# Language Syntax

The programming language of Anglican is a subset of Clojure, extended with a few special forms that make it a probabilistic programming language. These forms are sample for drawing a samples from distributions and observe for conditioning on data. There are other special forms — mem, store, and retrieve — which make writing probabilistic programs easier.

The following documentation is quite terse because Anglican is, to a large extent, intentionally syntactially indistiguishable from Clojure.
Clojure reference materials, obtainable from the web via standard search procedures, are as essential to programming in Anglican as is this Anglican language documentation. The key to Anglican knowing Clojure and understanding defquery, the interface between Clojure and Anglican.

This interface is meant to be as transparent as possible and for as much Clojure functionality to work inside defquery as possible.

In general the documentation that follows indicates functionality relative to Clojure. For instance, the absence of an explicit statement of existence means that the Clojure language feature probably isn’t supported in Anglican.

## A First Example Program

Anglican programs reside in Clojure source code modules, and are delimited by defquery (a macro). In order to enable the Anglican language in a Clojure module, at the minimum namespaces anglican.runtime and anglican.emit must be used. A simple way to do this is to write

(ns example
(:use [anglican emit runtime]))


in the beginning of a Clojure module, for example ‘example.clj’. Clojure namespacing is notably complex; arguably the best strategy for writing a namespace that includes Anglican functionality is to copy namespace declarations from the provided examples.

The rest of the module may contain one or more Anglican programs, aka queries. Queries are delimited by the keyword defquery, followed by the query name and the program text. If there is only one program in a namespace, it customarily bears the same name as the namespace:

(defquery example
(let [bet (sample (beta 5 3))]
(observe (flip bet) true)
(> bet 0.7)))


defquery assigns a value to example that allows it to be passed as the argument to a doquery. doquery uses inference algorithms to produce lazy sequence of weighted samples that characterize the conditional distribution of the value of the expression in the tail position of the defquery, here (> bet 0.7).

Syntactically defquery denotes a joint distribution, observe’s denote which random variables’ values are known, and the value of the last, “return” expression, is the variable whose conditional distribution is of interest.

What follows is a description of what program text can go inside defquery, i.e. a description of the Anglican language. Note that while the source code inside defquery intentionally bears a great deal of resemblance to Clojure, it is not Clojure code, it is Anglican code.

## Core Anglican syntax

The Anglican language is a subset of Clojure. Within defquery, let, if, when, cond, case, and, or, fn forms are supported (others may be in the future but are not now). In let bindings and fn argument lists, vector destructuring (but not hash map destructuring) is supported. Compound literals for vectors, hash maps, and sets are supported just like in Clojure.

## recur

Anglican is stackless, therefore recur is unnecessary, no recursive call can lead to stack overflow; Recursive calls to functions should be used instead. However, loop/recur is provided for convenience as a way to express loops. recur outside of loop will lead to unpredictable behaviour and hard-to-catch errors.

## Core library

All of Clojure’s core library, except for higher-order functions (functions that accept other functions as arguments) is available in Anglican. In addition, the following higher-order functions are implemented: map, reduce, filter, some, repeatedly, comp, partial.

## Definitions outside of defquery

The border between Clojure and Anglican is subtle and usually will pose no problem to most programmers, however, some confusion can arise from the fact that Anglican programs are macro compiled into CPS-style Clojure functions. This means that some wrapping of “native” Clojure functions needs to happen in order to use them in Anglican. Errors arising due to misunderstanding this boundary crop up in the form of “wrong number of argument” exceptions. Carefully following the guidance in this section will resolve most if not all such difficulties.

Data variables may be defined outside of defquery using def and used inside defquery. Anglican functions outside of defquery may be defined using defm (with the same syntax as defn, albeit with a single arity only). Their bodies may use the same subset of Clojure as defquery, as well as probabilistic and state access forms. defm-defined functions can be called from Anglican without restrictions.

Functions defined outside of defquery using defn may use the full Clojure syntax but no Anglican extensions, and must be declared primitive using with-primitive-procedures:

(with-primitive-procedures [name ...]
body)


Where name ... is the list of primitive procedures. The names can be namespace-qualified, but will be seen unqualified in the lexical scope of the form. For example,

(with-primitive-procedures [clojure.string/capitalize]
(defquery foo
(capitalize "hello")))


Denotes the dirac distribution over Hello (capitalized).

## Probabilistic forms

In Anglican there are two probabilistic forms: sample and observe.

• (sample distribution) returns a sample from a distribution.
• (observe distribution value) returns value; critical produces conditioning side-effect. It does this by adding the value of (observe* distribution value) (see below) to the log probability of the trace.

## State access and modification

Functions can be memoized using mem, which accepts a function object as its argument. If the argument is a named fn form, self-recursive calls will call the memoized version of the function. For example, every fact call in the following code

(defquery fact
(let [fact (mem (fn fact [n]
(if (= n 1) 1
* n (fact (- n 1)))))]
[(fact 1) (fact 2) (fact 3) (fact 4)]))


will reuse previous computation.

Values can be stored in the state using store, values stored during the same run of the program can be retrieved using retrieve. The syntax is

• (store key ... value) stores value at key ... in the state.
• (retrieve key ...) retrieves and returns the value stored at key .... key ... can be a sequence of any length.

For example:

(defquery customer
(store :customer 4 :age 18)
(retrieve :customer 4 :age))


will return be 18 in :result.

## Distributions

Distributions are Clojure implementations of a distribution protocol, consisting of two methods:

(sample* distribution) accepts a distribution instance and returns a sample from the distribution. The Anglican sample uses this method to generate random variable values.

(observe* distribution value) accepts a distribution and a value and returns log probability of the value given the distribution. Anglican inference backends stop at observe statements and often use the return value from calling observe* with the same original arguments as the observe call to effect conditioning.

The core runtime library provides the following distribution constructors which can be used either in Clojure or Anglican, remembering the difference between sample and sample*, and, observe and observe*:

(bernoulli p) constructs a single binomial trial. Calling sample on the returned distribution instance generates 1 with probability p and 0 with probability 1-p.

(beta a b) constructs a Beta distribution with pseudocounts a and b. Calling sample on the returned distribution instance generates a double on interval [0,1).

(binomial n p) constructs a Binomial distribution with success probability p and number of trials n. Calling sample on the returned distribution instance generates a long on the interval [0 ... n].

(categorical pairs) constructs a categorical distribution parameterized by a list of pairs (val p). Calling sample on the returned distribution instance generates val-k with probability p-k.

(dirichlet [alpha-1 … alpha-K]) constructs a Dirichlet distribution parameterized by a vector of pseudocounts alpha. Calling sample on the returned distribution instance generates a vector of probabilities prob such that (sum prob) = 1.0 and (count prob) = (count alpha).

(discrete p) constructs a discrete distribution parameterized by a list probabilities p. Calling sample on the returned distribution instance generates a long in the range [0 ... K-1], with K = (count p). The result k is returned with probability (nth p k).

(exponential l) constructs an exponential distribution with with rate parameter l. Calling sample on the returned distribution instance generates a double in the domain [0, Inf).

(flip p) constructs a single binomial trial. Calling sample on the returned distribution instance generates true with probability p and false with probability 1-p.

(gamma a b) constructs a Gamma distribution with shape a and rate b. Calling sample on the returned distribution instance generates a double on the domain (0, Inf).

(mvn mean cov) constructs a Multivariate normal distribution with mean mean and covariance matrix cov. Calling sample on the returned distribution instance generates a double vector of the same size as mean.

(normal mean std) constructs a normal distribution with mean mean and standard deviation std.

(poisson l) constructs a Poisson distribution with rate l. Calling sample on the returned distribution instance generates a non-negative long.

(uniform-continuous min max) constructs a uniform continuous distribution. Calling sample on the returned distribution instance generates a double in the domain [min, max].

(uniform-discrete min max) constructs a uniform discrete distribution. Calling sample on the returned distribution instance generates a long from the range [min ... max-1].

(wishart n V) constructs a Wishart distribution with n degrees of freedom and scale matrix V. Calling sample on the returned distribution instance generates a matrix of double of the same size as V.

In addition, so-called random processes are provided by the runtime, including CRP (Chinese Restaurant Process), DP (Dirichlet Process), and GP (Gaussian Process). Random processes implement the random-process protocol, consisting of two methods:

(produce process) accepts a process instance and returns a distribution object (which can be passed as a parameter to observe and sample) corresponding to the current state of the process.

(absorb process value) updates the process by incorporating a value sampled or observed from the distribution produced by the processes.

The following random process constructors are included into the core runtime library:

(CRP alpha) is a Chinese restaurant process with concentration alpha.

(DP alpha H) is a Dirichlet process with concentration alpha over base distribution H.

(GP m k) is a Gaussian process with mean function m and covariance function k.

Other distributions and processes can be defined by the user. The definition can be placed into Clojure modules containing Anglican programs. A user-defined distribution is specified using defdist:

(defdist dirac
"Dirac distribution"
[x]  ; distribution parameters
[]   ; auxiliary bindings
(sample* [this] x)
(observe* [this value] (if (= x value) 0.0 (- (/ 1.0 0.0)))))


Similarly, a user-defined random process is specified using defproc:

(defproc DSD
"discrete-symmetric-dirichlet process"
[alpha N]                                ; process parameters
[counts (vec (repeat N (double alpha)))] ; auxiliary bindings
(produce [this] (discrete counts))
(absorb [this sample]
(DSD alpha N (update-in counts [sample] + 1.))))


Constructors of user-defined distributions and processes must be declared primitive using with-primitive-procedures.