is this test useful?
(D−,T+) (D−,T−) (D+,T+) (D+,T−)What percent of the time when my test comes up true am I winner?D+ ∩T+
T+= 0.9×0.005
0.9×0.005+0.995×0.05 = 8.3%
=P(T + |D+)P(D+)
P(T+)
P(A|B) =P(B|A)P(A)
P(B)posterior =
likelihood× priorprobability of data
Andrei Barbu (MIT) Probability August 2018
![Page 79: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/79.jpg)
Is this test useful?Is this test useful?
(D−,T+) (D−,T−) (D+,T+) (D+,T−)What percent of the time when my test comes up true am I winner?D+ ∩T+
T+= 0.9×0.005
0.9×0.005+0.995×0.05 = 8.3% =P(T + |D+)P(D+)
P(T+)
P(A|B) =P(B|A)P(A)
P(B)posterior =
likelihood× priorprobability of data
Andrei Barbu (MIT) Probability August 2018
![Page 80: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/80.jpg)
Is this test useful?Is this test useful?
(D−,T+) (D−,T−) (D+,T+) (D+,T−)What percent of the time when my test comes up true am I winner?D+ ∩T+
T+= 0.9×0.005
0.9×0.005+0.995×0.05 = 8.3% =P(T + |D+)P(D+)
P(T+)
P(A|B) =P(B|A)P(A)
P(B)posterior =
likelihood× priorprobability of data
Andrei Barbu (MIT) Probability August 2018
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Human-Oriented RoboticsProf. Kai Arras
Social Robotics LabCommon Probability Distributions
Bernoulli Distribution
• Given a Bernoulli experiment, that is, a yes/no experiment with outcomes 0 (“failure”)or 1 (“success”)
• The Bernoulli distribution is a discrete proba-bility distribution, which takes value 1 with success probability and value 0 with failure probability 1 –
• Probability mass function
• Notation
0 10
0.5
1
1−h
h
Parameters• : probability of
observing a success
Expectation•
Variance•
37
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Binomial Distribution
• Given a sequence of Bernoulli experiments
• The binomial distribution is the discrete probability distribution of the number of successes m in a sequence of N indepen-dent yes/no experiments, each of which yields success with probability
• Probability mass function
• Notation
Human-Oriented RoboticsProf. Kai Arras
Social Robotics LabCommon Probability Distributions
0 10 20 30 400
0.1
0.2
h = 0.5, N = 20h = 0.7, N = 20h = 0.5, N = 40
m
Parameters• N : number of trials• : success probability
Expectation•
Variance•
38
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Gaussian Distribution
• Most widely used distribution for continuous variables
• Reasons: (i) simplicity (fully represented by only two moments, mean and variance) and (ii) the central limit theorem (CLT)
• The CLT states that, under mild conditions, the mean (or sum) of many independently drawn random variables is distributed approximately normally, irrespective of the form of the original distribution
• Probability density function
−4 −2 0 2 40
0.5
1
µ = 0, m = 1 µ = −3, m = 0.1 µ = 2, m = 2
Human-Oriented RoboticsProf. Kai Arras
Social Robotics LabCommon Probability Distributions
Parameters• : mean• : variance
Expectation•
Variance•
46
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Gaussian Distribution
• Notation
• Called standard normal distribution for µ = 0 and = 1
• About 68% (~two third) of values drawn from a normal distribution are within a range of ±1 standard deviations around the mean
• About 95% of the values lie within a range of ±2 standard deviations around the mean
• Important e.g. for hypothesis testing
Human-Oriented RoboticsProf. Kai Arras
Social Robotics LabCommon Probability Distributions
Parameters• : mean• : variance
Expectation•
Variance•
47
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Human-Oriented RoboticsProf. Kai Arras
Social Robotics LabCommon Probability Distributions
Multivariate Gaussian Distribution
• For d-dimensional random vectors, the multivariate Gaussian distribution is governed by a d-dimensional mean vector and a D x D covariance matrix that must be symmetric and positive semi-de#nite
• Probability density function
• Notation
Parameters• : mean vector• : covariance matrix
Expectation•
Variance•
48
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Human-Oriented RoboticsProf. Kai Arras
Social Robotics LabCommon Probability Distributions
Multivariate Gaussian Distribution
• For d = 2, we have the bivariate Gaussian distribution
• The covariance matrix (often C) deter-mines the shape of the distribution (video)
Parameters• : mean vector• : covariance matrix
Expectation•
Variance•
49
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Human-Oriented RoboticsProf. Kai Arras
Social Robotics LabCommon Probability Distributions
Multivariate Gaussian Distribution
• For d = 2, we have the bivariate Gaussian distribution
• The covariance matrix (often C) deter-mines the shape of the distribution (video)
Parameters• : mean vector• : covariance matrix
Expectation•
Variance•
49
![Page 88: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/88.jpg)
Human-Oriented RoboticsProf. Kai Arras
Social Robotics LabCommon Probability Distributions
Multivariate Gaussian Distribution
• For d = 2, we have the bivariate Gaussian distribution
• The covariance matrix (often C) deter-mines the shape of the distribution (video)
Parameters• : mean vector• : covariance matrix
Expectation•
Variance•
Sour
ce [1
]
50
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Human-Oriented RoboticsProf. Kai Arras
Social Robotics LabCommon Probability Distributions
Poisson Distribution
• Consider independent events that happen with an average rate of over time
• The Poisson distribution is a discrete distribution that describes the probability of a given number of events occurring in a !xed interval of time
• Can also be de#ned over other intervals such as distance, area or volume
• Probability mass function
• Notation
0 5 10 15 200
0.1
0.2
0.3
0.4
h = 1h = 4h = 10
Parameters• : average rate of events
over time or space
Expectation•
Variance•
45
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Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 91: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/91.jpg)
Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 92: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/92.jpg)
Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 93: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/93.jpg)
Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 94: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/94.jpg)
Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 95: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/95.jpg)
Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 96: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/96.jpg)
Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 97: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/97.jpg)
Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 98: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/98.jpg)
Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 99: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/99.jpg)
Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?
∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 100: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/100.jpg)
Bayesian updatesBayesian updates
P(θ|X) = P(X|θ)P(θ)P(X)
P(θ): I have some prior over how good a player is: informative vsuninformative.
P(X|θ): I think dart throwing is a stochastic process, every player hasan unknown mean.
X: I observe them throwing darts.
P(X): Across all parameters this is how likely the data is.
Normalization is usually hard to compute, but it’s often not needed.
Say P(θ) is a normal distribution with mean 0 and high variance.
And P(X|θ) is also a normal distribution.
What’s the best estimate for this player’s performance?∂∂θ logP(θ|X) = 0
Andrei Barbu (MIT) Probability August 2018
![Page 101: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/101.jpg)
Graphical modelsGraphical models
So far we’ve talked about independence, conditioning, andobservation.
A toolkit to discuss these at a higher level of abstraction.
Andrei Barbu (MIT) Probability August 2018
![Page 102: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/102.jpg)
Graphical modelsGraphical models
So far we’ve talked about independence, conditioning, andobservation.
A toolkit to discuss these at a higher level of abstraction.
Andrei Barbu (MIT) Probability August 2018
![Page 103: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/103.jpg)
Graphical modelsGraphical models
So far we’ve talked about independence, conditioning, andobservation.
A toolkit to discuss these at a higher level of abstraction.
Andrei Barbu (MIT) Probability August 2018
![Page 104: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/104.jpg)
Graphical modelsGraphical models
So far we’ve talked about independence, conditioning, andobservation.
A toolkit to discuss these at a higher level of abstraction.
Andrei Barbu (MIT) Probability August 2018
![Page 105: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/105.jpg)
Speech recognition: Naïve bayesSpeech recognition: Naïve bayes
Break down a speech signal into parts.
Recover the original speech
Andrei Barbu (MIT) Probability August 2018
![Page 106: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/106.jpg)
Speech recognition: Naïve bayesSpeech recognition: Naïve bayes
Break down a speech signal into parts.
Recover the original speech
Andrei Barbu (MIT) Probability August 2018
![Page 107: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/107.jpg)
Speech recognition: Naïve bayesSpeech recognition: Naïve bayes
Break down a speech signal into parts.
Recover the original speech
Andrei Barbu (MIT) Probability August 2018
![Page 108: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/108.jpg)
Speech recognition: Naïve bayesSpeech recognition: Naïve bayes
Create a set of features, each sound is composed of combinations offeatures.
P(c|X) ∝∏
KP(Xk|c)P(c)
Andrei Barbu (MIT) Probability August 2018
![Page 109: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/109.jpg)
Speech recognition: Naïve bayesSpeech recognition: Naïve bayes
Create a set of features, each sound is composed of combinations offeatures.
P(c|X) ∝∏
KP(Xk|c)P(c)
Andrei Barbu (MIT) Probability August 2018
![Page 110: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/110.jpg)
Speech recognition: Naïve bayesSpeech recognition: Naïve bayes
Create a set of features, each sound is composed of combinations offeatures.
P(c|X) ∝∏
KP(Xk|c)P(c)
Andrei Barbu (MIT) Probability August 2018
![Page 111: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/111.jpg)
Speech recognition: Gaussian mixture modelSpeech recognition: Gaussian mixture model
Andrei Barbu (MIT) Probability August 2018
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Speech recognition: Gaussian mixture modelSpeech recognition: Gaussian mixture model
Andrei Barbu (MIT) Probability August 2018
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Speech recognition: Gaussian mixture modelSpeech recognition: Gaussian mixture model
Andrei Barbu (MIT) Probability August 2018
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Speech recognition: Hidden Markov modelSpeech recognition: Hidden Markov model
Andrei Barbu (MIT) Probability August 2018
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Speech recognition: Hidden Markov modelSpeech recognition: Hidden Markov model
Andrei Barbu (MIT) Probability August 2018
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SummarySummary
Probabilities defined in terms of events
Random variables and their distributions
Reasoning with probabilities and Bayes’ rule
Updating our knowledge over time
Graphical models to reason abstractly
A quick tour of how we would build a more complex model
Andrei Barbu (MIT) Probability August 2018
![Page 117: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/117.jpg)
SummarySummary
Probabilities defined in terms of events
Random variables and their distributions
Reasoning with probabilities and Bayes’ rule
Updating our knowledge over time
Graphical models to reason abstractly
A quick tour of how we would build a more complex model
Andrei Barbu (MIT) Probability August 2018
![Page 118: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/118.jpg)
SummarySummary
Probabilities defined in terms of events
Random variables and their distributions
Reasoning with probabilities and Bayes’ rule
Updating our knowledge over time
Graphical models to reason abstractly
A quick tour of how we would build a more complex model
Andrei Barbu (MIT) Probability August 2018
![Page 119: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/119.jpg)
SummarySummary
Probabilities defined in terms of events
Random variables and their distributions
Reasoning with probabilities and Bayes’ rule
Updating our knowledge over time
Graphical models to reason abstractly
A quick tour of how we would build a more complex model
Andrei Barbu (MIT) Probability August 2018
![Page 120: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/120.jpg)
SummarySummary
Probabilities defined in terms of events
Random variables and their distributions
Reasoning with probabilities and Bayes’ rule
Updating our knowledge over time
Graphical models to reason abstractly
A quick tour of how we would build a more complex model
Andrei Barbu (MIT) Probability August 2018
![Page 121: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/121.jpg)
SummarySummary
Probabilities defined in terms of events
Random variables and their distributions
Reasoning with probabilities and Bayes’ rule
Updating our knowledge over time
Graphical models to reason abstractly
A quick tour of how we would build a more complex model
Andrei Barbu (MIT) Probability August 2018
![Page 122: Intro to Probability - MachinesIntro to Probability Andrei Barbu Andrei Barbu (MIT) Probability August 2018](https://reader035.vdocument.in/reader035/viewer/2022062603/5f02eaf37e708231d406a73b/html5/thumbnails/122.jpg)
SummarySummary
Probabilities defined in terms of events
Random variables and their distributions
Reasoning with probabilities and Bayes’ rule
Updating our knowledge over time
Graphical models to reason abstractly
A quick tour of how we would build a more complex model
Andrei Barbu (MIT) Probability August 2018