Expression

abstract class Expression<Value> < DelayExpression

Lazy expression.

• Value: Value type.

• x: Fixed value, or nil to evaluate.

The simplest use case of lazy expressions is to simply construct them and then call value() to evaluate them. Evaluations are memoized, so that subsequent calls to value() do not require re-evaluation of the expression, they simply return the memoized value. Objects may be reused in multiple expressions, e.g. two expressions with a common subexpression can share that subexpression.

More elaborate use cases include computing gradients and applying Markov kernels. For this purpose operations are separated into closed world and open world operations. The world is that of lazy expressions.

1. The quintessential open world operation is value(). The evaluated value may be used in any context, including eager-mode expressions and I/O operations. Consequently, the evaluated value of the expression must be made constant to avoid the inconsistency (and possibly incorrectness) of different evaluated values being used in different contexts.

2. Closed world operations include get(), pilot(), grad(), and move(). The caller of such member functions guarantees that the value is only used in the world of lazy expressions. The evaluated value of the expression can remain variable and still produce consistent results---it may, for example, be moved with a Markov kernel.

To keep track of whether an evaluated value is open world (constant) or closed world (variable), expressions are stateful:

• An Expression is initially considered variable.
• Once an open world operation is performed on the Expression, it and any subexpressions are considered constant.

This particularly affects the grad() and move() operations:

• grad() will compute gradients with respect to any variable Random objects but not constant objects, and similarly,
• move() will apply a Markov kernel to any variable Random objects but not constant objects.

Attention

Unless you are working on something like a Markov kernel, or other closed-world feature, you want to use value(), not get() or pilot(). Doing otherwise may risk correctness. Consider:

if x.value() >= 0.0 {
doThis();
} else {
doThat();
}


This is correct usage. Using get() or pilot() instead of value() here may result in a subsequent move on the value of x (by e.g. MoveParticleFilter) that switches the sign of the evaluated value, but does not change the branch taken, resulting in an invalid trace and incorrect results. This is because the evaluated value of the expression is not being used in the world of lazy expressions here: the if statement is not lazily evaluated.

Assignments

Name Description
Value Value assignment.

Member Variables

Name Description
x:Value? Memoized value.

Member Functions

Name Description
hasValue Does this have a value?
setValue Set the value.
setPilot Set the pilot value.
distribution If this is a Random, get the distribution associated with it, if any, otherwise nil.
value Evaluate, open world.
pilot Evaluate, closed world, before a call to grad().
get Evaluate, closed world.
move Move and re-evaluate.
compare Evaluate log-ratio of proposal probability densities after move.
count Make constant, as though calling value(), but without re-evaluating the expression.
constant Update counts, as though calling pilot(), but without re-evaluating the expression.

Member Function Details

compare

final function compare(gen:Integer, x:DelayExpression, κ:Kernel) -> Real

Evaluate log-ratio of proposal probability densities after move.

• gen: Generation limit.
• x: Starting state, $x$.
• κ: Markov kernel, $\kappa(\mathrm{d}x^\prime \mid x)$.

Returns: The log-ratio of proposal probability densities after move, $\log q(x \mid x^\prime) - \log q(x^\prime \mid x)$. This object is considered to represent the proposed state $x^\prime$.

The generation limit gen works to truncate the recursion, as for grad().

constant

function constant()

Update counts, as though calling pilot(), but without re-evaluating the expression.

count

function count(gen:Integer)

Make constant, as though calling value(), but without re-evaluating the expression.

distribution

function distribution() -> Distribution<Value>?

If this is a Random, get the distribution associated with it, if any, otherwise nil.

get

final function get() -> Value

Evaluate, closed world.

Returns: The evaluated value of the expression.

get() is similar to pilot() except that it does not update counts in anticipation of a following call too grad(). It can be used at any time to retrieve the evaluated value of the expression, evaluating it if necessary, without disrupting the state.

Evaluate gradients. Gradients are computed with respect to all variables (Random objects in the pilot or gradient state).

• Gradient: Gradient type. Must be one of Real, Real[_] or Real[_,_].

• gen: Generation limit.

• d: Upstream gradient. For an initial call, this should be the unit for the given type, e.g. 1.0, a vector of ones, or the identity matrix.

The generation limit gen is used to truncate the recursion. Any expressions that have been assigned a generation number less of than gen (usually at the time they are evaluated with pilot()), are considered constant for the purposes of gradient evaluation. As the default generation is zero, a value of less than or equal to zero here will not truncate the recursion.

grad() must be called as many times as pilot() was previously called. This is because subexpressions may be shared. The calls to pilot() are used to determine how many times the object occurs as a subexpression. The subsequent calls to grad() accumulate upstream gradients this number of times before recursing. This is an important optimization for many use cases, such as computing gradients through Bayesian updates.

Reverse-mode automatic differentiation is used. The previous calls to pilot() constitute the forward pass, the subsequent calls to grad() constitute the backward pass. If the expression tree encodes

x_n = f(x_0) = (f_n \circ \cdots \circ f_1)(x_0),

and this particular node encodes one of those functions $x_i = f_i(x_{i-1})$, the argument to the function is

\frac{\partial (f_n \circ \cdots \circ f_{i+1})} {\partial x_i}\left(x_i\right),

it computes

\frac{\partial (f_n \circ \cdots \circ f_{i})} {\partial x_{i-1}}\left(x_{i-1}\right),

and passes the result to its child, which encodes $f_{i-1}$, to continue the computation. The variable (Random object) that encodes $x_0$ keeps the final result.

Evaluate gradient for an element of a vector.

• gen: Generation limit.
• i: Element index.

final function grad(gen:Integer, d:Real, i:Integer, j:Integer)

Evaluate gradient for an element of a matrix.

• gen: Generation limit.
• i: Row index.
• j: Column index.

hasValue

function hasValue() -> Boolean

Does this have a value?

Returns: true if a value has been evaluated with value(), pilot() or get(), false otherwise.

Note that this differs from isConstant(), which will only return true for a value that has been evaluated with value().

move

final function move(gen:Integer, κ:Kernel) -> Value

Move and re-evaluate. Variables are updated with a Markov kernel, possibly using gradient information, and the expression re-evaluated with these new values.

• gen: Generation limit.
• κ: Markov kernel, $\kappa(\mathrm{d}x^\prime \mid x)$.

Returns: The evaluated value of the expression.

The generation limit gen works to truncate the recursion, as for grad().

pilot

final function pilot(gen:Integer) -> Value

Evaluate, closed world, before a call to grad().

• gen: Generation number.

Returns: The evaluated value of the expression.

If the expression has not yet been evaluated (excepting a previous call to get()), the generation number gen is assigned to it for future use.

pilot() may be called multiple times, which accumulates a count. If the intent is to compute gradients, grad() must subsequently be called the same number of times. This is because subexpressions may be shared. The calls to pilot() are used to determine how many times the object occurs as a subexpression. The subsequent calls to grad() accumulate upstream gradients this number of times before recursing. This is an important optimization for many use cases, such as computing gradients through Bayesian updates.

setPilot

function setPilot(x:Value)

Set the pilot value.

• x: Pilot value.

setValue

function setValue(x:Value)

Set the value.

• x: Value.

value

final function value() -> Value

Evaluate, open world.

Returns: The evaluated value of the expression.

An expression is considered constant once value() has been called on it. All subexpressions are necessarily also then constant, including any Random objects that occur, which are no longer considered variables for the purpose of grad() and move().

If this is not the intended behavior, consider get() or pilot().