Bayesian Network | AI Playgrounds

Network:

Inference:

Drag nodes to rearrange. Edges flow parent → child. Green = clamped TRUE, red = clamped FALSE.

Evidence: click T / F / ? per variable

Quick scenarios:

Posterior: P(variable = TRUE | evidence)

🔗 d-separation checker (are X and Y independent given evidence?)

Pick two variables and click "Check". The algorithm runs Bayes-Ball on the DAG using the current evidence as the conditioning set. Two variables are independent (d-separated) when no active path connects them.

X: Y:

(select X and Y, then click Check)

🎯 Try these scenarios first

Pick a scenario, predict what should happen, then apply it and compare the result with your prediction.

Scenario 1: Start from the prior probabilities

Core question: What does the network believe before any evidence is entered?

Run and watch: Clear all evidence and read the burglary, earthquake, alarm, and call probabilities.

Predict first: Predict which events should be rarest and which probability should be largest.

Explain afterward: Explain how the conditional probability tables create the calm baseline.

Scenario 2: One phone call changes burglary belief

Core question: How can evidence travel against the direction of an arrow?

Run and watch: Set John calls to true, then compare the burglary posterior with its prior probability.

Predict first: Predict whether the burglary probability will rise a little or by many times.

Explain afterward: Explain why a possible effect can provide diagnostic evidence about its cause.

Scenario 3: Two independent reports strengthen the evidence

Core question: Why do two calls change belief more than one call?

Run and watch: Set both John and Mary calls to true while leaving the possible causes unobserved.

Predict first: Predict whether the second report will add weak or strong evidence.

Explain afterward: Explain how two conditionally independent reports corroborate the alarm.

Scenario 4: An earthquake explains away a burglary

Core question: How can adding evidence make a hypothesis less likely?

Run and watch: Keep both calls true, then also set earthquake to true and watch burglary probability.

Predict first: Predict the direction of the burglary update before applying the earthquake evidence.

Explain afterward: Explain how the observed earthquake already accounts for the alarm and calls.

Scenario 5: Wet grass has two possible causes

Core question: How does explaining away appear in a different network?

Run and watch: Switch to the sprinkler network, observe wet grass, then add evidence that it rained.

Predict first: Predict what will happen to the sprinkler probability after rain is confirmed.

Explain afterward: Explain why the two causes compete to explain the same wet grass.

✨ Featured comparison

Featured comparison

Turn evidence on one shared effect into a competition between causes. This is the fastest way to see explaining away.

Set evidence on the effect node, then compare what happens when one cause is also observed.

🧭 Visual explanation

Live state to watch

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📚 Lesson tour (5 steps)

Step 1/5: -

📝 Worksheet (3 questions)

Q1. Burglary network. Click "John & Mary both call". Note P(Burglary). Now click "Both call + earthquake". P(Burglary) goes:

Up: a second alarm cause makes a burglary even more likely. Down: the earthquake explains away the alarm, so burglary is less needed. Stays the same: burglary and earthquake are independent, so extra evidence can't change it.

Q2. Switch to Sprinkler network. Set Wet=T. Now ALSO set Rain=T. P(Sprinkler=T) is:

Higher than with wet grass alone: two confirmed causes stack up. Lower than with just wet grass: rain explains the wet grass, so the sprinkler is less needed. Unchanged: once the grass is wet, learning about rain tells you nothing about the sprinkler.

Q3. Burglary network, "Both call" evidence. Switch inference method from Exact to Gibbs sampling with 5000 samples. The posterior for P(Burglary) compared to Exact is:

An exact match: Gibbs sampling computes the same posterior as enumeration. Close but noisy: Gibbs approximates the posterior, and the estimate wobbles a bit each re-sample. Wildly different: sampling methods can't recover this posterior at all.

🧪 Student response packet

Set evidence, compare posteriors, then explain how conditional dependence and explaining away changed the inference.

Predict

Observe

Explain

Transfer

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↓ Read the explanation

A Bayesian network represents a joint probability distribution with a directed acyclic graph. In the classic burglary network, burglary (B) and earthquake (E) can trigger an alarm (A), and John (J) and Mary (M) may call after hearing it. The graph and its conditional probability tables let you ask questions such as P(B | J): the probability of a burglary given that John called.

What the arrows encode

An arrow points from a parent variable to a child whose probability table is conditioned on that parent. Here, the alarm table depends on burglary and earthquake, while each call table depends on the alarm. An arrow may be given a causal interpretation when the model and data justify it, but the mathematical role of the graph is to encode factorization and conditional independence.

The safe rule is the local Markov property: given its parents, a node is conditionally independent of its non-descendants. A missing arrow by itself does not prove that two arbitrary variables are independent; the conclusion depends on the full graph and on which variables are observed.

The explaining-away effect

Select John and Mary both call. The two calls make an alarm more plausible, which raises the posterior probability of burglary. Then add evidence that an earthquake occurred. The burglary probability falls because the earthquake already provides an explanation for the alarm and the calls.

Burglary and earthquake are independent before evidence in this network. After observing their common effect, they become dependent: evidence for one cause can reduce belief in the other. This collider pattern is called explaining away.

Exact inference and Gibbs sampling

Exact inference sums or eliminates the possible values of unobserved variables to normalize the posterior. Enumeration is easy in this five-variable network, but its worst-case work grows exponentially with the number and structure of variables. More efficient exact methods, such as variable elimination or junction-tree algorithms, exploit the graph structure but can still be expensive.

Gibbs sampling approximates the posterior by repeatedly resampling one unobserved variable from its conditional distribution given the current values of the others. More samples usually reduce visible sampling noise, although successive samples are correlated and convergence depends on the model. A finite sample is an estimate, not an exact answer.

What this model leaves out

Larger Bayesian networks may learn their probability tables or even parts of their structure from data. Practical inference can use variable elimination, belief propagation, Monte Carlo methods, or variational approximations. This applet isolates the central idea: a graph specifies how a joint distribution factors, and evidence changes beliefs throughout that structure.

For teachers

Curriculum: Directed graphical models, conditional probability tables, conditional independence, explaining away, and exact versus sampling-based inference.

Pre-exploration prompts:

Does an arrow always establish causation, or can it represent a probabilistic dependency?
If an alarm sounds, which possible causes become more likely?
Could learning one cause make another cause less likely?

Post-exploration prompts:

Observe John calling, then observe both calls. Why does the second call add evidence?
Add earthquake evidence. Which posterior changes most, and why?
Run Gibbs sampling several times at 1,000 samples, then at 50,000. Separate sampling variation from a change in the underlying model.
Raise the burglary prior in the table. Which conclusions change because the prior changed rather than because the evidence changed?

Misconceptions to test: every arrow is causal; no edge always means independence; sampling is exact inference performed more slowly.

贝叶斯网络用有向无环图表示联合概率分布。在经典的入室盗窃网络中，入室盗窃 B 和地震 E 都可能触发警报 A，John J 和 Mary M 听到警报后可能打电话。图结构与条件概率表可以回答 P(B | J) 之类的问题，也就是在 John 打来电话的条件下发生入室盗窃的概率。

箭头编码了什么

箭头从父变量指向子变量，子变量的概率表以父变量为条件。在本网络中，警报表取决于入室盗窃和地震，而两人的来电表都取决于警报。当模型假设与数据支持时，箭头可以被赋予因果解释；从数学上看，图的作用是编码概率分解和条件独立关系。

更准确的规则是局部马尔可夫性质：给定一个节点的父节点后，该节点与其非后代条件独立。仅仅缺少一条箭头，并不能证明任意两个变量独立；结论取决于完整图结构以及哪些变量已被观测。

解释消除效应

选择 John 和 Mary 都来电。两次来电使警报发生的可能性更高，因此入室盗窃的后验概率上升。随后再加入发生地震的证据。入室盗窃的概率会下降，因为地震已经为警报和来电提供了一个解释。

在这个网络中，入室盗窃和地震在没有证据时相互独立。观测到它们的共同结果后，两者会变得相关：支持一个原因的证据可能降低对另一个原因的信念。这种碰撞节点结构产生的现象称为解释消除。

精确推理与 Gibbs 采样

精确推理通过对未观测变量的可能取值进行求和或消元，计算并归一化后验概率。在这个五变量网络中，枚举很容易；在一般情况下，计算量会随变量数量和图结构呈指数增长。变量消元或连接树等更高效的精确方法会利用图结构，但仍可能十分昂贵。

Gibbs 采样通过反复更新一个未观测变量来近似后验分布，每次都依据其他变量当前取值下的条件分布重新采样。更多样本通常会减少可见的采样波动，不过相邻样本彼此相关，而且收敛情况取决于模型。有限样本给出的是估计值，不是精确答案。

这个模型省略了什么

更大的贝叶斯网络可能从数据中学习概率表，甚至学习部分图结构。实际推理会使用变量消元、信念传播、蒙特卡洛方法或变分近似。本工具集中呈现核心思想：图规定联合分布如何分解，而证据会沿着这一结构改变各变量的信念。

教师指导

课程目标：有向图模型、条件概率表、条件独立、解释消除，以及精确推理与采样推理的区别。

探索前问题：

箭头是否总能证明因果关系，还是也可以表示概率依赖？
警报响起时，哪些可能原因会变得更可信？
得知一个原因后，另一个原因是否可能变得不那么可信？

探索后问题：

先观测 John 来电，再观测两人都来电。第二次来电为什么增加了证据？
加入地震证据。哪个后验概率变化最大，为什么？
用 1,000 个样本多次运行 Gibbs 采样，再改为 50,000 个样本。区分采样波动与模型本身发生变化。
提高表中的入室盗窃先验概率。哪些结论因为先验改变而变化，而不是因为证据改变？

需要检验的误解：每条箭头都表示因果；没有边就总是独立；采样只是速度较慢的精确推理。