GuidesWork with edge probabilities

Work with edge probabilities

Hold a model's directed-edge probabilities over named variables, read them in the form a decoder needs, and see where they disagree with any single graph. You'll also turn bootstrap frequencies into the same object.

Run the program#

The matrix below is made up for four variables. Read values[i, j] as "how likely is an edge from names[i] to names[j]".

import numpy as np

from ergodic import EdgeProbabilities, tabular
from ergodic.discovery import bootstrap

# 1. Scores a model returned for four variables: rows the source, columns the target.
names = ("rain", "sprinkler", "wet", "slippery")
values = np.array(
    [
        [0.00, 0.45, 0.91, 0.06],
        [0.40, 0.00, 0.83, 0.05],
        [0.04, 0.07, 0.00, 0.88],
        [0.58, 0.02, 0.10, 0.00],
    ]
)
probs = EdgeProbabilities(names, values, kind="independent", provenance={"model": "example"})
print(probs)

# 2. Every edge above 0.5, as a diagnostic graph.
print(probs.threshold(0.5))

# 3. Three exclusive states per pair: no edge, one way, the other way.
pair = probs.to_pair_states()
print(pair.to_frame().round(3))

# 4. The chance each pair is adjacent, either way.
print(probs.existence().round(3))

# 5. Bootstrap frequencies of a discovery method, as edge probabilities.
rng = np.random.default_rng(0)
x = rng.normal(size=500)
y = 0.8 * x + rng.normal(size=500)
z = 0.8 * y + rng.normal(size=500)
data = tabular({"x": x, "y": y, "z": z})
freq = EdgeProbabilities.from_bootstrap(bootstrap(data, method="ges", n_bootstrap=20, seed=0))
print(freq)
print(freq.to_frame())

EdgeProbabilities(independent, 4 variables, 4 directed cells above 0.5, uncalibrated)
MixedGraph(kind=mixed, nodes=4, edges=[rain <-- slippery, rain --> wet, slippery <-- wet, sprinkler --> wet])
target      rain  sprinkler    wet  slippery
source
rain       0.000      0.329  0.907     0.026
sprinkler  0.268      0.000  0.820     0.049
wet        0.004      0.013  0.000     0.868
slippery   0.565      0.019  0.013     0.000
[[0.    0.598 0.91  0.591]
 [0.598 0.    0.832 0.068]
 [0.91  0.832 0.    0.882]
 [0.591 0.068 0.882 0.   ]]
EdgeProbabilities(frequency, 3 variables, 0 directed cells above 0.5, uncalibrated)
target      x    y      z
source
x       0.000  0.5  0.025
y       0.500  0.0  0.500
z       0.025  0.5  0.000

• Step 1. kind="independent" says each number is a marginal on its own, so both directions of a pair may be high at once. The object stores a read-only copy, checks the diagonal is zero and every value is in [0, 1], and keeps provenance as you gave it.
• Step 2. Four edges pass 0.5, and three of them, rain --> wet --> slippery --> rain, form a cycle. threshold shows that on purpose: it returns a mixed graph that may be cyclic, not a DAG.
• Step 3. to_pair_states conditions each pair on "not both directions": for p = P[i, j] and q = P[j, i] the states are (1 - p)(1 - q), p(1 - q) and q(1 - p), renormalized. Each pair's two cells now sum to at most 1; the rest is the no-edge mass. The ranking of directions is kept.
• Step 4. existence adds the two directions of each pair in that view. rain and wet are adjacent with probability 0.91, sprinkler and slippery with 0.068.
• Step 5. On a chain x -> y -> z, GES returns undirected edges, because the data can't tell the directions apart. from_bootstrap splits each undirected edge's frequency evenly between the two directions, so x and y get 0.5 each way.

Read the object#

The three kinds read the numbers differently:

• independent: one marginal per directed edge. AVICI and CDFM return these.
• pair_state: three exclusive states per pair, so P[i, j] + P[j, i] <= 1. Building one that breaks this raises.
• frequency: the share of runs that drew each directed edge. One edge per pair per run, so the same bound holds.

threshold draws a pair whose two directions both pass as an undirected X --- Y edge, because a graph holds one edge per pair. Read it as "adjacent, direction unresolved".

from_bootstrap drops bidirected and circle-marked glyphs, which say something other than "this direction", and records the dropped share in provenance["dropped_frequency"]. Pass undirected="drop" to drop undirected edges too. It reads results over a table's variables; a window-graph bootstrap from PCMCI is refused.

Next steps#