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CS 4110
Probabilistic Programming
Probabilistic Programming
It's not about writing software.
CS 4110 Probabilistic Programming Probabilistic Programming It's - - PowerPoint PPT Presentation
CS 4110 Probabilistic Programming Probabilistic Programming It's - - PowerPoint PPT Presentation
CS 4110 Probabilistic Programming Probabilistic Programming It's not about writing software. Probabilistic Programming Probabilistic programming is a tool for statistical modeling. OR A probabilistic programming language is a plain old
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Probabilistic Programming
Probabilistic programming is a tool for statistical modeling. OR A probabilistic programming language is a plain old programming language with rand(3) and a suite of fancy analysis tools for understanding its probabilistic behavior.
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An Example Model
PL+stats PL stats
Takes CS 4242 Takes CS 4110 Takes CS 4780 Paper 2 Relevant Paper 3 Relevant Paper 1 Relevant Cloud of Unknowing
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A Model for Humans
Interest in PL Interest in Stats Takes CS 4242 Takes CS 4110 Takes CS 4780
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A Model for Humans
Interest in PL Interest in Stats Takes CS 4242 Takes CS 4110 Takes CS 4780 Busy?
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A Model for Humans
Interest in PL Interest in Stats Takes CS 4242 Takes CS 4110 Takes CS 4780 Busy? Interest in PL Interest in Stats Paper 2 Relevant Paper 3 Relevant Paper 1 Relevant
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A Model for Humans
Pr[ANIPS|Istats ∧ B] = 0.3 Pr[ANIPS|Istats ∧ ¬B] = 0.8 Pr[ANIPS|¬Istats] = 0.1 Pr[ADagstuhl|Istats ∧ IPL] = 0.3 Pr[ADagstuhl|Istats ∧ IPL ∧ ¬B] = 0.8 Pr[ADagstuhl|¬(Istats ∨ IPL)] = 0.1 R1 ∼ IPL ∧ Istats R3 ∼ Istats R2 ∼ IPL
… …
Whither reuse? Whither abstraction? Whither intermediate variables?
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Writing even this tiny model feels like drudgery.
(and we haven’t even gotten to the hard part yet)
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- What and Why
- The Basics and Examples
- Applications
- Current Problems
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webppl.org
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Our First Probabilistic Program
var b = flip(0.5); b ? "yes" : "no"
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Enumeration
var roll = function () { var die1 = randomInteger(6) + 1; var die2 = randomInteger(6) + 1; return die1 + die2; } Enumerate(roll)
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Our Basic Model in webppl
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Conditioning
var roll = function () { var die1 = randomInteger(6) + 1; var die2 = randomInteger(6) + 1; if (!(die1 === 4 || die2 === 4)) { factor(-Infinity); } return die1 + die2; } Enumerate(roll)
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Conditioning
- n Observations
// Discard any executions that // don’t sum to 10. var out = die1 + die2; if (out !== 10) { factor(-Infinity); } // Return the values on the dice. return [die1, die2];
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Recommending Papers
// Require my class attendance. var att = attendance(i_pl, i_stats, busy); require(att.cs4110 && att.cs4242 && !att.cs4780); return relevance(i_pl, i_stats);
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Inference Algorithms
Enumerate is the simplest possible inference strategy.
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- What and Why
- The Basics and Examples
- Applications
- Current Problems
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TrueSkill
// prior distributions, the hypothesis
let skill() = random (Gaussian(10.0,20.0)) let Alice,Bob,Cyd = skill(),skill(),skill()
// observe the evidence
let performance player = random (Gaussian(player,1.0))
- bserve (performance Alice > performance Bob) //Alice beats Bob
- bserve (performance Bob > performance Cyd) //Bob beats Cyd
- bserve (performance Alice > performance Cyd) //Alice beats Cyd
// return the skills Alice,Bob,Cyd
Measure Transformer Semantics for Bayesian Machine Learning
Johannes Borgstr¨
- m
Andrew D. Gordon Michael Greenberg James Margetson Jurgen Van Gael
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webppl Vision Demo
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Forestdb.org
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- What and Why
- The Basics and Examples
- Applications
- Current Research
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R2
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R2’s weakest preconditions
var die1 = randomInteger(7) + 1; var die2 = randomInteger(7) + 1; // Discard any executions that // don’t sum to 10. var out = die1 + die2; require(out === 10);
wasted work!
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R2’s weakest preconditions
var die1 = randomInteger(7) + 1; var die2 = randomInteger(7) + 1; require( (die1 == 3 && die2 == 7) || …); var out = die1 + die2; require(out === 10);
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R2’s weakest preconditions
var die1 = randomInteger(7) + 1; var die2 = randomInteger(7) + 1; require( (die1 == 3 && die2 == 7) || …); var out = die1 + die2;
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Probabilistic assertions: design goals
Work on a messy, mainstream language (C and C++) Efficiently check statistical properties of the output We don’t care about conditioning
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assert e p
, p, c
e must hold with probability p at confidence c
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passert e, p, c
float obfuscated(float n) { return n + gaussian(0.0, 1000.0); } float average_salary(float* salaries) { total = 0.0; for (int i = 0; i < COUNT; ++i) total += obfuscated(salaries[i]); avg = total / len(salaries); p_avg = ...; }
Bayesian network IR
✓
distribution extraction
via symbolic execution statistical
- ptimizations
verification
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Bayesian network IR
✓
distribution extraction
via symbolic execution statistical
- ptimizations
verification
passert e, p, c
float obfuscated(float n) { return n + gaussian(0.0, 1000.0); } float average_salary(float* salaries) { total = 0.0; for (int i = 0; i < COUNT; ++i) total += obfuscated(salaries[i]); avg = total / len(salaries); p_avg = ...; }
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Distribution extraction: random draws are symbolic
b = a + gaussian(0.0, 1.0)
a 4.2
symbolic heap
a 4.2 b
G0,1
4.2 +
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input: a = 4.2 b = gaussian(0.0, 1.0)
a
4.2
b
G0,1
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input: a = 4.2 b = gaussian(0.0, 1.0) c = a + b
a
4.2
b
G0,1
c
+
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input: a = 4.2 b = gaussian(0.0, 1.0) c = a + b d = c + b
a
4.2
b
G0,1
c
+
d
+
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input: a = 4.2 b = gaussian(0.0, 1.0) c = a + b d = c + b
a
4.2
b
G0,1
c
+
d
+
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input: a = 4.2 b = gaussian(0.0, 1.0) c = a + b d = c + b if b > 0.5 e = 2.0 else e = 4.0
a
4.2 G0,1
c
+
d
+ ?
e
if
2.0 4.0
then else
b
> 0.5
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input: a = 4.2 b = gaussian(0.0, 1.0) c = a + b d = c + b if b > 0.5 e = 2.0 else e = 4.0 passert e <= 3.0, 0.9, 0.9
a
4.2 G0,1
c
+
d
+ ?
e
if
2.0 4.0
then else
b
> 0.5 ≤ 3.0
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passert e, p, c
float obfuscated(float n) { return n + gaussian(0.0, 1000.0); } float average_salary(float* salaries) { total = 0.0; for (int i = 0; i < COUNT; ++i) total += obfuscated(salaries[i]); avg = total / len(salaries); p_avg = ...; }
Bayesian network IR
✓
distribution extraction
via symbolic execution statistical
- ptimizations
verification
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statistical property passert verifier
- ptimization
X ∼ G ( µX , σ2
X)
Y ∼ G ( µY , σ2
Y )
Z = X + Y ⇒ Z ∼ G ( µX + µY , σ2
X + σ2 Y )
X ∼ U(a, b) Y = cX ⇒ Y ∼ U(ca, cb) X ∼ U(a, b) Y ∼ X ≤ c a ≤ c ≤ b ⇒ Y ∼ B ✓c − a b − a ◆
X
1
, X
2
, . . . , X
n
∼ D Y = X
i
X
i
⇒ Y ∼ G ( n µ
D
, n σ
2 D
)
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passert e, p, c
float obfuscated(float n) { return n + gaussian(0.0, 1000.0); } float average_salary(float* salaries) { total = 0.0; for (int i = 0; i < COUNT; ++i) total += obfuscated(salaries[i]); avg = total / len(salaries); p_avg = ...; }
Bayesian network IR
✓
distribution extraction
via symbolic execution statistical
- ptimizations
verification