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https://github.com/SHoltzen/dice

Exact inference for discrete probabilistic programs. (Research code, more documentation and ergonomics to come)
https://github.com/SHoltzen/dice

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Exact inference for discrete probabilistic programs. (Research code, more documentation and ergonomics to come)

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`dice` is a probabilistic programming language focused on *fast exact inference*

for *discrete* probabilistic programs. For more information for how `dice` works

see the research article [here](https://arxiv.org/abs/2005.09089). To cite

`dice`, please use:

```

@article{holtzen2020dice,

  title={Scaling Exact Inference for Discrete Probabilistic Programs},

  author={Holtzen, Steven and {Van den Broeck}, Guy and Millstein, Todd},

  journal={Proc. ACM Program. Lang. (OOPSLA)},

  publisher={ACM},

  doi={https://doi.org/10.1145/342820},

  year={2020}

}

```



# Installation



## Docker



A [docker](https://www.docker.com/) image is available, and can be installed

with:



```

docker pull sholtzen/dice

```



## Building From Source



The following steps set up the environment for building `dice`.

First install `opam` (version 2.0 or higher) following the instructions

[here](https://opam.ocaml.org/doc/Install.html).

Then, install `rust` following the commands [here](https://rustup.rs/).

Then, run the following in your

terminal:



```

opam init                  # must be performed before installing opam packages

opam switch create 4.09.0  # switch to use OCaml version 4.09

eval `opam config env`     # optional: add this line to your .bashrc

source $HOME/.cargo/env    # set up rust environment

git submodule update --init --recursive    # populate the rsdd subdirectory

opam install . --deps-only # install dependencies

```



### Building 



First follow the steps for installation. Then, the following build commands are

available:



* `dune build`: builds the project from source in the current directory.

* `dune exec dice`: runs the `dice` executable.

* `dune test`: runs the test suite

* `dune exec dicebench`: runs the benchmark suite.



# Quick Start 



We will start with a very simple example. Imagine you have two (unfair) coins

labeled `a` and `b`. Coin `a` has a 30% probability of landing on heads, and

coin `b` has a 80% chance of landing on heads. You flip both coins and observe

that one of them lands heads-side up. What is the probability that 

coin `a` landed heads-side up?



We can encode this scenario in `dice` as the following program:



```

let a = flip 0.3 in 

let b = flip 0.8 in

let tmp = observe a || b in 

a

```



The syntax of `dice` is similar to OCaml. Breaking down the elements of this

program:



* The expression `let x = e1 in e2` creates a local variable `x` with value

  specified by `e1` and makes it available inside of `e2`.

* The expression `flip 0.3` is true with probability 0.3 and false with

  probability 0.8. This is how we model our coin flips: a value of true 

  represents a coin landing heads-side up in this case.

* The expression `observe a || b` conditions either `a` or `b` to be true. This

  expression returns `true`. `dice` supports logical conjunction (`||`),

  conjunction (`&&`), equality (`<=>`), negation (`!`), and exclusive-or (`^`).

* The program returns `a`.



You can find this program in `resources/example.dice`, and then you can run it

by using the `dice` executable:



```

> dice resources/example.dice

Value	Probability

true	0.348837

false	0.651163

```



This output shows that `a` has a 34.8837% chance of landing on heads.



## Optimizations



The Dice compiler has the following built-in optimizations and alternative run-time modes 

that are activated with the following flags:



* `-determinism`: replaces deterministic probabilistic choices with non-random choices (i.e., `flip 1.0` becomes `true`). It is recommended that this flag be enabled for most cases.

* `-eager-eval`: changes the compilation order to avoid substitution during compilation. Can perform faster than the default compilation order on certain cases.

* `-flip-lifting`: removes redundant `flip` expressions from certain classes of programs -- can increase performance.



## Datatypes

In addition to Booleans, `dice` supports integers, tuples, and lists.



### Tuples



Tuples are pairs of values. The following simple example shows tuples being

used:



```

let a = (flip 0.3, (flip 0.8, false)) in

fst (snd a)

```



Breaking this program down:



* `(flip 0.3, (flip 0.8, false))` creates a tuple.

* `snd e` and `fst e` access the first and second element of `e` respectively.



Running this program:



```

> dice resources/tuple-ex.dice

Value	Probability

true	0.800000

false	0.200000

```



### Unsigned Integers



`dice` supports distributions over unsigned integers. An example program:



```

let x = discrete(0.4, 0.1, 0.5) in 

let y = int(2, 1) in 

x + y

```



Breaking this program down:



* `discrete(0.4, 0.1, 0.5)` creates a random integer that is 0 with probability 0.4, 

   1 with probability 0.1, and 2 with probability 0.3.

* `int(2, 1)` creates a 2-bit integer constant with value 1. All integer

  constants in `dice` must specify their size.

* `x + y` adds `x` and `y` together. All integer operations in `dice` are

  performed modulo the size (i.e., `x + y` is implicitly modulo 4 in this

  case). `dice` supports the following integer operations: `+`, `*`, `/`, `-`, 

  `==`, `!=`, `<`, `<=`, `>`, `>=`.



Running this program:



```

> dice resources/int-ex.dice

Value	Probability

0	0.

1	0.4

2	0.1

3	0.5

```



Various distributions over integers have their own syntax. For instance,

* `uniform(3, 2, 6)` creates a random 3-bit integer, containing a uniform distribution

  over the integers 2, 3, 4, 5.

* `binomial(3, 4, 0.5)` creates a random 3-bit integer, containing a binomial distribution

  with parameters `n=4`, `p=0.5`



### Lists



`dice` supports distributions over lists, possibly of different lengths.



```

let xs = [flip 0.2, flip 0.4] in

if flip 0.5 then (head xs) :: xs else tail xs

```



Breaking this program down:



- `[flip 0.2, flip 0.4]` creates a list of Booleans with two elements.

- `head xs` returns the first element of `xs` and `tail xs` returns a list of everything after the first element.

- `x :: xs` returns a list with `x` added to the front of `xs`.



Running this program:



```

> dice -max-list-length 3 resources/list-ex.dice

Value   Probability

[]      0.

[true]  0.2

[false] 0.3

[true, true]    0.

[true, false]   0.

[false, true]   0.

[false, false]  0.

[true, true, true]      0.04

[true, true, false]     0.06

[true, false, true]     0.

[true, false, false]    0.

[false, true, true]     0.

[false, true, false]    0.

[false, false, true]    0.16

[false, false, false]   0.24

```



## Functions



`dice` supports functions for reusing code. A key feature of `dice` is that 

functions are *compiled once and then reused* during inference.



A simple example program:



```

fun conjoinall(a: bool, b: (bool, bool)) {

  a && (fst b) && (snd b)

}

conjoinall(flip 0.5, (flip 0.1, true))

```



Breaking this program down:



* A function is declared using the syntax `fun name(arg1: type1, arg2: type2, ...) { body }`. 

* A program starts by listing all of its functions. Then, the program has a main body after

  the functions that is run when the program is executed. In this program, the main 

  body is `conjoinall(flip 0.5, (flip 0.1, true))`.

* Right now recursion is not supported.

* Functions do not have `return` statements; they simply return whatever the

  last expression that evaluated returns.



Result of running this program:



```

Value	Probability

true	0.050000

false	0.950000

```



## Caesar Cipher



Here is a more complicated example that shows how to use many `dice` features

together to model a complicated problem.

We will decrypt text that was

encrypted using a [Caesar cipher](http://practicalcryptography.com/ciphers/caesar-cipher/). We can decrypt

text that was encrypted using a Caesar cipher by [frequency analysis](https://en.wikipedia.org/wiki/Frequency_analysis):

using our knowledge of the rate at which English characters are typically in order to 

infer what the underlying key must be.



Consider the following simplified scenario. Suppose we have a 4-letter language called `FooLang`

consisting of the letters `A`, `B`, `C`, and `D`. Suppose that for this language,

the letter `A` is used 50% of the time when spelling a word, `B` is used 25% of the

time, and `C` and `D` are both used 12.5% of the time.



Now, we want to infer the most likely key given after seeing some encrypted

text, using knowledge of the underlying frequency of letter usage. Initially we

assume that all keys are equally likely. Then, we observe some encrypted text:

say the string `CCCC`. Intuitively, the most likely key should be 2: since `A`

is the most common letter, the string `CCCC` is most likely the encrypted string

`AAAA`. Let's use `dice` to model this.



The following program models this scenario in `dice`:



```

fun sendChar(key: int(2), observation: int(2)) {

  let gen = discrete(0.5, 0.25, 0.125, 0.125) in    // sample a FooLang character

  let enc = key + gen in                            // encrypt the character

  observe observation == enc

}

// sample a uniform random key: A=0, B=1, C=2, D=3

let key = discrete(0.25, 0.25, 0.25, 0.25) in

// observe the ciphertext CCCC

let tmp = sendChar(key, int(2, 2)) in

let tmp = sendChar(key, int(2, 2)) in

let tmp = sendChar(key, int(2, 2)) in

let tmp = sendChar(key, int(2, 2)) in

key

```



Now we break this down. First we look at the `sendChar` function:



* It takes two arguments: `key`, which is the underlying secret encryption key, and

  `observation`, which is the observed ciphertext.

* The characters `A,B,C,D` are encoded as integers.

* A random character `gen` is sampled according to the underlying distribution

  of characters in `FooLang`.

* Then, `gen` is encrypted by adding the key (remember, addition occurs modulo 4

  here).

* Then, ciphertext character is observed to be equal to the encrypted character.



Next, in the main program body, we sample a uniform random key and encrypt the

string `CCCC`. Running this program:



```

> dice resources/caesar-ex.dice

Value	Probability

0	0.003650

1	0.058394

2	0.934307

3	0.003650

```



This matches our intuition that `2` is the most likely key.



## More Examples



More example `dice` programs can be found in the source directories:



* The `test/Test.ml` file contains many test case programs.

* The `benchmarks/` directory contains example programs that are run during

  benchmarks.



# Syntax



The parser for `dice` is written in [menhir](http://gallium.inria.fr/~fpottier/menhir/) and can be found in `lib/Parser.mly`. The

complete syntax for `dice` in is:



```

ident := ['a'-'z' 'A'-'Z' '_'] ['a'-'z' 'A'-'Z' '0'-'9' '_']*

binop := +, -, *, /, <, <=, >, >=, ==, !=, &&, ||, <=>, ^, ::



expr := 

   (expr)

   | ident

   | true

   | false

   | int (size, value)

   | discrete(list_of_probabilities) 

   | uniform(size, start, stop)

   | binomial(size, n, p)

   | expr  expr

   | (expr, expr)

   | fst expr

   | snd expr

   | ! expr

   | flip probability

   | observe expr

   | if expr then expr else expr

   | let ident = expr in expr

   | [ expr (, expr)* ]

   | [] : type

   | head expr

   | tail expr

   | length expr



type := bool | (type, type) | int(size) | list(type)

arg := ident: type

function := fun name(arg1, ...) { expr }



program := expr 

        | function program

```