higher-order probabilistic programming

Higher-Order Probabilistic ProgrammingA Tutorial at POPL 2019

Part I

Ugo Dal Lago(Based on joint work with Flavien Breuvart, Raphaëlle Crubillé, Charles

Grellois, Davide Sangiorgi,. . . )

POPL 2019, Lisbon, January 14th

Probabilistic Models

I The environment is supposed not to behave deterministically, butprobabilistically.

I Crucial when modeling uncertainty.I Useful to handle complex domains.I Example:

q0 q1

q2 q3

14

34 1

12

12

13

23

I Abstractions:I (Labelled) Markov Chains.



I Crucial when modeling uncertainty.

I Useful to handle complex domains.I Example:

q0 q1

q2 q3

14

34 1

12

12

13

23




I Crucial when modeling uncertainty.I Useful to handle complex domains.

I Example:

q0 q1

q2 q3

14

34 1

12

12

13

23




I Crucial when modeling uncertainty.I Useful to handle complex domains.I Example:

q0 q1

q2 q3

14

34 1

12

12

13

23



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BO

TIC

S


ART

IFIC

IAL

INT

ELL

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NAT

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Randomized Computation

I Algorithms and automata are assumed to have the ability to sample from adistribution [dLMSS1956,R1963].

I This is a powerful tool when solving computational problems.I Example:

I Abstractions:I Randomized algorithms;I Probabilistic Turing machines.I Labelled Markov chains.



I This is a powerful tool when solving computational problems.

I Example:




I This is a powerful tool when solving computational problems.I Example:



01 1 1 0

s

· · · · · · δ(s, 0) = {(q, 1,←)12 , (r, 0,→)

23 }

01 1 1 0

r

· · · · · ·

11 1 1 0

q

· · · · · ·

12

12


01 1 1 0

s

· · · · · · δ(s, 0) = {(q, 1,←)12 , (r, 0,→)

23 }

01 1 1 0

r

· · · · · ·

11 1 1 0

q

· · · · · ·

23

13


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What Algorithms Compute

I Deterministic ComputationI For every input x, there is at most one output y any algorithm A produces when

fed with x.

I As a consequence:

A ; JAK : N⇀ N.

I Randomized ComputationI For every input x, any algorithm A outputs y with a probability 0 ≤ p ≤ 1.

I As a consequence:

A ; JAK : N→ D(N).I The distribution JAK(n) sums to anything between 0 and 1, thus accounting for

the probability of divergence.

What Algorithms Compute

I Deterministic ComputationI For every input x, there is at most one output y any algorithm A produces when

fed with x.

I As a consequence:

A ; JAK : N⇀ N.I Randomized Computation

I For every input x, any algorithm A outputs y with a probability 0 ≤ p ≤ 1.

I As a consequence:

A ; JAK : N→ D(N).I The distribution JAK(n) sums to anything between 0 and 1, thus accounting for

the probability of divergence.

Higher-Order Computation

I Mainly useful in programming.

I Functions are first-class citizens:I They can be passed as arguments;I They can be obtained as results.

I Motivations:I Modularity;I Code reuse;I Conciseness.

I Example:

I Models:I λ-calculus


I Mainly useful in programming.I Functions are first-class citizens:

I They can be passed as arguments;I They can be obtained as results.

I Motivations:I Modularity;I Code reuse;I Conciseness.

I Example:

I Models:I λ-calculus


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Higher-Order Probabilistic Computation?

Does it Make Sense?

What Kind of MetatheoryDoes it Have?

Interesting Research Problems?

This Tutorial

1. Motivating Examples.2. A λ-Calculus Foundation.3. Operational and Denotational Semantics.4. Termination and Resource Analysis.

A webpage: http://www.cs.unibo.it/~dallago/HOPP/

http://www.cs.unibo.it/~dallago/HOPP/

MergeSort (1)

MergeSort (2)

The Structure of MergeSort

merge sort

merge sort

merge sort

MergeSort, HO

QuickSort

The Structure of MergeSort

quick sort

quick sort

quick sort

QuickSort, HO

Randomized QuickSort (1)

Randomized QuickSort (2)

Random Walk

0 1 2 n· · ·

p 1 − p

Two Kinds of Random Walks

The Coupon Collector

I A brand distributes a large quantity of coupons, each labelled withi ∈ {1, . . . , n}.

I Every day, you collect one coupon at a local store. Any label i has probability1n to occur.

I You win a prize when you collect a set of n coupons, each with a distinct label.I Example: if n = 5, you could get the following coupons:

3, 1, 5, 2, 3, 1, 5, 2, 3, 1, 5, 2, . . .

I Are you guaranteed to win the prize with probability 1? After how many days,on the average?




I You win a prize when you collect a set of n coupons, each with a distinct label.

I Example: if n = 5, you could get the following coupons:

3, 1, 5, 2, 3, 1, 5, 2, 3, 1, 5, 2, . . .





I You win a prize when you collect a set of n coupons, each with a distinct label.I Example: if n = 5, you could get the following coupons:

3, 1, 5, 2, 3, 1, 5, 2, 3, 1, 5, 2, . . .


Beyond Randomized Algorithms: The Grass Problem

I You get up in the morning, and notice that the grass in your garden is wet.

I You know that there is 30% probability that it rains during the night.I The sprinkler is activated once every two days.I You further know that:

I If it rains, there is 90% probability that the grass is wet the following morning.I If the sprinkler is activated, there is 80% probability that the grass is wet the

following morning.I In any other case, there is anyway a 10% probability that the grass is wet.

I In other words, you are in a situation like the following:rain

sprinkler

grass is wet

I Now: how likely it is that it rained?


I You get up in the morning, and notice that the grass in your garden is wet.I You know that there is 30% probability that it rains during the night.

I The sprinkler is activated once every two days.I You further know that:




sprinkler

grass is wet



I You get up in the morning, and notice that the grass in your garden is wet.I You know that there is 30% probability that it rains during the night.I The sprinkler is activated once every two days.

I You further know that:I If it rains, there is 90% probability that the grass is wet the following morning.I If the sprinkler is activated, there is 80% probability that the grass is wet the



sprinkler

grass is wet



I You get up in the morning, and notice that the grass in your garden is wet.I You know that there is 30% probability that it rains during the night.I The sprinkler is activated once every two days.I You further know that:




sprinkler

grass is wet


The Grass Model

Thank You!

Questions?

higher-order probabilistic programming

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