get_intermediate_order_log_likelihood ignores edge weights when computing transition probabilities #340

I found this while looking into #172. Below is a write up by Claude

What happens

When we compute the likelihood of observed paths under a multi-order model, every step of a path is
scored with the probability of taking that transition. That probability should be the observed
relative frequency: an edge's weight divided by the total weight of all edges leaving the same node.

For the intermediate orders we currently drop the weights:

# multi_order_model.py, in get_intermediate_order_log_likelihood
transition_probabilities = self.layers[order].transition_probabilities()[...]

Graph.transition_probabilities() falls back to counting each outgoing edge once when no edge
attribute is given, so every alternative at a node is treated as equally likely, regardless of how
often it was actually observed. The highest-order term in the same file already does it correctly:

# in get_mon_log_likelihood
transition_probabilities = self.layers[max_order].transition_probabilities(edge_attr="edge_weight")

How to reproduce

Two paths starting at the same node, observed a different number of times:

paths = PathData(IndexMap(list("abcde")))
for _ in range(3):
    paths.append_walk(("a", "b", "c"))
paths.append_walk(("a", "d", "e"))

m = MultiOrderModel.from_path_data(paths, max_order=2)
m.get_mon_log_likelihood(paths.data, max_order=2)

a → b was observed 3 times and a → d once, so the first-order probabilities are 3/4 and 1/4:

expected: 4*log(4/12) + 3*log(3/4) + 1*log(1/4) = -6.6438
actual:   4*log(4/12) + 3*log(1/2) + 1*log(1/2) = -7.1670

Why the existing tests do not catch it

Intermediate-order terms only exist for max_order >= 2, and they are only wrong when a node has
outgoing edges with different weights. In all current test fixtures the alternatives at a branching
node are observed equally often, so uniform and relative-frequency probabilities coincide.

Impact

get_mon_log_likelihood(max_order=k) is too low for every k >= 2, which propagates to
likelihood_ratio_test and estimate_order. Because only the higher-order side of each comparison
carries intermediate terms, the test statistic comes out too small — so the bias is towards
underestimating the optimal order on data with uneven branching.

Suggested fix

Pass edge_attr="edge_weight", matching the highest-order term. Worth checking at the same time
whether mode="diffusion" (where edge weights are probabilities rather than counts) needs different
handling here.