Operational semantics for disintegration

Transcription

1 Operational semantics for disintegration Chung-chieh Shan (Indiana University) Norman Ramsey (Tufts University) Mathematical Foundations of Programming Semantics

2 What is semantics for? 1. Language for composing generative stories do {x normal 0 10; (monadic bind, unit) 2. Guarantee that terms denote distributions (product measure tricky) 2

3 What is semantics for? 1. Language for composing generative stories do {x normal 0 10; normal x 1} (monadic bind, unit) 2. Guarantee that terms denote distributions (product measure tricky) 2

4 What is semantics for? 1. Language for composing generative stories do {x normal 0 10; return e x } (monadic bind, unit) 2. Guarantee that terms denote distributions (product measure tricky) 2

5 What is semantics for? 1. Language for composing generative stories do {x normal 0 10; return e x } (monadic bind, unit) 2. Guarantee that terms denote distributions do {x m; do {y n; return (x, y)}} (product measure tricky) 2

6 What is semantics for? 1. Language for composing generative stories do {x normal 0 10; return e x } (monadic bind, unit) 2. Guarantee that terms denote distributions do {x m; do {y n; return (x, y)}} = do {x m; y n; return (x, y)} (product measure tricky) 2

7 What is semantics for? 3. Equational reasoning by semantics-preserving rewriting! Fubini do {x m; y n; return (x, y)} = do {y n; x m; return (x, y)} Conjugacy do {x normal 0 1; factor e x2 ; return x} = do {factor (1/ 3); x normal 0 3; return x} 3

8 What is semantics for? 3. Equational reasoning by semantics-preserving rewriting! Fubini do {x m; y n; return (x, y)} = do {y n; x m; return (x, y)} Conjugacy do {x normal 0 1; factor e x2 ; return x} = do {factor (1/ 3); x normal 0 3; return x} 3

9 What is semantics for? 3. Equational reasoning by semantics-preserving rewriting! Fubini do {x m; y n; return (x, y)} = do {y n; x m; return (x, y)} Conjugacy do {x normal 0 1; factor e x2 ; return x} = do {factor (1/ 3); x normal 0 3; return x} (convenient factor expresses non-(sub)probability measures) 3

10 Two program transformations specified semantically 1. Simplification m = m 2. Disintegration m = do {t m 1 ; x m 2 ; return (t, x)} (typically t appears free in m 2 ) 4

11 Two program transformations specified semantically 1. Simplification m = m 2. Disintegration m = do {t m 1 ; x m 2 ; return (t, x)} (typically t appears free in m 2 ) 4

12 Two program transformations specified semantically 1. Simplification m = m 2. Disintegration m = do {t m 1 ; x m 2 ; return (t, x)} (typically t appears free in m 2 ) 4

13 Two program transformations specified semantically 1. Simplification m = m 2. Disintegration m = do {t m 1 ; x m 2 ; return (t, x)} = do {t m 1 ; m 2 } (typically t appears free in m 2 ) 4

14 What is disintegration for? m = do {t m 1 ; x m 2 ; return (t, x)} = do {t m 1 ; m 2 } 5

15 What is disintegration for? m = do {t m 1 ; x m 2 ; return (t, x)} = do {t m 1 ; m 2 } Disintegration defines conditional distributions t x do {x normal 0 10; t normal x 1; return (t, x)} -- latent cause -- observed effect 5

16 What is disintegration for? m = do {t m 1 ; x m 2 ; return (t, x)} = do {t m 1 ; m 2 } Disintegration defines conditional distributions t x do {x normal 0 10; t normal x 1; return (t, x)} -- latent cause -- observed effect = do {t normal 0 101; x normal (t 100/101) (10/ 101); return (t, x)} 5

17 What is disintegration for? m = do {t m 1 ; x m 2 ; return (t, x)} = do {t m 1 ; m 2 } Disintegration defines conditional distributions t x do {x normal 0 10; t normal x 1; return (t, x)} -- latent cause -- observed effect = do {t normal 0 101; normal (t 100/101) (10/ 101)} 5

18 What is disintegration for? m = do {t m 1 ; x m 2 ; return (t, x)} = do {t m 1 ; m 2 } Disintegration allows an uncountable space of observations 1 do {x uniform 0 1; y uniform 0 1; t ; return (t, (x, y))} 0 1 6

19 What is disintegration for? m = do {t m 1 ; x m 2 ; return (t, x)} = do {t m 1 ; m 2 } Disintegration allows an uncountable space of observations 1 do {x uniform 0 1; y uniform 0 1; t return (y 2 x); return (t, (x, y))} 0 1 6

20 What is disintegration for? m = do {t m 1 ; x m 2 ; return (t, x)} = do {t m 1 ; m 2 } Disintegration allows an uncountable space of observations 1 do {x uniform 0 1; y uniform 0 1; t return (y/x); return (t, (x, y))} 0 1 6

21 From denotation to operation Denotation: equivalence relation Operation: directed graph So denotation justifies soundness of operation 7

22 From denotation to operation Denotation: equivalence relation Operation: directed graph So denotation justifies soundness of operation 7

23 From denotation to operation Denotation: equivalence relation Operation: directed graph So denotation justifies soundness of operation This setup is typical for execution, but what about disintegration? Rest of this talk: a disintegrator that justifies its work, step by step A contextual operational semantics, implemented in PLT Redex 7

24 Syntax Term e, m ::= do {factor e; m} do {x m; m} do {x m; m} return e uniform e e normal e e lebesgue (e, e) fst e snd e number e e e/e x (not shown: conditionals) 8

25 Syntax Stratified syntax during disintegration State s ::= do {factor e; s} do {x m; s} do {delimit; t} Thread t ::= do {factor e; t} do {x m; t} do {x m; m} Term e, m ::= do {factor e; m} do {x m; m} return e uniform e e normal e e lebesgue (e, e) fst e snd e number e e e/e x (not shown: conditionals) 8

26 Example disintegration do {delimit; r do {x uniform 0 1; y uniform 0 1; return (y/x, (x, y))} t return (fst r); Stage 1: Isolate (evaluate) Stage 2: Invert (unevaluate) 9

27 Example disintegration do {delimit; r do {x uniform 0 1; y uniform 0 1; return (y/x, (x, y))}; t return (fst r); 9

28 Example disintegration (step 1) bind do {delimit; do {delimit; r do {x uniform 0 1; x uniform 0 1; y uniform 0 1; r do {y uniform 0 1; return (y/x, (x, y))}; return (y/x, (x, y))}; t return (fst r); t return (fst r); 9

29 Example disintegration (step 1) bind do {delimit; do {delimit; r do {x uniform 0 1; x uniform 0 1; y uniform 0 1; r do {y uniform 0 1; return (y/x, (x, y))}; return (y/x, (x, y))}; t return (fst r); t return (fst r); do {delimit; H; r do {x m 1 ; m 2 }; P[r]} bind do {delimit; H; x m 1 ; r m 2 ; P[r]} H is a heap of bindings P[r] is a program that demands r 9

30 Example disintegration (step 1) do {delimit; x uniform 0 1; r do {y uniform 0 1; return (y/x, (x, y))}; t return (fst r); 9

31 Example disintegration (step 2) do {delimit; x uniform 0 1; r do {y uniform 0 1; return (y/x, (x, y))}; t return (fst r); 10

32 Example disintegration (step 2) do {delimit; x uniform 0 1; r do {y uniform 0 1; return (y/x, (x, y))}; t return (fst r); bind do {delimit; x uniform 0 1; y uniform 0 1; r return (y/x, (x, y)); t return (fst r); do {delimit; H; r do {x m 1 ; m 2 }; P[r]} bind do {delimit; H; x m 1 ; r m 2 ; P[r]} 10

33 Example disintegration (step 3) do {delimit; x uniform 0 1; y uniform 0 1; r return (y/x, (x, y)); t return (fst r); return do {delimit; x uniform 0 1; y uniform 0 1; r return (y/x, (x, y)); t return (fst (y/x, (x, y))); do {delimit; H; r return v; P[r]} return do {delimit; H; r return v; P[v]} v is head normal form for lazy evaluation 11

34 Example disintegration (step 4) do {delimit; x uniform 0 1; y uniform 0 1; r return (y/x, (x, y)); t return (fst (y/x, (x, y))); fst do {delimit; x uniform 0 1; y uniform 0 1; r return (y/x, (x, y)); t return (y/x); do {delimit; P[fst (e 1, e 2 )]} fst do {delimit; P[e 1 ]} 12

35 Example disintegration (step 5) do { delimit; x uniform 0 1; y uniform 0 1; r return (y/x, (x, y)); t return (y/x); Fubini do {x uniform 0 1; delimit; x return x ; y uniform 0 1; r return (y/x, (x, y)); t return (y/x); do { delimit; H; x uniform v 1 v 2 ; P[x]} Fubini do {x uniform v 1 v 2 ; delimit; H; x return x ; P[x]} 13

36 Example disintegration (step 6) do {x uniform 0 1; delimit; x return x ; y uniform 0 1; r return (y/x, (x, y)); t return (y/x); return do {x uniform 0 1; delimit; x return x ; y uniform 0 1; r return (y/x, (x, y)); t return (y/x ); do {delimit; H; r return v; P[r]} return do {delimit; H; r return v; P[v]} 14

37 Example disintegration (step 7) do {x uniform 0 1; delimit; x return x ; y uniform 0 1; r return (y/x, (x, y)); t return (y/x ); un/ do {x uniform 0 1; delimit; z uniform (0/x ) (1/x ); y return (z x ); x return x ; r return (y/x, (x, y)); t return (z); do {delimit; H 1 ; y uniform v 1 v 2 ; H 2 ; t return E[y/v]; m} un/ do {delimit; z uniform (v 1 /v) (v 2 /v); y return (z v); H 1 ; H 2 ; t return E[z]; m} 15

38 Example disintegration (step 8) do {x uniform 0 1; delimit; z uniform (0/x ) (1/x ); y return (z x ); x return x ; r return (y/x, (x, y)); t return z; do {delimit; H 1 ; z m 1 ; H 2 ; t return z; m 2 } unreturn do {x uniform 0 1; delimit; t uniform (0/x ) (1/x ); z return t; y return (z x ); x return x ; r return (y/x, (x, y)); unreturn do {delimit; H 1 ; t m 1 ; z return t; H 2 ; m 2 } 16

39 Example disintegration (step 9) done do { x uniform 0 1; delimit; t uniform (0/x ) (1/x ); z return t; y return (z x ); x return x ; r return (y/x, (x, y)); do {t lebesgue; x uniform 0 1; factor if t is between 0/x and 1/x then x else 0; z return t; y return (z x ); x return x ; r return (y/x, (x, y)); 17

40 Summary Disintegration is useful: defines conditional distributions allows an uncountable space of observations Disintegrator is useful: generates steps by operational semantics justifies steps by denotational semantics resembles lazy evaluator + change of variables leaves verification conditions behind for exchanging integrals 18