Day 10: Latent Geometry and AME

Where Today Starts

Day 9 represented unobserved relationship patterns in two ways:

Two Questions Organize the Day

What Might the Missing Pattern Represent?

The same latent term can do very different jobs:

• Summarize unmeasured proximity between actors
• Represent complementary source and target roles
• Help the model handle connected observations
• Measure a relationship pattern we want to study

The Models Answer Different Questions

A Distance Model Starts With One Claim

\[ \operatorname{logit}\Pr(Y_{ij}=1) =\alpha+x_{ij}^{\mathsf T}\beta-\lVert z_i-z_j\rVert \]

The model runs a familiar idea backward: ties go in, and positions that make those ties plausible come out.

Read the Distance Equation in Pieces

What Makes One Fitted Map Better Than Another?

\[ p(Y\mid Z,\alpha,\beta) =\prod_{i<j}p_{ij}^{y_{ij}}(1-p_{ij})^{1-y_{ij}} \]

• An observed tie rewards a map that gives the pair a high \(p_{ij}\).
• An observed non-tie rewards a map that gives the pair a low \(p_{ij}\).
• Moving one actor changes its distance from everyone else, so the model must compromise across the whole partner list.
• The posterior combines this likelihood with priors on the coefficients, positions, and their spread.

Estimation Runs the Distance Story Backward

We observe who relates to whom. The estimator searches for coefficients and actor positions that make the complete pattern plausible.

1. Define the outcome, risk set, predictors, and map dimension.
2. Start with provisional coefficients and positions.
3. Compare their fitted probabilities with every tie and non-tie.
4. Update the coefficients and positions in turn.
5. Retain many plausible states rather than only one map.

What the Distance Posterior Represents

\[ p(\alpha,\beta,Z,\Sigma\mid Y,X) \propto \underbrace{p(Y\mid X,\alpha,\beta,Z)}_{\text{network fit}} \times \underbrace{p(\alpha,\beta,Z,\Sigma)}_{\text{prior structure}} \]

• \(\alpha\) and \(\beta\) describe the baseline and measured associations.
• \(Z\) contains the actor positions.
• \(\Sigma\) describes how dispersed those positions can be.
• The likelihood rewards ties with high probabilities and non-ties with low probabilities.

How MCMC Explores the Posterior

Imagine reconstructing a seating chart from observed relationships:

1. Begin with provisional positions, coefficients, and variance quantities.
2. Hold the positions fixed and update the other model pieces.
3. Hold those pieces fixed and move actors using their complete partner lists.
4. Record the complete current explanation and repeat.

Diagnostics Answer a Narrow Question

Distance Buys Clustering and Imposes a Restriction

The Gade Application Uses Recorded Joint Operations

Gade et al. (2019) provide the cooperation data.

This Is a Focused Reanalysis, Not a Replication

• An undirected tie means at least one recorded joint operation during the full period.
• An operation involving several organizations creates several pair ties.
• The original article models a square-root count outcome; today uses a binary whole-period projection.
• A tie means recorded tactical cooperation, not ideological agreement or a stable coalition.

Fitting the Gade Map Is a Repeated Conditional Update

1. Choose a two-dimensional Euclidean space.
2. Start with provisional organization positions and an intercept.
3. Update the intercept given the current positions.
4. Update positions given the current intercept and observed ties.
5. Repeat, discard burn-in, and retain 4,000 draws.
6. Align equivalent maps before summarizing positions.

Check the Chain Before Reading the Map

The focused check follows two quantities that do not change when the map rotates:

• The intercept, which controls the baseline tie tendency at zero distance
• The latent-position variance, which describes how spread out the map is

In the matching seeded run, both effective sample sizes exceed 1,500.

Read the Gade Map as a Probability Surface

Does the Geometry Reproduce the Observed Clustering?

What the Gade Map Adds

What we would report:

• The relationship and period
• The two-dimensional distance assumption
• Sampling diagnostics
• Invariant distances or fitted probabilities
• Posterior predictive performance and remaining uncertainty

Distance Becomes Awkward for Complementary Roles

Suppose two groups mostly connect across groups:

A Longer Run Cannot Repair the Wrong Representation

Ask two questions in order:

1. Did the sampler explore the fitted model? Check the trace, effective sample size, and stability across starts.
2. Can the fitted model represent the pattern economically? Compare observed and fitted features after the computational check passes.

The Geometry Itself Was Chosen

Before seeing the network, we chose:

• Two dimensions
• A flat Euclidean space
• No curvature
• Distance as the relationship rule

Dimension, geometry, and curvature require an argument and sensitivity checks (Lubold, Chandrasekhar, and McCormick 2023).

Similarity Can Mean Two Different Things

Start With the Regression We Already Know

For an ordered relationship from \(i\) to \(j\):

\[ g\!\left(E[Y_{ij}\mid\cdot]\right) =\eta_{ij} =\beta_0+\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta}. \]

The SRM Adds Broad Source and Target Differences

\[ \eta_{ij} =\beta_0+\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta} +a_i+b_j \]

A Factor Is an Interaction With Unknown Inputs

If source capability \(c_i\) and target vulnerability \(r_j\) are measured, and their main effects are already in \(\mathbf{x}_{ij}\), an ordinary regression can include

\[ \delta\,c_i r_j. \]

The rank-one factor model uses the same multiplication:

\[ \delta\,u_i v_j, \]

but \(u_i\) and \(v_j\) are not measured variables. The model learns them from repeated source-target patterns in the relationship matrix.

One Source Can Fit Two Targets Differently

With one latent dimension and \(\delta=1\):

Rank Is the Number of Latent Interactions

\[ \eta_{ij} =\beta_0+\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta}+a_i+b_j +\underbrace{\sum_{r=1}^{R}\delta_r u_{ir}v_{jr}}_{\text{\(R\) latent interactions}} \]

Equivalently,

\[ \eta_{ij} =\beta_0+\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta}+a_i+b_j +\mathbf{u}_i^{\mathsf T}D\mathbf{v}_j. \]

A Movie Matrix Makes the Factor Idea Visible

Zero is a recorded score in this toy example. It is not a missing value.

SVD Separates Rows, Columns, and Pattern Strength

\[ M=UDV^{\mathsf T} \]

Rank Two Nearly Reconstructs the Movie Matrix

Every Fitted Cell Is a Compatibility Score

Split each singular value across the two sides:

\[ P=U_RD_R^{1/2},\qquad Q=V_RD_R^{1/2},\qquad \widehat m_{ij}=\mathbf p_i^{\mathsf T}\mathbf q_j. \]

For Emily and Star Wars:

\[ \widehat m_{\text{Emily, Star Wars}} =p_{\text{Emily},1}q_{\text{Star Wars},1} +p_{\text{Emily},2}q_{\text{Star Wars},2}. \]

Carry the Same Product Into a Directed Network

In a directed network, the same actors appear on both sides of the matrix, but the jobs differ:

\[ z_{ij}\approx u_i^{\mathsf T}v_j. \]

The product says whether this particular source profile fits this particular target profile. Additive effects still handle broad source activity and target exposure.

SVD Gives the Scaffold, Not the Network Fit

The Factor Term Adds Compatibility

For a directed relationship:

\[ \gamma(u_i,v_j)=u_i^{\mathsf T}v_j \]

Multiply matching coordinates and add them:

• A large positive result raises the fitted relationship score.
• A value near zero adds little.
• A negative result lowers the score.

What Factor-Model Estimation Learns

The model learns these pieces together:

• \(\beta\): associations with measured predictors
• \(a_i\) and \(b_j\): actors that are broadly high as sources or targets
• \(U\) and \(V\): recurring source-target compatibility patterns
• Variance and reciprocity parameters: remaining dependence and uncertainty

\[ p(\beta,a,b,U,V,\Sigma\mid Y,X) \propto p(Y\mid X,\beta,a,b,U,V,\Sigma)\,p(\beta,a,b,U,V,\Sigma) \]

For a probit AME model, the sampler first represents each zero or one with a latent continuous relationship score. Given those scores, it cycles through regression, actor-effect, factor, and variance updates.

Eigenvalue Signs Describe the Symmetric Surface

For a symmetric eigenmodel:

\[ \gamma(u_i,u_j)=u_i^{\mathsf T}\Lambda u_j \]

The Planted Simulations Show Why the Sign Matters

Choose the Representation That Matches the Question

Check prediction, posterior reproduction, stability, and interpretability. A more flexible model has to earn the extra complexity.

Same Data, A More Specific Question

Day 9 used the ICEWS panel to ask which states appeared in many above-threshold relationships as the coded source or target.

Day 10 asks what the additive scores could not:

The outcome equals one when an ordered state pair has more than 20 coded material-conflict events in a year. It measures high coded event volume, not conflict initiation or severity.

There Is Enough Movement to Ask

The Dynamic AME Keeps Five Jobs Separate

\[ Y^*_{ijt} =\alpha_t+\mathbf{x}_{ijt}^{\mathsf T}\boldsymbol\beta +a_{it}+b_{jt} +\mathbf{u}_{it}^{\mathsf T}\mathbf{v}_{jt} +e_{ijt}. \]

Broad Involvement Is Not a Bilateral Relationship

netify Keeps the Panel Aligned

netify() receives the ordered state-pair-year table and keeps the design choices visible:

1. source and target preserve direction.
2. year defines the 13 network slices.
3. symmetric = FALSE keeps Iran-Syria separate from Syria-Iran.
4. missing_to_zero = FALSE prevents an unobserved dyad from becoming a recorded zero.

to_lame() returns 13 aligned outcome matrices and 13 aligned predictor arrays.

What Dynamic ALS Is Trying to Do

The estimator has two connected jobs:

1. Reconstruct which ordered state pairs cross the event threshold in each year.
2. Keep neighboring years connected unless the data support a sustained change.

The baseline, measured predictors, broad actor paths, and pair-specific factor surface divide the first job. Smoothing penalties handle the second.

What Dynamic ALS Minimizes

At each binary-response update, the estimator approximately minimizes

\[ \sum_{t,i\ne j}w_{ijt}(z_{ijt}-\widehat\eta_{ijt})^2 +\lambda_{ab}\,\text{actor jumps} +\lambda_{uv}\,\text{factor jumps} +\lambda_\alpha\,\text{intercept jumps}. \]

1. Build a working response \(z_{ijt}\) and weight \(w_{ijt}\) from current probabilities.
2. Update coefficients, actor paths, and factor paths in turn.
3. Recalculate probabilities and repeat until the objective barely changes.

The Actual lame() Call

fit_r2 <- lame(
  Y = Y_icews,
  Xdyad = X_icews,
  family = "binary",
  symmetric = FALSE,
  R = 2,
  dynamic_ab = TRUE,
  dynamic_uv = TRUE,
  dynamic_beta = "intercept",
  dynamic_beta_kind = "rw1",
  method = "als",
  als_stability = "validation",
  bootstrap = 100,
  seed = 6886
)

All four validation starts converged to essentially the same fitted probability surface. The 100 bootstrap refits also completed successfully.

Rank Two Is a Practical Choice

The test hides both directions and all 13 years for each unordered pair.

The Annual Shape Is Mostly Right

What the Iran-Syria Paths Show

1. Some states appear in above-threshold relationships with many partners, but the Iran-Syria directions still follow their own distinct paths after accounting for that broad involvement.
2. Rank two improves held-pair prediction and makes simulated movement much more realistic than additive effects alone.
3. Iran toward Syria and Syria toward Iran both rise sharply, but not at exactly the same time.
4. The remaining reciprocity and transition misses limit the claim.

A Published Directed Application: Conflict in Nigeria

Dorff, Gallop, and Minhas (2020) study directed ACLED battle ties among 37 Nigerian armed organizations from 2000 through 2016.

The factors place organizations in similar roles when they repeatedly fight the same kinds of opponents, even when they operate in different places and pursue different projects. The wider post-2009 increase in fitted battle risk is a system-level descriptive result, not a randomized effect of Boko Haram’s uprising.

Latent Adjustment Does Not Settle Omitted-Variable Bias

Suppose the latent outcome contains an omitted relational pattern \(W\):

\[ Y^*=\beta X+\gamma W+e, \qquad W=\alpha X+z. \]

Substitution gives

\[ Y^*=(\beta+\gamma\alpha)X+\gamma z+e. \]

AME can represent patterned dependence. It cannot turn that dependence adjustment into automatic confounding control (Minhas et al. 2022).

What We Learned About Iran and Syria

1. Question: When high-volume conflictual relationships change, is the change broad across a state’s partners or concentrated in particular directed pairs?
2. Validation: Rank two improves complete-pair prediction and reproduces much more observed movement than the dynamic SRM.
3. Relationship pattern: Iran toward Syria and Syria toward Iran rise at different times, and the fitted probabilities preserve that difference.
4. Boundary: Reciprocity remains weak in several years, and four transition intervals still miss.