Exact Inference in Bayesian Networks

• This notebook teaches how to compute exact posteriors $P(X \mid \mathbf{e})$ in a discrete Bayesian network
• Concepts are built on the the burglary-alarm network and the query $P(Burglary \mid JohnCalls, MaryCalls)$
• The flow is:
  • Inference by enumeration (brute force)
  • Variable elimination (caching)
  • Irrelevant-variable pruning
  • Complexity and limits

Imports¶

Part 1: Setting Up the Inference Problem¶

Cell 1.1: The burglary alarm network and its CPTs¶

Goal:

• Ground every later inference step in one concrete network: the burglary-alarm story and its five conditional probability tables
• The five variables are:
  • Root causes: $Burglary$, $Earthquake$ (blue)
  • The $Alarm$ (orange)
  • The calls: $JohnCalls$, $MaryCalls$ (purple)
• The joint factorizes into five small CPTs:
  $$P(B,E,A,J,M) = P(B)\,P(E)\,P(A \mid B,E)\,P(J \mid A)\,P(M \mid A)$$

  (1)

Implementation: cell1_1_show_network_and_cpts()

• Draws the DAG colored by role, prints the priors $P(B)$/$P(E)$, the $P(Alarm \mid B, E)$ table, and the two call CPTs

• The joint over five variables factorizes into five small CPTs
• Calls depend on the world only through $Alarm$: they are conditionally independent of $Burglary$ and $Earthquake$ given $Alarm$
• Storing 5 small CPTs (10 independent numbers) is far cheaper than the full joint ($2^5 - 1 = 31$ numbers)
• Inference never touches the full joint directly: it works with the factored form, a product of the small CPTs attached to each node

Cell 1.2: Query, evidence, and hidden variables¶

Goal:

• Build intuition for which variables must be summed out, by letting students pick the query and mark any subset of nodes as evidence

Implementation: cell1_2_query_roles_widget()

• Recolors the DAG by role, query/evidence/hidden, for the current choice of Query X and observed nodes, via _read_query()
• Restates the query in math and counts the $2^{|\mathbf{Y}|}$ hidden terms in the Comments panel

Usage

• Inputs

  • Query X: which node is being asked about
  • observe <node>: one checkbox plus True/False value per node, marking it as evidence

• Panels

  • Roles for the current query: the DAG recolored by role: query (blue), evidence (red), hidden (grey)
  • Comments: the query restated in math, and the hidden-term count

Guided usage

• Mark every node except Query X as evidence
  • Observe no node stays hidden, and the term count in Comments drops to $2^0 = 1$: nothing needs to be summed out
• Uncheck every evidence box, leaving only Query X set
  • Observe every other node turns hidden, and the term count climbs to 2⁴: the posterior we want, $P(X \mid \mathbf{e})$, still has to marginalize all of them away, they cannot simply be dropped

Part 2: Inference by Enumeration¶

Cell 2.1: From conditional to joint via normalization¶

Goal:

• Derive why a conditional query reduces to summing the joint and rescaling by a normalization constant $\alpha$

Implementation: cell2_1_normalization_widget()

• Fixes the canonical query $P(Burglary \mid JohnCalls, MaryCalls)$ and computes both the unnormalized joint and the normalized posterior with _enumerate_posterior()
• show normalization switches the middle panel between the two views

Usage

• Inputs

  • show normalization: toggle between the unnormalized joint $P(B, j, m)$ and the normalized posterior $P(B \mid j, m)$

• Panels

  • Derivation: the three equation lines from $P(X \mid e)$ down to the alarm-network sum
  • Unnormalized joint/Normalized posterior: bars for $B$=T/F in whichever view is toggled on
  • Comments: the unnormalized values, $\alpha$, and the resulting posterior

Guided usage

• Toggle show normalization off, then on
  • Observe the bars change scale (they no longer sum to 1) but keep the same relative height: rescaling by $\alpha$ never changes which value is more likely, only the units
• Read $\alpha$ off the Comments panel
  • Observe $\alpha = 1 / \sum_x P(x, e)$: exactly the constant that makes the posterior sum to 1, computed without ever touching $P(e)$ directly

Cell 2.2: Computing the posterior by enumeration¶

Goal:

• Compute the exact posterior by hand, summing CPT products over the hidden variables, and confirm it against the pgmpy engine

Implementation: cell2_2_enumeration_widget()

• Enumerates every hidden assignment and its CPT product with _enumerate_posterior(), for the current Query X/evidence
• Cross-checks the result against pgmpy’s VariableElimination reference

Usage

• Inputs

  • Query X: which node is being asked about
  • observe <node>: one checkbox plus True/False value per node, marking it as evidence

• Panels

  • Sum for X=T/Sum for X=F: every hidden assignment and its CPT product, for each query value
  • Posterior: the resulting bars, with the pgmpy reference overlaid in dashed red
  • Comments: hidden-variable count, row count summed, and whether the two engines agree

Guided usage

• Mark one more node as evidence, then another
  • Observe the row count in Comments halve each time: one fewer hidden variable means one fewer factor of 2 in $2^{|\mathbf{Y}|}$
• Compare the enumeration bars against the dashed pgmpy reference for several different queries
  • Observe they always coincide: hand enumeration and the library engine compute the same posterior, e.g. the famous result $P(Burglary \mid j,m) \approx 0.284$

Cell 2.3: Visualizing the enumeration tree¶

Goal:

• Expose the enumeration computation as a tree, to see the repeated subexpressions that motivate variable elimination

Implementation: cell2_3_enumeration_tree_widget()

• Draws the two-level branch tree over the hidden Earthquake/Alarm values, for the fixed query $P(Burglary \mid j, m)$
• Colors each leaf’s shared $P(j \mid a)P(m \mid a)$ factor by the value of Alarm on that branch, when highlighting is on

Usage

• Inputs

  • Sum order: which hidden variable branches first, Earthquake or Alarm
  • highlight repeated work: color leaves by their shared factor

• Panels

  • Enumeration evaluation tree: the branch tree, leaves labeled by their CPT product
  • Comments: the operation count and why the repeats are wasteful

Guided usage

• Turn on highlight repeated work
  • Observe leaves under different first-level branches share the same color whenever Alarm agrees: the identical $P(j \mid a)P(m \mid a)$ factor is recomputed in more than one branch
• Switch Sum order between the two options
  • Observe the tree’s shape changes, but the same repeated-factor pattern persists either way: caching it once, instead of recomputing it per branch, is the one idea behind variable elimination

Part 3: Variable Elimination¶

Cell 3.1: Factors as the unit of computation¶

Goal:

• Introduce the factor, a table over a subset of variables, as the single data structure variable elimination manipulates

Implementation: cell3_1_factor_operations_widget()

• Renders the chosen CPT as a Factor table, then marginalizes out Sum out with pgmpy’s factor.marginalize()

Usage

• Inputs

  • Factor: which CPT to view as a factor, labeled by its scope
  • Sum out: which variable in that factor’s scope to marginalize away

• Panels

  • Before: the selected factor’s table and scope
  • After summing out <var>: the same factor once that variable is marginalized out
  • Comments: how many rows the table shrank by, and the two operations inference needs

Guided usage

• Pick a factor with a scope of 3 variables, then sum out one of them
  • Observe the row count drop from 8 to 4: summing out one variable halves the table, one dimension at a time
• Switch Factor to a CPT with only 1 variable in scope
  • Observe Sum out leaves a (scalar) factor: summing out every remaining variable collapses the table to a single number

Cell 3.2: Variable elimination step by step¶

Goal:

• Walk through variable elimination on the alarm query, one elimination at a time, and compare its operation count against enumeration

Implementation: cell3_2_variable_elimination_widget()

• Advances step through _variable_elimination_steps(), which eliminates hidden variables in the chosen Order and caches each new factor
• Tallies multiplications and additions used so far, against the fixed enumeration operation count

Usage

• Inputs

  • Order: which hidden variable, Earthquake or Alarm, is eliminated first
  • elimination step: how many variables have been eliminated so far, 0-2

• Panels

  • Active factors (step N): the current factor scopes
  • New factor: the factor created at this step, or “no elimination yet” at step 0
  • Posterior and op count: the running posterior next to the enumeration-vs-VE operation comparison
  • Comments: order, step, and both operation counts

Guided usage

• Advance elimination step from 0 to 2
  • Observe the active-factor list shrink by one each step, while the posterior bars never change: the final answer is fixed from the start, only the work to reach it is staged
• Compare the VE operation total in Comments against the enumeration count
  • Observe VE finishes with noticeably fewer operations, by reusing each cached factor instead of recomputing it per branch

Cell 3.3: Pruning irrelevant variables¶

Goal:

• Show that a variable which is not an ancestor of the query or evidence can be deleted before any computation, for free

Implementation: cell3_3_pruning_widget()

• Builds an extended network with an extra Neighbor leaf, then shades nodes by whether they are the query, evidence, or an ancestor of either
• prune irrelevant variables compares the posterior on the full network against the posterior on the ancestor-only subgraph

Usage

• Inputs

  • Query X: which node is being asked about, over the extended network
  • observe <node>: one checkbox plus True/False value per node
  • prune irrelevant variables: toggle between the full and pruned network

• Panels

  • Relevant (green) vs irrelevant (grey): the DAG shaded by relevance to the current query/evidence
  • Full vs pruned posterior: bars for both networks, side by side
  • Comments: the irrelevant nodes and whether the two posteriors match

Guided usage

• Leave Neighbor unobserved and toggle prune irrelevant variables on and off
  • Observe the posterior bars stay identical either way, while Comments lists Neighbor as irrelevant: an unobserved effect node with no bearing on the query contributes nothing
• Mark Neighbor as evidence
  • Observe it turns green (relevant) and pruning stops removing it: an observed node is never irrelevant, however far it sits from the query

Part 4: Complexity and Limits¶

Cell 4.1: Complexity: polytrees vs general graphs¶

Goal:

• Make concrete that exact inference is cheap on tree-shaped networks and blows up on densely connected ones

Implementation: cell4_1_complexity_widget()

• Draws a chain (polytree) and a fully connected graph of the same n, then plots $O(n)$ vs $O(2^n)$ operation-count curves with Order quality scaling the constant

Usage

• Inputs

  • n: number of nodes in both example graphs, 3-10
  • Structure: which curve the marked operating point sits on, polytree or fully connected
  • Order quality: good or bad, scaling the cost constant

• Panels

  • Polytree (n=N): a chain network of the chosen size
  • Dense network (n=N): a fully connected network of the same size
  • Cost vs network size: the two cost curves on a log y-axis, with the current operating point marked
  • Comments: the operating-point cost and the role of order quality

Guided usage

• Raise n from 3 toward 10 with Structure set to fully connected
  • Observe the marked point climb the exponential curve, while the same raise under polytree barely moves the point on the linear curve
• Switch Order quality from good to bad
  • Observe the operating-point cost jump higher at the same n: a worse elimination order inflates the constant in front of both curves

Cell 4.2: When exact inference breaks down¶

Goal:

• Summarize the three regimes where exact inference is the right tool, or is not, and motivate the approximate methods covered next

Implementation: cell4_2_breakdown_widget()

• Looks up the chosen Scenario in a fixed regime table and highlights its row and recommendation

Usage

• Inputs

  • Scenario: Small discrete polytree, Large dense discrete network, or Continuous variables

• Panels

  • Regimes for exact inference: the three-regime comparison table, with the current scenario’s row highlighted
  • Continuous case: a small DAG sketch showing $\sum_y \to \int dy$
  • Comments: the recommendation banner, EXACT or APPROXIMATE, and why

Guided usage

• Step through all three Scenario options
  • Observe only Small discrete polytree recommends EXACT; both other scenarios recommend APPROXIMATE, for two different reasons: an exponential cost blowup, or a sum that is not even a finite sum
• Compare this recommendation table against Cell 4.1’s cost curves
  • Observe the Large dense discrete network row is exactly the regime where Cell 4.1’s exponential curve makes exact inference impractical