Scala概率编程语言Figaro教程：构建与推理

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图灵(Tutorial)是关于Scala概率编程语言Figaro的详尽指南。它由Avi Pfeffer、Brian Ruttenberg、Michael Howard等人合作，由Charles River Analytics提供，旨在帮助用户充分利用Scala的强大功能构建和解决概率建模问题。本教程从入门介绍开始，逐步深入到安装配置、编写Hello World示例，以及如何构建复杂的模型。在第一章中，读者会了解什么是Figaro，它作为一个概率编程框架，如何通过Scala语言简洁地表达概率逻辑，使开发者能够专注于业务模型的构建。章节二介绍了基础安装过程，确保读者能够在本地环境中运行和实验。接下来的部分主要围绕图灵的核心概念展开： 1. **元素与原子元素**：图灵提供了基本的元素类型，如随机变量（Atomics）作为概率编程的基本单元，它们可以表示离散或连续的概率分布。 2. **复合元素与链**：图灵支持组合元素，如联合、条件和贝叶斯网络结构（Chain），这些有助于构建复杂的概率关系网络。 3. **apply和过程与容器**：讲解了如何通过apply函数执行图灵程序，以及如何组织代码，如定义过程和容器来管理和重用计算逻辑。 4. **创建模型**：章节详细介绍了如何使用Figaro创建模型，包括基本模型的实例化、条件和约束、类与实例、可变字段、多个宇宙（用于处理不确定性）、命名、引用和多值引用，以及聚合概念。 5. **推理机制**：图灵提供了多种推理方法，如计算概率范围、证据断言、精确的变量消除算法、近似信念传播、懒惰因子分解、重要性采样、Metropolis-Hastings Markov Chain Monte Carlo (MCMC)算法，以及Gibbs Sampling MCMC，这些都是进行统计推断的关键技术。每个推理方法都有相应的步骤和调试技巧，如定义Metropolis-Hastings的提案策略和调试方法，以确保模型正确性和性能。这个教程是一个全面的资源，不仅适合初学者快速上手，也适合经验丰富的开发者深入了解图灵语言及其在实际概率建模中的应用。通过学习，读者将能够运用Figaro在各种场景下进行概率分析和决策支持。

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C R E AT I N G M O D E L S

The previous section described the basic building blocks of Figaro

models. Out of these building blocks, a wide variety of models can

be created. This section describes how to build a range of models.

4.1 basic models

One of the ﬁrst things you can do with an element is to assign it to a

Scala value:

val burglary = Flip(0.01)

val in Figaro

represents an

immutable value.

When a thing is

assigned to a val,

data inside the thing

can change but the

reference stored in

the val is constant

A val represents a ﬁeld (in this case burglary) that takes on an im-

mutable value (in this case the element Flip(0.01)). A ﬁeld is not a

variable; its value cannot be changed (Note that the scala assignment

of the ﬁeld burglary cannot change, but the value of the Figaro ele-

ment that is assigned to it, Flip(0.01), can change). You can use the

value of a ﬁeld by referring to its name:

val alarm = If(burglary, Flip(0.9), Flip(0.1))

Recall that an element deﬁnes a process that probabilistically pro-

duces a value. If an element is referred to multiple times, it must

produce the same value everywhere it appears. Consider:

val x = Flip(0.5)

val y = x === x

Although we don’t know the value, x must produce the same value

on both sides of the equality test. Therefore, y produces the value true

with probability 1.0. In contrast, in:

val y = Flip(0.5) === Flip(0.5)

the left and right hand sides are distinct elements (each call pro-

duces a new Flip), so they need not produce the same value. There-

fore, y will produce true with probability 0.5.

With the tools we have deﬁned so far, we can easily create a Bayesian

network. In the following code, CPD is a library element (based on

Chain) that makes it easy to deﬁne conditional probability distribu-

tions:

4.2 conditions and constraints 15

4.2 conditions and constraints

So far, we have described models that generate the values of elements.

It is also possible to inﬂuence the values of elements by imposing

conditions or constraints on them.

A condition represents something the value of the element must sat-

isfy. Only values that satisfy the condition are possible. Every element

has a condition, which is a function from a value of the element to

a Boolean. If the element is of type Element[T], the condition is of

type T => Boolean. Conditions can have multiple purposes. One is

to assert evidence, by specifying something that is known about an

element. Alternatively, a condition can specify a structural property

of a model, for example, that only one of two teams playing a game

can be the winner.

The default condition of an element returns true for all values. The

condition can be changed using setCondition:

val x1 = Select(0.1 -> 1, 0.2 -> 2, 0.3 -> 3, 0.4 -> 4)

x1.setCondition((i: Int) => i == 1 || i == 4)

which says that x1 must have value 1 or 4. We can add a condition

on top of existing conditions using the addCondition method. For

example, the following code says that not only must x1 equal 1 or 4,

it must also be odd:

x1.addCondition((i: Int) => i % 2 == 1)

The observe method provides an easy way to specify a condition

that only allows a single value. For example, to specify that x1 must

have the value 2, we can use:

x1.observe(2)

Note that using observe will remove all previous conditions on an

element.

A constraint provides a way to specify a potential or weighting over

an element. It is a function from a value of the element to a Double,

so if the element has type Element[T], the constraint is of type T =>

Double.

Constraint values should always be non-negative. Also, although

it is not strictly enforced, we recommend that constraint values be

at most 1. Some algorithms compute upper bounds on probabilities

and need to assume an upper bound on constraint values. An upper

bound of 1 is assumed, so if the actual value can be higher, these

algorithms will be incorrect. For algorithms that don’t compute upper

bounds in this way, it doesn’t matter. Currently, the only algorithms

that compute upper bounds like this are the lazy factored inference

4.2 conditions and constraints 16

algorithms. If you have a constraint value greater than 1, a warning

will be issued.

Constraints serve multiple purposes in Figaro. One is to specify

soft evidence on an element. For example, if in the above Bayesian

network we think we heard John call but we’re not sure, we might

introduce the constraint:

johnCalls.setConstraint((b: Boolean) =>

if (b) 1.0; else 0.1)

This line will have the effect of making John calling 10 times more

likely than not, all else being equal. Another purpose of constraints is

to deﬁne some probabilistic relationships conveniently that are more

difﬁcult to express without them. Consider the following example, in

which we are modeling the process of ﬁrms bidding for a contract

and one of them being selected as the winner.

This example is

found in Firms.scala

import com.cra.figaro.language.

import com.cra.figaro.library.atomic.

import com.cra.figaro.library.compound.If

class Firm {

val efficient = Flip(0.3)

val bid = If(efficient, continuous.Uniform(5, 15),

continuous.Uniform(10, 20))

}

val firms = Array.fill(20)(new Firm)

val winner = discrete.Uniform(firms:

)

val winningBid = Chain(winner, (f: Firm) => f.bid)

winningBid.setConstraint((d: Double) => 20 − d)

This example shows some new Scala features. First, we have a class

deﬁnition (the Firm class). A class creates a type that can be instanti-

ated to create instances. The Firm class has two ﬁelds, efficient and

bid. Note that bid makes use of continuous.Uniform. This is the con-

tinuous uniform element deﬁned in the library.atomic.continuous

package, but we did not import the members of this package, only the

members of the library.atomic package. The reason we did things

this way is that later in the example, we use the discrete uniform, and

we want to be explicit about which uniform element we mean at each

point.

Once we have deﬁned the Firm class, we create an array named

firms consisting of 20 instances of Firm.Array.fill(20)(new Firm)

creates an array ﬁlled with the result of 20 different invocations of new

Firm, each of which creates a separate instance of Firm (and separate

Figaro elements in each class). We then deﬁne the winner to be one

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