docs/reference_mutual_information_metrics.md

Mutual Information Metrics for Collapse Detection

This document provides a comprehensive explanation of the mutual information (MI) based metrics used in RAGEN for detecting training collapse phenomena.

1. Overview: Two Core Metrics

We focus on two diagnostic quantities:

Quantity	Meaning	Diagnostic Metric
Within-input variability	How much reasoning varies under the same input	$H(Z \mid X)$
Across-input dependence	How much reasoning still depends on the input	$I(X; Z)$

Key Insight: We compute MI under the batch's empirical input distribution (uniform over prompts), not the true $p(x)$. This is exactly what's needed for diagnosing whether reasoning remains input-dependent inside a training batch.

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2. Design Decision: Partitioning $X$ and $Z$

The choice of how to partition the sequence into conditioning context $X$ and reasoning $Z$ is crucial for meaningful collapse detection. The partition answers: "Which segment of generation depends on which segment of input context?"

2.1 Design Goal

We want to measure whether the reasoning content becomes increasingly input-independent (i.e., ignoring the environment state and producing generic outputs), while also tracking how much variability remains under the same input.

2.2 Recommended Partition

For a typical agent turn with structure:

[System Prompt] [User: State] [Assistant:] <think> reasoning content </think> <answer> action </answer>

We define:

Variable	Content	Rationale
$X$	System prompt + User turn (state) + Assistant prefix + <think> tag	Everything the model sees before generating reasoning content
$Z$	Reasoning content tokens (between <think> and </think>, excluding both tags)	The actual reasoning we want to measure dependency for

2.3 Why Include <think> in $X$?

The <think> tag should be part of $X$ (conditioning context), not $Z$ (reasoning):

1. Semantic role: <think> is a control token meaning "start generating reasoning" — it's a boundary marker, not reasoning content itself.

2. Near-constant token: <think> appears identically in every sample, so including it in $Z$ would:

   • Add no discriminative information between prompts
   • Dilute entropy/MI statistics with high-probability constant tokens

3. Clean separation: With <think> in $X$, the partition becomes: "everything before reasoning starts" vs "reasoning content itself"

2.4 Why Exclude </think> from $Z$?

The </think> closing tag should also be excluded from $Z$:

1. Structural boundary: Like <think>, it's a format token, not reasoning content.

2. Format stability signal: If </think> is included in $Z$, MI/entropy metrics would conflate:

   • Reasoning content dependency (what we want)
   • Format stability (whether the model reliably closes tags)

3. Cleaner interpretation: Excluding both tags means $Z$ purely measures "does the reasoning content depend on the input state?"

2.5 Implementation Mapping

In the codebase, this corresponds to:

Field	Content
first_turn_prompt_ids	Tokens up to and including <think>
first_turn_reasoning_ids	Reasoning content tokens only (no <think>, no </think>)

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3. Notation and Definitions

3.1 Random Variables

Symbol	Description
$X$	Input context: system prompt + user turn + assistant prefix + <think>
$Z$	Reasoning content tokens (between <think> and </think>, excluding tags)
$x_j$	The $j$-th unique prompt in the batch, $j \in {1, \ldots, N}$
$z_{i,k}$	The $k$-th reasoning sample for trajectory $i$
$N$	Number of unique prompts in the batch
$K$	Number of reasoning samples per prompt (group size)

3.2 Probability Distributions

Symbol	Definition	Description
$p(z \mid x)$	$\prod_{t=1}^{T} p_\theta(z_t \mid x, z_{1:t-1})$	Conditional probability of reasoning $z$ given prompt $x$ under policy $\pi_\theta$
$p_{\text{mix}}(z)$	$\frac{1}{N} \sum_{j=1}^{N} p(z \mid x_j)$	Marginal probability under uniform prompt mixture
$\hat{p}(x)$	$\frac{1}{N}$	Empirical (uniform) distribution over batch prompts

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4. Core Information-Theoretic Quantities

4.1 Conditional Entropy $H(Z \mid X)$

Definition: The expected uncertainty in the reasoning $Z$ given the prompt $X$.

$$H(Z\mid X)=-{\mathbb{E}}_{x\sim \widehat{p}(x)}{\mathbb{E}}_{z\sim p(z\mathrm{\mid }x)}[\log p(z\mid x)]H(Z\; \backslash mid\; X)\; =\; -\backslash mathbb\{E\}\_\{x\; \backslash sim\; \backslash hat\{p\}(x)\}\; \backslash mathbb\{E\}\_\{z\; \backslash sim\; p(z|x)\}\; \backslash left[\; \backslash log\; p(z\; \backslash mid\; x)\; \backslash right]$$

Estimation: Using sampled (prompt, reasoning) pairs:

$$\widehat{H}(Z\mid X)=-\frac{1}{NK}\sum _{i,k}\log p(z_{i,k}\mid x_i)\backslash hat\{H\}(Z\; \backslash mid\; X)\; =\; -\backslash frac\{1\}\{NK\}\; \backslash sum\_\{i,k\}\; \backslash log\; p(z\_\{i,k\}\; \backslash mid\; x\_i)$$

Interpretation:

• High $H(Z \mid X)$: Model generates diverse responses for each prompt (stochastic policy)
• Low $H(Z \mid X)$: Model generates deterministic/repetitive responses for each prompt

Code Reference (collapse_metrics.py:675-700):

conditional_entropy = -matched.mean().item()  # H(Z|X) estimate

4.2 Marginal Entropy $H(Z)$

Definition: The total entropy of reasoning under the marginal distribution.

$$H(Z)=-{\mathbb{E}}_{z\sim p_{\text{mix}}(z)}[\log p_{\text{mix}}(z)]H(Z)\; =\; -\backslash mathbb\{E\}\_\{z\; \backslash sim\; p\_\{\backslash text\{mix\}\}(z)\}\; \backslash left[\; \backslash log\; p\_\{\backslash text\{mix\}\}(z)\; \backslash right]$$

Estimation: Using the mixture distribution:

$$\widehat{H}(Z)=-\frac{1}{NK}\sum _{i,k}\log p_{\text{mix}}(z_{i,k})\backslash hat\{H\}(Z)\; =\; -\backslash frac\{1\}\{NK\}\; \backslash sum\_\{i,k\}\; \backslash log\; p\_\{\backslash text\{mix\}\}(z\_\{i,k\})$$

where:

$$p_{\text{mix}}(z)=\frac{1}{N}\sum _{j=1}^{N}p(z\mid x_j)p\_\{\backslash text\{mix\}\}(z)\; =\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{j=1\}^\{N\}\; p(z\; \backslash mid\; x\_j)$$

Code Reference (collapse_metrics.py:675-700):

reasoning_entropy = -marginal.mean().item()  # H(Z) estimate

4.3 Mutual Information $I(X; Z)$

Definition: The amount of information that the reasoning $Z$ contains about the prompt $X$.

$$I(X;Z)=H(Z)-H(Z\mid X)I(X;\; Z)\; =\; H(Z)\; -\; H(Z\; \backslash mid\; X)$$

Equivalently:

$$I(X;Z)={\mathbb{E}}_{x,z}[\log \frac{p(z\mid x)}{p_{\text{mix}}(z)}]I(X;\; Z)\; =\; \backslash mathbb\{E\}\_\{x,\; z\}\; \backslash left[\; \backslash log\; \backslash frac\{p(z\; \backslash mid\; x)\}\{p\_\{\backslash text\{mix\}\}(z)\}\; \backslash right]$$

Estimation:

$$\widehat{I}(X;Z)=\frac{1}{NK}\sum _{i,k}[\log p(z_{i,k}\mid x_i)-\log p_{\text{mix}}(z_{i,k})]\backslash hat\{I\}(X;\; Z)\; =\; \backslash frac\{1\}\{NK\}\; \backslash sum\_\{i,k\}\; \backslash left[\; \backslash log\; p(z\_\{i,k\}\; \backslash mid\; x\_i)\; -\; \backslash log\; p\_\{\backslash text\{mix\}\}(z\_\{i,k\})\; \backslash right]$$

Interpretation:

• High $I(X; Z)$: Reasoning is input-dependent (healthy)
• Low $I(X; Z)$: Reasoning has weak input dependence
• Upper Bound: $I(X; Z) \leq H(X) = \log N$ (when $X$ is uniform)

Practical Note on Negative Values:

• The true mutual information satisfies $I(X; Z) \geq 0$.
• Our logged mi_estimate and mi_seq_estimate are finite-sample Monte Carlo estimates, not exact MI.
• Because they average noisy sample terms of the form $\log p(z \mid x) - \log p_{\text{mix}}(z)$, they can temporarily dip below zero when the true MI is near zero or the sampled batch is noisy.
• In practice, a small negative value should usually be read as "approximately zero input dependence within estimation noise," not as a violation of information theory.

Code Reference (collapse_metrics.py:563-590):

def _compute_mi_estimate(self, matched, marginal, N_prompts):
    mi = matched.mean().item() - marginal.mean().item()
    return {
        "collapse/mi_estimate": mi,
        "collapse/mi_upper_bound": math.log(N_prompts),
    }

---

5. Computation Pipeline

5.1 Cross Log-Probability Matrix

For each reasoning $z_{i,k}$ and each prompt $x_j$, we compute the cross log-probability:

$${\mathrm{\ell }}_j(z_{i,k})=\log p(z_{i,k}\mid x_j)=\sum _{t=1}^T\log p_{\theta }(z_{i,k,t}\mid x_j,z_{i,k,1:t-1})\backslash ell\_j(z\_\{i,k\})\; =\; \backslash log\; p(z\_\{i,k\}\; \backslash mid\; x\_j)\; =\; \backslash sum\_\{t=1\}^\{T\}\; \backslash log\; p\_\backslash theta(z\_\{i,k,t\}\; \backslash mid\; x\_j,\; z\_\{i,k,1:t-1\})$$

This forms a matrix $\mathbf{L} \in \mathbb{R}^{NK \times N}$ where:

• Rows index (trajectory, sample) pairs
• Columns index unique prompts

Code Reference (collapse_metrics.py:452-546):

def _compute_cross_log_probs(self, ...):
    """
    For each reasoning z_{i,k} and each prompt x_j:
    1. Construct sequence [x_j | z_{i,k}]
    2. Compute teacher-forcing log prob
    3. Sum over reasoning tokens → ℓ_j(z_{i,k})
    """
    cross_log_probs = torch.zeros(NK, N, device=device)      # per-token mean
    cross_log_probs_sum = torch.zeros(NK, N, device=device)  # per-sequence sum

5.2 Matched vs Marginal Log-Probabilities

Matched: Log-probability of reasoning under its true prompt: $${\text{matched}}_{i,k}={\mathrm{\ell }}_i(z_{i,k})=\log p(z_{i,k}\mid x_i)\backslash text\{matched\}\_\{i,k\}\; =\; \backslash ell\_i(z\_\{i,k\})\; =\; \backslash log\; p(z\_\{i,k\}\; \backslash mid\; x\_i)$$

Marginal: Log-probability under uniform prompt mixture: $${\text{marginal}}_{i,k}=\log p_{\text{mix}}(z_{i,k})=\log (\frac{1}{N}\sum _{j=1}^N\exp ({\mathrm{\ell }}_j(z_{i,k})))\backslash text\{marginal\}\_\{i,k\}\; =\; \backslash log\; p\_\{\backslash text\{mix\}\}(z\_\{i,k\})\; =\; \backslash log\; \backslash left(\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{j=1\}^\{N\}\; \backslash exp(\backslash ell\_j(z\_\{i,k\}))\; \backslash right)$$

Using log-sum-exp for numerical stability: $${\text{marginal}}_{i,k}={\text{logsumexp}}_j({\mathrm{\ell }}_j(z_{i,k}))-\log N\backslash text\{marginal\}\_\{i,k\}\; =\; \backslash text\{logsumexp\}\_j(\backslash ell\_j(z\_\{i,k\}))\; -\; \backslash log\; N$$

Code Reference (collapse_metrics.py:548-561):

def _compute_log_prob_stats(self, cross_log_probs, col_ids):
    NK, N = cross_log_probs.shape
    matched = cross_log_probs[torch.arange(NK), col_ids]  # diagonal elements
    marginal = torch.logsumexp(cross_log_probs, dim=1) - math.log(N)
    return matched, marginal

---

6. Per-Token vs Per-Sequence Metrics

We compute two variants of each metric:

Variant	Normalization	Use Case
Per-token (_est)	Divide by sequence length	Length-invariant comparison
Per-sequence (_seq_est)	Sum over tokens	Total information content

6.1 Per-Token (Length-Normalized)

$${\bar{\mathrm{\ell }}}_j(z)=\frac{1}{T}\sum _{t=1}^T\log p(z_t\mid x_j,z_{1:t-1})\backslash bar\{\backslash ell\}\_j(z)\; =\; \backslash frac\{1\}\{T\}\; \backslash sum\_\{t=1\}^\{T\}\; \backslash log\; p(z\_t\; \backslash mid\; x\_j,\; z\_\{1:t-1\})$$

This reduces length bias when comparing reasoning of different lengths.

6.2 Per-Sequence (Sum)

$${\mathrm{\ell }}_j(z)=\sum _{t=1}^T\log p(z_t\mid x_j,z_{1:t-1})\backslash ell\_j(z)\; =\; \backslash sum\_\{t=1\}^\{T\}\; \backslash log\; p(z\_t\; \backslash mid\; x\_j,\; z\_\{1:t-1\})$$

This captures total log-probability without normalization.

Base Metric Suffixes:

• collapse/mi_estimate — Per-token MI
• collapse/mi_seq_estimate — Per-sequence MI
• collapse/conditional_entropy_est — Per-token $H(Z|X)$
• collapse/conditional_entropy_seq_est — Per-sequence $H(Z|X)$
• collapse/reasoning_entropy_est — Per-token $H(Z)$
• collapse/reasoning_entropy_seq_est — Per-sequence $H(Z)$

These are the raw suffixes produced inside _compute_metrics_for_pairs. In logged outputs, they are usually namespaced as collapse_first_turn_sample/<suffix> or collapse_trajectory_sample/<suffix>.

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7. Additional Diagnostic Metrics

7.1 Retrieval Accuracy

Definition: Fraction of samples where the highest cross-log-probability matches the true prompt. If multiple prompts are identical (same tokenized prompt text), they are treated as equivalent columns, and any of those columns counts as correct for retrieval accuracy and chance.

$$\text{Acc}=\frac{1}{NK}\sum _{i,k}\mathbf{1}[\arg \underset{j}{\max }{\mathrm{\ell }}_j(z_{i,k})=i]\backslash text\{Acc\}\; =\; \backslash frac\{1\}\{NK\}\; \backslash sum\_\{i,k\}\; \backslash mathbf\{1\}\backslash left[\; \backslash arg\backslash max\_j\; \backslash ell\_j(z\_\{i,k\})\; =\; i\; \backslash right]$$

Interpretation:

• High Accuracy ($\approx 1$): Reasoning is highly prompt-specific
• Chance Level ($\approx 1/N$): Reasoning is prompt-independent

Code Reference (collapse_metrics.py:592-673):

def _compute_retrieval_accuracy(self, cross_log_probs, col_ids, N_prompts):
    predicted_cols = torch.argmax(cross_log_probs, dim=1)
    correct = (predicted_cols == col_ids).float()
    accuracy = correct.mean().item()
    chance_level = 1.0 / N_prompts

Base Metric Suffixes:

• collapse/retrieval_accuracy — Top-1 accuracy
• collapse/retrieval_accuracy@k — Top-k accuracy (k ∈ {2, 4, 8})
• collapse/retrieval_chance_level — Expected accuracy under random guessing
• collapse/retrieval_above_chance — Accuracy improvement over chance
• collapse/retrieval_chance_level@k — Expected top-k accuracy under random guessing
• collapse/retrieval_above_chance@k — Top-k accuracy improvement over chance

7.2 MI Z-Score

Definition: Standardized MI using the marginal log-probability standard deviation.

$$\text{MI-ZScore}=\frac{\text{matched}-\text{marginal}}{{\sigma }_{\text{marginal}}+ϵ}\backslash text\{MI-ZScore\}\; =\; \backslash frac\{\backslash text\{matched\}\; -\; \backslash text\{marginal\}\}\{\backslash sigma\_\{\backslash text\{marginal\}\}\; +\; \backslash epsilon\}$$

where $\sigma_{\text{marginal}} = \text{std}(\text{marginal}_{i,k})$ and $\epsilon = 10^{-3}$ for stability.

Interpretation: Measures how many standard deviations the matched log-prob is above the marginal. More robust to scale changes during training.

Practical Note on Extreme Negative Z-Scores:

• Negative mi_zscore* values are normal; they simply mean the matched log-prob is below the marginal baseline on that batch.
• Very large-magnitude values, especially for mi_zscore_seq, often happen when marginal_std or marginal_std_seq becomes very small, so the normalization denominator is close to std_eps.
• When this happens, interpret mi_zscore* together with marginal_std* and mi_estimate rather than in isolation.

Code Reference (collapse_metrics.py:302-320):

marginal_std = marginal.std(unbiased=False)
metrics["collapse/mi_zscore"] = ((matched - marginal) / (marginal_std + self.std_eps)).mean().item()

7.3 EMA-Normalized MI Z-Score

To handle variance drift during training, we track an exponential moving average of the marginal standard deviation:

$${\sigma }_{\text{EMA}}^{(t)}=\alpha \cdot {\sigma }_{\text{EMA}}^{(t-1)}+(1-\alpha )\cdot {\sigma }_{\text{marginal}}^{(t)}\backslash sigma\_\{\backslash text\{EMA\}\}^\{(t)\}\; =\; \backslash alpha\; \backslash cdot\; \backslash sigma\_\{\backslash text\{EMA\}\}^\{(t-1)\}\; +\; (1\; -\; \backslash alpha)\; \backslash cdot\; \backslash sigma\_\{\backslash text\{marginal\}\}^\{(t)\}$$

where $\alpha = 0.9$ (default decay rate).

Base Metric Suffixes:

• collapse/marginal_std — Current batch marginal std
• collapse/marginal_std_seq — Current batch marginal std (per-sequence)
• collapse/marginal_std_ema — EMA of marginal std
• collapse/mi_zscore_ema — MI Z-score normalized by EMA std
• collapse/marginal_std_ema_seq — EMA of marginal std (per-sequence)
• collapse/mi_zscore_seq — MI Z-score (per-sequence)
• collapse/mi_zscore_ema_seq — MI Z-score normalized by EMA std (per-sequence)

---

8. Multi-Turn Sampling Strategies

For multi-turn trajectories, we support two sampling strategies:

8.1 Trajectory-Uniform Sampling

Probability: $\Pr(m, t) = \frac{1}{M} \cdot \frac{1}{T_m}$

• First sample trajectory $m$ uniformly
• Then sample turn $t$ uniformly within trajectory
• Each trajectory has equal weight regardless of length

Code Reference (collapse_metrics.py:780-813):

def _sample_trajectory_uniform(self, ...):
    """Each trajectory has equal weight regardless of length."""
    for _ in range(num_to_sample):
        m = np.random.randint(M)  # uniform over trajectories
        t = np.random.randint(turn_counts[m])  # uniform over turns

8.2 Turn-Uniform Sampling (Disabled by Default)

Probability: $\Pr(m, t) = \frac{1}{\sum_m T_m}$

• Uniform over all (trajectory, turn) pairs
• Longer trajectories contribute more samples

---

9. Summary of All Logged Metrics

The code logs metrics in two layers:

1. Sample-scoped diagnostic metrics: computed on sampled $(x, z)$ pairs, then namespaced by sampling strategy.
2. Global coverage / timing metrics: logged directly without an additional sample prefix.

9.1 W&B Namespace Patterns

Logged Key Pattern	When It Appears	Meaning
collapse_first_turn_sample/<suffix>	first_turn_enabled=True and first-turn data exists	Diagnostics computed on first-turn $(x, z)$ pairs
collapse_trajectory_sample/<suffix>	multi_turn_enabled=True and multi-turn data exists	Diagnostics computed on trajectory-uniform multi-turn samples
collapse_turn_sample/<suffix>	Code path exists, but currently disabled by default	Diagnostics computed on turn-uniform multi-turn samples
collapse/valid_thinking_rate	turn_counts_total and turn_counts are available	Fraction of valid reasoning turns among all turns
collapse/first_turn_num_total	First-turn metrics enabled and data exists	Number of first-turn candidates before filtering empty reasoning
collapse/first_turn_num_valid	First-turn metrics enabled and data exists	Number of first-turn samples with non-empty reasoning
collapse/first_turn_valid_rate	First-turn metrics enabled and data exists	Valid first-turn fraction
timing_s/collapse_multi_turn_step	multi_turn_enabled=True	Wall-clock time for the multi-turn collapse pass
timing_s/collapse_first_turn_step	first_turn_enabled=True	Wall-clock time for the first-turn collapse pass

The suffix tables below describe the metric families that can appear under collapse_first_turn_sample/, collapse_trajectory_sample/, and, if re-enabled, collapse_turn_sample/.

9.2 Core Information Metrics

Suffix	Formula / Definition	Typical Reading
mi_estimate	$\mathbb{E}[\log p(z \mid x) - \log p_{\text{mix}}(z)]$	Higher means stronger input dependence
mi_seq_estimate	Sequence-sum version of MI	Same as above, but not length-normalized
mi_upper_bound	$\log N$	Theoretical ceiling given $N$ unique prompts
conditional_entropy_est	$-\mathbb{E}[\log p(z \mid x)]$	Higher means more within-input variability
conditional_entropy_seq_est	Sequence-sum version of $H(Z \mid X)$	Total within-input uncertainty per sequence
reasoning_entropy_est	$-\mathbb{E}[\log p_{\text{mix}}(z)]$	Total marginal diversity across prompts
reasoning_entropy_seq_est	Sequence-sum version of $H(Z)$	Total marginal uncertainty per sequence
matched_log_prob_mean	$\mathbb{E}[\log p(z \mid x)]$	Less negative is better fit to the true prompt
marginal_log_prob_mean	$\mathbb{E}[\log p_{\text{mix}}(z)]$	Less negative means the response is broadly likely under the prompt mixture

Example W&B keys:

• collapse_first_turn_sample/mi_estimate
• collapse_trajectory_sample/conditional_entropy_est
• collapse_first_turn_sample/reasoning_entropy_seq_est

9.3 Retrieval Metrics

Suffix	Definition	Typical Reading
retrieval_accuracy	Top-1 prompt retrieval accuracy from cross log-probs	Higher means reasoning is more prompt-specific
retrieval_accuracy@2, @4, @8	Top-k retrieval accuracy	Higher means prompt identity is easier to recover
retrieval_chance_level	Expected top-1 accuracy under random guessing	Baseline for comparison
retrieval_chance_level@2, @4, @8	Expected top-k accuracy under random guessing	Top-k baseline
retrieval_above_chance	retrieval_accuracy - retrieval_chance_level	Positive margin over chance
retrieval_above_chance@2, @4, @8	Top-k accuracy minus top-k chance	Positive margin over chance

Example W&B keys:

• collapse_first_turn_sample/retrieval_accuracy
• collapse_trajectory_sample/retrieval_accuracy@4
• collapse_first_turn_sample/retrieval_above_chance@8

9.4 Variance-Normalized Metrics

Suffix	Definition	Typical Reading
marginal_std	$\text{std}(\text{marginal})$	Current-batch spread of marginal log-probs
marginal_std_seq	Sequence-sum version of marginal_std	Current-batch spread on total log-prob scale
marginal_std_ema	EMA of marginal_std	Smoothed normalization scale
marginal_std_ema_seq	EMA of marginal_std_seq	Smoothed sequence-scale normalization
mi_zscore	$(\text{matched} - \text{marginal}) / (\text{marginal_std} + \epsilon)$	Standardized MI, batch-normalized
mi_zscore_seq	Sequence-sum version of mi_zscore	Standardized sequence-scale MI
mi_zscore_ema	MI normalized by marginal_std_ema	More stable across training drift
mi_zscore_ema_seq	Sequence-sum version of mi_zscore_ema	Stable sequence-scale normalization

9.5 Directly Logged Coverage and Timing Metrics

Logged Key	Meaning
collapse/valid_thinking_rate	Share of valid reasoning turns among all recorded turns
collapse/first_turn_num_total	Count of first-turn entries before removing empty reasoning
collapse/first_turn_num_valid	Count of first-turn entries with non-empty reasoning
collapse/first_turn_valid_rate	first_turn_num_valid / first_turn_num_total
timing_s/collapse_multi_turn_step	Time spent computing multi-turn collapse metrics on this step
timing_s/collapse_first_turn_step	Time spent computing first-turn collapse metrics on this step

---

10. Configuration Parameters

Parameter	Default	Description
compute_freq	5	Compute metrics every N steps
micro_batch_size	128	Batch size for cross-scoring
first_turn_enabled	True	Compute first-turn metrics
multi_turn_enabled	True	Enable multi-turn sampling
num_samples	64	Number of $(x, z)$ pairs to sample
std_eps	1e-3	Stability constant for std normalization
ema_decay	0.9	EMA decay for cross-time std tracking

Configuration in base.yaml (base.yaml:135-139):

collapse_detection:
  compute_freq: 5
  micro_batch_size: 128
  first_turn_enabled: true
  multi_turn_enabled: true
  num_samples: 64

---

11. Mathematical Derivations

11.1 MI Estimation via Importance Sampling

The mutual information is:

$$I(X;Z)={\mathbb{E}}_{p(x,z)}[\log \frac{p(z\mid x)}{p(z)}]I(X;\; Z)\; =\; \backslash mathbb\{E\}\_\{p(x,z)\}\; \backslash left[\; \backslash log\; \backslash frac\{p(z\; \backslash mid\; x)\}\{p(z)\}\; \backslash right]$$

Under the empirical distribution $\hat{p}(x) = 1/N$ (uniform over batch prompts):

$$I(X;Z)={\mathbb{E}}_{x\sim \widehat{p}(x)}{\mathbb{E}}_{z\sim p(z\mathrm{\mid }x)}[\log \frac{p(z\mid x)}{p_{\text{mix}}(z)}]I(X;\; Z)\; =\; \backslash mathbb\{E\}\_\{x\; \backslash sim\; \backslash hat\{p\}(x)\}\; \backslash mathbb\{E\}\_\{z\; \backslash sim\; p(z|x)\}\; \backslash left[\; \backslash log\; \backslash frac\{p(z\; \backslash mid\; x)\}\{p\_\{\backslash text\{mix\}\}(z)\}\; \backslash right]$$

where $p_{\text{mix}}(z) = \sum_j \hat{p}(x_j) p(z \mid x_j) = \frac{1}{N} \sum_j p(z \mid x_j)$.

Monte Carlo estimate with $K$ samples per prompt:

$$\widehat{I}(X;Z)=\frac{1}{NK}\sum _{i=1}^N\sum _{k=1}^K[\log p(z_{i,k}\mid x_i)-\log p_{\text{mix}}(z_{i,k})]\backslash hat\{I\}(X;\; Z)\; =\; \backslash frac\{1\}\{NK\}\; \backslash sum\_\{i=1\}^\{N\}\; \backslash sum\_\{k=1\}^\{K\}\; \backslash left[\; \backslash log\; p(z\_\{i,k\}\; \backslash mid\; x\_i)\; -\; \backslash log\; p\_\{\backslash text\{mix\}\}(z\_\{i,k\})\; \backslash right]$$

11.2 Information-Theoretic Identity

The fundamental identity relating our metrics:

$$I(X;Z)=H(Z)-H(Z\mid X)I(X;\; Z)\; =\; H(Z)\; -\; H(Z\; \backslash mid\; X)$$

This means:

• If $H(Z|X)$ drops but $H(Z)$ stays constant → MI increases (good)
• If both $H(Z)$ and $H(Z|X)$ drop equally → MI stays constant
• If $H(Z) \to H(Z|X)$ → MI → 0 (input dependence vanishes)