Post on 27-Jul-2020
transcript
Running ProbabilisticRunning ProbabilisticRunning Probabilistic
Programs BackwardsPrograms BackwardsPrograms Backwards
Neil Toronto * Jay McCarthy † David Van Horn *
* University of Maryland † Vassar College
ESOP 2015
2015/04/14
RoadmapRoadmapRoadmap
• Probabilistic inference, and why it’s hard
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RoadmapRoadmapRoadmap
• Probabilistic inference, and why it’s hard
• Limitations of current probabilistic programming languages(PPLs)
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RoadmapRoadmapRoadmap
• Probabilistic inference, and why it’s hard
• Limitations of current probabilistic programming languages(PPLs)
• Contributions
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RoadmapRoadmapRoadmap
• Probabilistic inference, and why it’s hard
• Limitations of current probabilistic programming languages(PPLs)
• Contributions
Uncomputable, compositional ways to not limit language
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RoadmapRoadmapRoadmap
• Probabilistic inference, and why it’s hard
• Limitations of current probabilistic programming languages(PPLs)
• Contributions
Uncomputable, compositional ways to not limit language
Computable, compositional ways to not limit language
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Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
(let ([x (flip 0.5)]) x)
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Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
(let ([x (flip 0.5)]) x)
0.5
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Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
(let ([x (flip 0.5)]) x)
0.5
0.5
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Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
(let ([x (flip 0.5)][y (flip 0.5)])
(cons x y))
0.5
0.5 ⟨ , ⟩0.5 ⟨ , ⟩
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Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
(let ([x (flip 0.5)][y (flip 0.5)])
(cons x y))
0.5 0.5
0.5 ⟨ , ⟩ ⟨ , ⟩0.5 ⟨ , ⟩ ⟨ , ⟩
2222222222222222
Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
(let ([x (flip 0.5)][y (flip 0.5)])
(cons x y))
0.5 0.5
0.5 ⟨ , ⟩ ⟨ , ⟩0.5 ⟨ , ⟩ ⟨ , ⟩
2222222222222222
Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
(let* ([x (flip 0.5)][y (flip (if (equal? x heads) 0.5 0.3))])
(cons x y))
0.5 0.5
0.5 ⟨ , ⟩ ⟨ , ⟩0.5 ⟨ , ⟩ ⟨ , ⟩
2222222222222222
Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
(let* ([x (flip 0.5)][y (flip (if (equal? x heads) 0.5 0.3))])
(cons x y))
0.5 0.5
0.5 ⟨ , ⟩ ⟨ , ⟩0.5 ⟨ , ⟩ ⟨ , ⟩
0.3 0.72222222222222222
Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
0.5 0.5
0.5 ⟨ , ⟩ ⟨ , ⟩0.5 ⟨ , ⟩ ⟨ , ⟩
0.3 0.72222222222222222
Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
0.5 0.5
0.5 ⟨ , ⟩ ⟨ , ⟩0.5 ⟨ , ⟩ ⟨ , ⟩
0.3 0.72222222222222222
Programming Coin FlipsProgramming Coin FlipsProgramming Coin Flips
0.5
0.5 ⟨ , ⟩0.5 ⟨ , ⟩
0.32222222222222222
Stochastic Ray TracingStochastic Ray TracingStochastic Ray Tracing
sto·cha·stic /stō-ˈkas-tik/ adj. fancy word for "randomized"
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Stochastic Ray TracingStochastic Ray TracingStochastic Ray Tracing
sto·cha·stic /stō-ˈkas-tik/ adj. fancy word for "randomized"
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Stochastic Ray TracingStochastic Ray TracingStochastic Ray Tracing
ap·er·ture /ˈap-ə(r)-chər/ n. fancy word for "opening"
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Stochastic Ray TracingStochastic Ray TracingStochastic Ray Tracing
ap·er·ture /ˈap-ə(r)-chər/ n. fancy word for "opening"
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Stochastic Ray TracingStochastic Ray TracingStochastic Ray Tracing
Simulate projecting rays onto a sensor...
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Stochastic Ray TracingStochastic Ray TracingStochastic Ray Tracing
... and collect them to form an image
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Programming Stochastic Ray TracingProgramming Stochastic Ray TracingProgramming Stochastic Ray Tracing
• Normally thousands of lines of code
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Programming Stochastic Ray TracingProgramming Stochastic Ray TracingProgramming Stochastic Ray Tracing
• Normally thousands of lines of code
• Bears little resemblance to the physical process
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Programming Stochastic Ray TracingProgramming Stochastic Ray TracingProgramming Stochastic Ray Tracing
• Normally thousands of lines of code
• Bears little resemblance to the physical process
• In DrBayes, it’s simple physics simulation:
(define/drbayes (ray-plane-intersect p0 v n d) (let ([denom (- (dot v n))])
(if (> denom 0)(let ([t (/ (+ d (dot p0 n)) denom)]) (if (> t 0)
(collision t (vec+ p0 (vec* v t)) n)#f))
#f)))
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Programming Stochastic Ray TracingProgramming Stochastic Ray TracingProgramming Stochastic Ray Tracing
• Normally thousands of lines of code
• Bears little resemblance to the physical process
• In DrBayes, it’s simple physics simulation:
(define/drbayes (ray-plane-intersect p0 v n d) (let ([denom (- (dot v n))])
(if (> denom 0)(let ([t (/ (+ d (dot p0 n)) denom)]) (if (> t 0)
(collision t (vec+ p0 (vec* v t)) n)#f))
#f)))
• Other PPLs really aren’t up to this yet
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Programming Stochastic Ray TracingProgramming Stochastic Ray TracingProgramming Stochastic Ray Tracing
• Normally thousands of lines of code
• Bears little resemblance to the physical process
• In DrBayes, it’s simple physics simulation:
(define/drbayes (ray-plane-intersect p0 v n d) (let ([denom (- (dot v n))])
(if (> denom 0)(let ([t (/ (+ d (dot p0 n)) denom)]) (if (> t 0)
(collision t (vec+ p0 (vec* v t)) n)#f))
#f)))
• Other PPLs really aren’t up to this yet
• The issue is one of theory, not engineering effort
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Simpler ExampleSimpler ExampleSimpler Example
• Assume (random) returns a value uniformly in
3333333333333333
Simpler ExampleSimpler ExampleSimpler Example
• Assume (random) returns a value uniformly in
Density function for value of (random):
xxxxxxxxx
p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)
000000000 .25.25.25.25.25.25.25.25.25 .5.5.5.5.5.5.5.5.5 .75.75.75.75.75.75.75.75.75 111111111 1.251.251.251.251.251.251.251.251.25000000000
.25.25.25.25.25.25.25.25.25
.5.5.5.5.5.5.5.5.5
.75.75.75.75.75.75.75.75.75
111111111
1.251.251.251.251.251.251.251.251.25
3333333333333333
Simpler ExampleSimpler ExampleSimpler Example
• Assume (random) returns a value uniformly in
Density function for value of (random):
xxxxxxxxx
p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)
000000000 .25.25.25.25.25.25.25.25.25 .5.5.5.5.5.5.5.5.5 .75.75.75.75.75.75.75.75.75 111111111 1.251.251.251.251.251.251.251.251.25000000000
.25.25.25.25.25.25.25.25.25
.5.5.5.5.5.5.5.5.5
.75.75.75.75.75.75.75.75.75
111111111
1.251.251.251.251.251.251.251.251.25
3333333333333333
Simpler ExampleSimpler ExampleSimpler Example
• Assume (random) returns a value uniformly in
Density function for value of (random):
xxxxxxxxx
p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)p(x
)
000000000 .25.25.25.25.25.25.25.25.25 .5.5.5.5.5.5.5.5.5 .75.75.75.75.75.75.75.75.75 111111111 1.251.251.251.251.251.251.251.251.25000000000
.25.25.25.25.25.25.25.25.25
.5.5.5.5.5.5.5.5.5
.75.75.75.75.75.75.75.75.75
111111111
1.251.251.251.251.251.251.251.251.25
3333333333333333
Simpler ExampleSimpler ExampleSimpler Example
• Assume (random) returns a value uniformly in
Density function for value of (max 0.5 (random)):
xxxxxxxxx
pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)
000000000 .25.25.25.25.25.25.25.25.25 .5.5.5.5.5.5.5.5.5 .75.75.75.75.75.75.75.75.75 111111111 1.251.251.251.251.251.251.251.251.25000000000
.25.25.25.25.25.25.25.25.25
.5.5.5.5.5.5.5.5.5
.75.75.75.75.75.75.75.75.75
111111111
1.251.251.251.251.251.251.251.251.25
pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ???
3333333333333333
Simpler ExampleSimpler ExampleSimpler Example
• Assume (random) returns a value uniformly in
Density function for value of (max 0.5 (random)):
xxxxxxxxx
pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)
000000000 .25.25.25.25.25.25.25.25.25 .5.5.5.5.5.5.5.5.5 .75.75.75.75.75.75.75.75.75 111111111 1.251.251.251.251.251.251.251.251.25000000000
.25.25.25.25.25.25.25.25.25
.5.5.5.5.5.5.5.5.5
.75.75.75.75.75.75.75.75.75
111111111
1.251.251.251.251.251.251.251.251.25
pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ???
3333333333333333
Simpler ExampleSimpler ExampleSimpler Example
• Assume (random) returns a value uniformly in
Density function for value of (max 0.5 (random)):
xxxxxxxxx
pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)
000000000 .25.25.25.25.25.25.25.25.25 .5.5.5.5.5.5.5.5.5 .75.75.75.75.75.75.75.75.75 111111111 1.251.251.251.251.251.251.251.251.25000000000
.25.25.25.25.25.25.25.25.25
.5.5.5.5.5.5.5.5.5
.75.75.75.75.75.75.75.75.75
111111111
1.251.251.251.251.251.251.251.251.25
pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ??? pₘ(x) = ???
3333333333333333
Simpler ExampleSimpler ExampleSimpler Example
• Assume (random) returns a value uniformly in
Density function for value of (max 0.5 (random)):
xxxxxxxxx
pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)pₘ(x
)
000000000 .25.25.25.25.25.25.25.25.25 .5.5.5.5.5.5.5.5.5 .75.75.75.75.75.75.75.75.75 111111111 1.251.251.251.251.251.251.251.251.25000000000
.25.25.25.25.25.25.25.25.25
.5.5.5.5.5.5.5.5.5
.75.75.75.75.75.75.75.75.75
111111111
1.251.251.251.251.251.251.251.251.25
pₘ(x) doesn't exist pₘ(x) doesn't exist pₘ(x) doesn't exist pₘ(x) doesn't exist pₘ(x) doesn't exist pₘ(x) doesn't exist pₘ(x) doesn't exist pₘ(x) doesn't exist pₘ(x) doesn't exist
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What Can’t Densities Model?What Can’t Densities Model?What Can’t Densities Model?
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What Can’t Densities Model?What Can’t Densities Model?What Can’t Densities Model?
• Results of discontinuous functions (bounded measuring devices)
(let ([temperature (normal 99 1)]) (min 100 temperature))
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What Can’t Densities Model?What Can’t Densities Model?What Can’t Densities Model?
• Results of discontinuous functions (bounded measuring devices)
(let ([temperature (normal 99 1)]) (min 100 temperature))
• Variable-dimensional things (union types)
(if test? none (just x))
4444444444444444
What Can’t Densities Model?What Can’t Densities Model?What Can’t Densities Model?
• Results of discontinuous functions (bounded measuring devices)
(let ([temperature (normal 99 1)]) (min 100 temperature))
• Variable-dimensional things (union types)
(if test? none (just x))
• Infinite-dimensional things (recursion)
4444444444444444
What Can’t Densities Model?What Can’t Densities Model?What Can’t Densities Model?
• Results of discontinuous functions (bounded measuring devices)
(let ([temperature (normal 99 1)]) (min 100 temperature))
• Variable-dimensional things (union types)
(if test? none (just x))
• Infinite-dimensional things (recursion)
• In general: the distributions of program values
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Probability MeasuresProbability MeasuresProbability Measures
• Like already-integrated densities, but a primitive concept
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Probability MeasuresProbability MeasuresProbability Measures
• Like already-integrated densities, but a primitive concept
• Measure of (random) is , defined by
5555555555555555
Probability MeasuresProbability MeasuresProbability Measures
• Like already-integrated densities, but a primitive concept
• Measure of (random) is , defined by
5555555555555555
Probability MeasuresProbability MeasuresProbability Measures
• Like already-integrated densities, but a primitive concept
• Measure of (random) is , defined by
• Measure of (max 0.5 (random)) defined by
5555555555555555
Probability MeasuresProbability MeasuresProbability Measures
• Like already-integrated densities, but a primitive concept
• Measure of (random) is , defined by
• Measure of (max 0.5 (random)) defined by
This term assigns probability
5555555555555555
Probability MeasuresProbability MeasuresProbability Measures
• Like already-integrated densities, but a primitive concept
• Measure of (random) is , defined by
• Measure of (max 0.5 (random)) defined by
This term assigns probability
• Need a way to derive measures from code
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Probability Measures Via PreimagesProbability Measures Via PreimagesProbability Measures Via Preimages
• Interpret (max 0.5 (random)) as , defined
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Probability Measures Via PreimagesProbability Measures Via PreimagesProbability Measures Via Preimages
• Interpret (max 0.5 (random)) as , defined
• Derive measure of (max 0.5 (random)) as
6666666666666666
Probability Measures Via PreimagesProbability Measures Via PreimagesProbability Measures Via Preimages
• Interpret (max 0.5 (random)) as , defined
• Derive measure of (max 0.5 (random)) as
where
6666666666666666
Probability Measures Via PreimagesProbability Measures Via PreimagesProbability Measures Via Preimages
• Interpret (max 0.5 (random)) as , defined
• Derive measure of (max 0.5 (random)) as
where
• Factored into random and deterministic parts:
6666666666666666
Probability Measures Via PreimagesProbability Measures Via PreimagesProbability Measures Via Preimages
• Interpret (max 0.5 (random)) as , defined
• Derive measure of (max 0.5 (random)) as
where
• Factored into random and deterministic parts:
• In other words, compute measures of expressions by runningthem backwards
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Crazy Idea is Feasible If...Crazy Idea is Feasible If...Crazy Idea is Feasible If...
• Seems like we need:
7777777777777777
Crazy Idea is Feasible If...Crazy Idea is Feasible If...Crazy Idea is Feasible If...
• Seems like we need:
Standard interpretation of programs as pure functions from arandom source
7777777777777777
Crazy Idea is Feasible If...Crazy Idea is Feasible If...Crazy Idea is Feasible If...
• Seems like we need:
Standard interpretation of programs as pure functions from arandom source
Efficient way to compute preimage sets
7777777777777777
Crazy Idea is Feasible If...Crazy Idea is Feasible If...Crazy Idea is Feasible If...
• Seems like we need:
Standard interpretation of programs as pure functions from arandom source
Efficient way to compute preimage sets
Efficient representation of arbitrary sets
7777777777777777
Crazy Idea is Feasible If...Crazy Idea is Feasible If...Crazy Idea is Feasible If...
• Seems like we need:
Standard interpretation of programs as pure functions from arandom source
Efficient way to compute preimage sets
Efficient representation of arbitrary sets
Efficient way to compute areas of preimage sets
7777777777777777
Crazy Idea is Feasible If...Crazy Idea is Feasible If...Crazy Idea is Feasible If...
• Seems like we need:
Standard interpretation of programs as pure functions from arandom source
Efficient way to compute preimage sets
Efficient representation of arbitrary sets
Efficient way to compute areas of preimage sets
Proof of correctness w.r.t. standard interpretation
7777777777777777
Crazy Idea is Feasible If...Crazy Idea is Feasible If...Crazy Idea is Feasible If...
• Seems like we need:
Standard interpretation of programs as pure functions from arandom source
Efficient way to compute preimage sets
Efficient representation of arbitrary sets
Efficient way to compute areas of preimage sets
Proof of correctness w.r.t. standard interpretation
• WAT
7777777777777777
What About Approximating?What About Approximating?What About Approximating?
Conservative approximation with rectangles:
8888888888888888
What About Approximating?What About Approximating?What About Approximating?
Conservative approximation with rectangles:
8888888888888888
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Restricting preimages to rectangular subdomains:
9999999999999999
What About Approximating?What About Approximating?What About Approximating?
Sampling: exponential to quadratic (e.g. days to minutes)
10101010101010101010101010101010
What About Approximating?What About Approximating?What About Approximating?
Sampling: exponential to quadratic (e.g. days to minutes)
10101010101010101010101010101010
Contribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea Feasible
• Standard interpretation of programs as pure functions from arandom source
• Efficient way to compute preimage sets
• Efficient representation of arbitrary sets
• Efficient way to compute volumes of preimage sets
• Proof of correctness w.r.t. standard interpretation
11111111111111111111111111111111
Contribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea Feasible
• Standard interpretation of programs as pure functions from arandom source
• Efficient way to compute abstract preimage subsets
• Efficient representation of arbitrary sets
• Efficient way to compute volumes of preimage sets
• Proof of correctness w.r.t. standard interpretation
11111111111111111111111111111111
Contribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea Feasible
• Standard interpretation of programs as pure functions from arandom source
• Efficient way to compute abstract preimage subsets
• Efficient representation of abstract sets
• Efficient way to compute volumes of preimage sets
• Proof of correctness w.r.t. standard interpretation
11111111111111111111111111111111
Contribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea Feasible
• Standard interpretation of programs as pure functions from arandom source
• Efficient way to compute abstract preimage subsets
• Efficient representation of abstract sets
• Efficient way to sample uniformly in preimage sets
• Proof of correctness w.r.t. standard interpretation
11111111111111111111111111111111
Contribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea Feasible
• Standard interpretation of programs as pure functions from arandom source
• Efficient way to compute abstract preimage subsets
• Efficient representation of abstract sets
• Efficient way to sample uniformly in preimage sets
Efficient domain partition sampling
• Proof of correctness w.r.t. standard interpretation
11111111111111111111111111111111
Contribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea FeasibleContribution: Making This Crazy Idea Feasible
• Standard interpretation of programs as pure functions from arandom source
• Efficient way to compute abstract preimage subsets
• Efficient representation of abstract sets
• Efficient way to sample uniformly in preimage sets
Efficient domain partition sampling
Efficient way to determine whether a domain sample isactually in the preimage (just use standard interpretation)
• Proof of correctness w.r.t. standard interpretation
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How Many Meanings?How Many Meanings?How Many Meanings?
• Start with pure programs, then lift by threading a random store
11111111111111111111111111111111
How Many Meanings?How Many Meanings?How Many Meanings?
• Start with pure programs, then lift by threading a random store
• Nonrecursive, nonprobabilistic programs: , ,
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How Many Meanings?How Many Meanings?How Many Meanings?
• Start with pure programs, then lift by threading a random store
• Nonrecursive, nonprobabilistic programs: , ,
• Add 3 semantic functions for recursion and probabilistic choice
11111111111111111111111111111111
How Many Meanings?How Many Meanings?How Many Meanings?
• Start with pure programs, then lift by threading a random store
• Nonrecursive, nonprobabilistic programs: , ,
• Add 3 semantic functions for recursion and probabilistic choice
• Full development needs 2 more to transfer theorems frommeasure theory...
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How Many Meanings?How Many Meanings?How Many Meanings?
• Start with pure programs, then lift by threading a random store
• Nonrecursive, nonprobabilistic programs: , ,
• Add 3 semantic functions for recursion and probabilistic choice
• Full development needs 2 more to transfer theorems frommeasure theory...
• ... oh, and 1 more to collect information for Monte Carlointegration
11111111111111111111111111111111
How Many Meanings?How Many Meanings?How Many Meanings?
• Start with pure programs, then lift by threading a random store
• Nonrecursive, nonprobabilistic programs: , ,
• Add 3 semantic functions for recursion and probabilistic choice
• Full development needs 2 more to transfer theorems frommeasure theory...
• ... oh, and 1 more to collect information for Monte Carlointegration
Tally: 3+3+2+1 = 9 semantic functions, 11 or 12 rules each
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Enter Category TheoryEnter Category TheoryEnter Category Theory
• Moggi (1989): Introduces monads for interpreting effects
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Enter Category TheoryEnter Category TheoryEnter Category Theory
• Moggi (1989): Introduces monads for interpreting effects
• Other kinds of categories: idioms, arrows
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Enter Category TheoryEnter Category TheoryEnter Category Theory
• Moggi (1989): Introduces monads for interpreting effects
• Other kinds of categories: idioms, arrows
• Arrow defined by type constructor and thesecombinators:
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Enter Category TheoryEnter Category TheoryEnter Category Theory
• Moggi (1989): Introduces monads for interpreting effects
• Other kinds of categories: idioms, arrows
• Arrow defined by type constructor and thesecombinators:
• Arrows are always function-like
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Reducing ComplexityReducing ComplexityReducing Complexity
Function arrow: is just
11111111111111111111111111111111
Reducing ComplexityReducing ComplexityReducing Complexity
Function arrow: is just
11111111111111111111111111111111
Reducing ComplexityReducing ComplexityReducing Complexity
Function arrow: is just
11111111111111111111111111111111
Reducing ComplexityReducing ComplexityReducing Complexity
Function arrow: is just
11111111111111111111111111111111
Reducing ComplexityReducing ComplexityReducing Complexity
Function arrow: is just
11111111111111111111111111111111
Reducing ComplexityReducing ComplexityReducing Complexity
Function arrow: is just
11111111111111111111111111111111
Reducing ComplexityReducing ComplexityReducing Complexity
Function arrow: is just
11111111111111111111111111111111
Reducing ComplexityReducing ComplexityReducing Complexity
Function arrow: is just
11111111111111111111111111111111
Reducing ComplexityReducing ComplexityReducing Complexity
Function arrow: is just
11111111111111111111111111111111
Pair PreimagesPair PreimagesPair Preimages
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Pair PreimagesPair PreimagesPair Preimages
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Pair PreimagesPair PreimagesPair Preimages
:
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Pair PreimagesPair PreimagesPair Preimages
and :
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Pair PreimagesPair PreimagesPair Preimages
:
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Correctness Theorems For Low, Low PricesCorrectness Theorems For Low, Low PricesCorrectness Theorems For Low, Low Prices
• Define
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Correctness Theorems For Low, Low PricesCorrectness Theorems For Low, Low PricesCorrectness Theorems For Low, Low Prices
• Define
• Derive and others so that distributes; e.g.
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Correctness Theorems For Low, Low PricesCorrectness Theorems For Low, Low PricesCorrectness Theorems For Low, Low Prices
• Define
• Derive and others so that distributes; e.g.
• Distributive properties makes proving this very easy:
Theorem (correctness). For all , .
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Correctness Theorems For Low, Low PricesCorrectness Theorems For Low, Low PricesCorrectness Theorems For Low, Low Prices
• Define
• Derive and others so that distributes; e.g.
• Distributive properties makes proving this very easy:
Theorem (correctness). For all , .
In English: computes preimages under .
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Correctness Theorems For Low, Low PricesCorrectness Theorems For Low, Low PricesCorrectness Theorems For Low, Low Prices
• Define
• Derive and others so that distributes; e.g.
• Distributive properties makes proving this very easy:
Theorem (correctness). For all , .
In English: computes preimages under .
• Other correctness proofs are similarly easy: prove 5 distributiveproperties
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Correctness Theorems For Low, Low PricesCorrectness Theorems For Low, Low PricesCorrectness Theorems For Low, Low Prices
• Define
• Derive and others so that distributes; e.g.
• Distributive properties makes proving this very easy:
Theorem (correctness). For all , .
In English: computes preimages under .
• Other correctness proofs are similarly easy: prove 5 distributiveproperties
• Can add (random) and recursion to all semantics in one shot
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AbstractionAbstractionAbstraction
Rectangle: An interval or union of intervals, a subset of , or for rectangles and
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AbstractionAbstractionAbstraction
Rectangle: An interval or union of intervals, a subset of , or for rectangles and
• Easy representation; easy intersection, join (whichoverapproximates union), empty test, etc.
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AbstractionAbstractionAbstraction
Rectangle: An interval or union of intervals, a subset of , or for rectangles and
• Easy representation; easy intersection, join (whichoverapproximates union), empty test, etc.
• Define (and therefore ) by replacing sets and setoperations with rectangles and rectangle operations
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AbstractionAbstractionAbstraction
Rectangle: An interval or union of intervals, a subset of , or for rectangles and
• Easy representation; easy intersection, join (whichoverapproximates union), empty test, etc.
• Define (and therefore ) by replacing sets and setoperations with rectangles and rectangle operations
• Recursion is somewhat tricky—requires fine control overrecursion depth or if choices
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In Theory...In Theory...In Theory...
Theorem (sound). computes overapproximations of thepreimages computed by .
• Consequence: Sampling in abstract preimages doesn’t leaveanything out
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In Theory...In Theory...In Theory...
Theorem (sound). computes overapproximations of thepreimages computed by .
• Consequence: Sampling in abstract preimages doesn’t leaveanything out
Theorem (decreasing). never returns preimages larger thanthe given subdomain.
• Consequence: Refining abstract preimage sets never results in aworse approximation
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In Theory...In Theory...In Theory...
Theorem (sound). computes overapproximations of thepreimages computed by .
• Consequence: Sampling in abstract preimages doesn’t leaveanything out
Theorem (decreasing). never returns preimages larger thanthe given subdomain.
• Consequence: Refining abstract preimage sets never results in aworse approximation
Theorem (monotone). is monotone.
• Consequence: Partitioning and then refining never results in aworse approximation 14141414141414141414141414141414
In Practice...In Practice...In Practice...
Theorems prove this always works:
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In Practice...In Practice...In Practice...
Theorems prove this always works:
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In Practice...In Practice...In Practice...
Theorems prove this always works:
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In Practice...In Practice...In Practice...
Theorems prove this always works:
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In Practice...In Practice...In Practice...
Theorems prove this always works:
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In Practice...In Practice...In Practice...
Theorems prove this always works:
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In Practice...In Practice...In Practice...
Theorems prove this always works:
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In Practice...In Practice...In Practice...
Theorems prove this always works:
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In Practice...In Practice...In Practice...
Theorems prove this always works:
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Program Domain ValuesProgram Domain ValuesProgram Domain Values
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Program Domain ValuesProgram Domain ValuesProgram Domain Values
• Program inputs are infinite binary trees:
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Program Domain ValuesProgram Domain ValuesProgram Domain Values
• Program inputs are infinite binary trees:
• Every expression in a program is assigned a node
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Program Domain ValuesProgram Domain ValuesProgram Domain Values
• Program inputs are infinite binary trees:
• Every expression in a program is assigned a node
• Implemented using lazy trees of random values
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Program Domain ValuesProgram Domain ValuesProgram Domain Values
• Program inputs are infinite binary trees:
• Every expression in a program is assigned a node
• Implemented using lazy trees of random values
• No probability density for domain, but there is a measure 16161616161616161616161616161616
Example: Stochastic Ray TracingExample: Stochastic Ray TracingExample: Stochastic Ray Tracing
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Example: Probabilistic VerificationExample: Probabilistic VerificationExample: Probabilistic Verification
(struct/drbayes float-any ())(struct/drbayes float (value error))
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Example: Probabilistic VerificationExample: Probabilistic VerificationExample: Probabilistic Verification
(struct/drbayes float-any ())(struct/drbayes float (value error))
(define/drbayes (flsqrt x) (if (float-any? x)
x(let ([v (float-value x)]
[e (float-error x)]) (cond [(negative? v) (float-any)]
[(zero? v) (float 0 0)][else (float (sqrt v)
(+ (- 1 (sqrt (- 1 e)))(* 1/2 epsilon)))]))))
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Example: Probabilistic VerificationExample: Probabilistic VerificationExample: Probabilistic Verification
(struct/drbayes float-any ())(struct/drbayes float (value error))
(define/drbayes (flsqrt x) (if (float-any? x)
x(let ([v (float-value x)]
[e (float-error x)]) (cond [(negative? v) (float-any)]
[(zero? v) (float 0 0)][else (float (sqrt v)
(+ (- 1 (sqrt (- 1 e)))(* 1/2 epsilon)))]))))
• Idea: sample e where (> (float-error e) threshold)
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Example: Probabilistic VerificationExample: Probabilistic VerificationExample: Probabilistic Verification
(struct/drbayes float-any ())(struct/drbayes float (value error))
(define/drbayes (flsqrt x) (if (float-any? x)
x(let ([v (float-value x)]
[e (float-error x)]) (cond [(negative? v) (float-any)]
[(zero? v) (float 0 0)][else (float (sqrt v)
(+ (- 1 (sqrt (- 1 e)))(* 1/2 epsilon)))]))))
• Idea: sample e where (> (float-error e) threshold)
• Verified flhypot, flsqrt1pm1, flsinh in Racket’s mathlibrary, as well as others
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Examples: Other Inference TasksExamples: Other Inference TasksExamples: Other Inference Tasks
• Typical Bayesian inference
Hierarchical models
Bayesian regression
Model selection
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Examples: Other Inference TasksExamples: Other Inference TasksExamples: Other Inference Tasks
• Typical Bayesian inference
Hierarchical models
Bayesian regression
Model selection
• Atypical
Programs that halt with probability < 1, or never halt
Probabilistic context-free grammars with context-sensitiveconstraints
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SummarySummarySummary
• Probabilistic inference is hard, so PPLs have been popping up
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SummarySummarySummary
• Probabilistic inference is hard, so PPLs have been popping up
• Interpreting every program requires measure theory
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SummarySummarySummary
• Probabilistic inference is hard, so PPLs have been popping up
• Interpreting every program requires measure theory
• Defined a semantics that computes preimages
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SummarySummarySummary
• Probabilistic inference is hard, so PPLs have been popping up
• Interpreting every program requires measure theory
• Defined a semantics that computes preimages
• Measuring abstract preimages or sampling in them carries outinference
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SummarySummarySummary
• Probabilistic inference is hard, so PPLs have been popping up
• Interpreting every program requires measure theory
• Defined a semantics that computes preimages
• Measuring abstract preimages or sampling in them carries outinference
• Can do a lot of cool stuff that’s normally inaccessible
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https://github.com/ntoronto/drbayes
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