Statistical Methods¶

This document provides a comprehensive reference for the statistical methods employed in Truthound for drift detection, anomaly detection, and distributional analysis.

Table of Contents¶

Overview
Drift Detection Methods
Anomaly Detection Methods
Distribution Analysis
Statistical Thresholds
Method Selection Guide
Mathematical Foundations
References

1. Overview¶

Truthound implements a suite of statistical methods for data quality validation:

Category	Methods	Primary Use Case
Drift Detection	14 methods (auto, ks, psi, chi2, js, kl, wasserstein, cvm, anderson, hellinger, bhattacharyya, tv, energy, mmd)	Distribution comparison between datasets
Anomaly Detection	Multiple methods	Outlier and anomaly identification
Distribution Analysis	Various methods	Statistical characterization

All methods are optimized for Polars LazyFrame execution, enabling efficient processing of large-scale datasets.

2. Drift Detection Methods¶

2.1 Kolmogorov-Smirnov Test (KS)¶

The Kolmogorov-Smirnov test measures the maximum distance between two empirical cumulative distribution functions.

Mathematical Definition:

D = max|F₁(x) - F₂(x)|

Where F₁ and F₂ are the empirical CDFs of the two samples.

Usage:

# KS test requires numeric columns
drift = th.compare(baseline, current, method="ks", columns=["age", "salary", "score"])

Note: KS test only works with numeric columns. For mixed data types, use method="auto".

Characteristics:

Aspect	Description
Column Type	Numeric only
Best For	Continuous numeric distributions
Sensitivity	Shape and location differences
Output	D-statistic (0-1), p-value
Threshold	p-value < 0.05 indicates drift

Interpretation:

p-value	Interpretation
p < 0.05	Significant difference (reject null hypothesis)
p >= 0.05	No significant difference

2.2 Population Stability Index (PSI)¶

PSI quantifies how much a variable's distribution has shifted between two samples. Widely used in credit scoring and model monitoring.

Mathematical Definition:

PSI = Σ (Pᵢ - Qᵢ) × ln(Pᵢ / Qᵢ)

Where Pᵢ and Qᵢ are the proportions in bin i for baseline and current distributions.

Usage:

# PSI requires numeric columns
drift = th.compare(baseline, current, method="psi", columns=["age", "salary", "score"])

Note: PSI only works with numeric columns. For mixed data types, use method="auto".

Interpretation:

PSI Value	Interpretation	Action
< 0.1	No significant shift	None required
0.1 - 0.25	Moderate shift	Monitor closely
> 0.25	Significant shift	Investigation required

Characteristics: - Numeric columns only - Automatic decile binning - Smoothing applied to prevent division by zero - Industry standard for model monitoring

2.3 Chi-Square Test¶

The chi-square test assesses independence between observed and expected categorical frequencies.

Mathematical Definition:

χ² = Σ (Oᵢ - Eᵢ)² / Eᵢ

Where Oᵢ is the observed frequency and Eᵢ is the expected frequency.

Usage:

drift = th.compare(baseline, current, method="chi2")

Characteristics:

Aspect	Description
Best For	Categorical variables
Output	χ²-statistic, p-value
Degrees of Freedom	k - 1 (where k is number of categories)

2.4 Jensen-Shannon Divergence (JS)¶

Jensen-Shannon divergence is a symmetrized and smoothed version of KL divergence.

Mathematical Definition:

JS(P||Q) = 0.5 × KL(P||M) + 0.5 × KL(Q||M)

Where M = 0.5 × (P + Q).

Usage:

drift = th.compare(baseline, current, method="js")

Interpretation:

JS Value	Interpretation
JS ≈ 0	Distributions are identical
JS < 0.1	Very similar distributions
0.1 <= JS < 0.3	Moderate difference
JS >= 0.3	Significant difference

Properties: - Symmetric: JS(P||Q) = JS(Q||P) - Bounded: 0 <= JS <= 1 (when using log base 2) - Metric: Square root of JS is a valid distance metric

2.5 Kullback-Leibler Divergence (KL)¶

KL divergence measures information loss when approximating one distribution with another.

Mathematical Definition:

KL(P||Q) = Σ P(x) × log(P(x) / Q(x))

Usage:

# KL divergence requires numeric columns
drift = th.compare(baseline, current, method="kl", columns=["age", "salary", "score"])

Note: KL divergence only works with numeric columns. For categorical data or symmetric divergence, use method="js" (Jensen-Shannon).

Interpretation:

KL Value	Interpretation
KL ≈ 0	Distributions are identical
KL < 0.1	Very similar distributions
0.1 <= KL < 0.2	Moderate difference
KL >= 0.2	Significant difference

Properties: - Asymmetric: KL(P||Q) ≠ KL(Q||P) - Non-negative: KL >= 0, with KL = 0 iff P = Q - Unbounded: Can be infinite if Q(x) = 0 where P(x) > 0 - Numeric columns only

2.6 Wasserstein Distance (Earth Mover's Distance)¶

Wasserstein distance measures the minimum "work" required to transform one distribution into another.

Mathematical Definition (1D case):

W(P, Q) = ∫|F_P(x) - F_Q(x)| dx

Where F_P and F_Q are the cumulative distribution functions.

Usage:

# Wasserstein distance requires numeric columns
drift = th.compare(baseline, current, method="wasserstein", columns=["age", "salary", "score"])

Note: Wasserstein distance only works with numeric columns. The statistic is normalized by baseline standard deviation for comparability.

Interpretation:

Normalized Wasserstein	Interpretation
W < 0.05	Very similar distributions
0.05 <= W < 0.1	Minor shift
0.1 <= W < 0.2	Moderate shift
W >= 0.2	Significant shift

Characteristics: - Numeric columns only - Metric (satisfies triangle inequality) - Meaningful even when distributions have non-overlapping support - Interpretable as "work" needed to move probability mass - Normalized by baseline standard deviation for scale independence

2.7 Cramér-von Mises Test¶

Cramér-von Mises is an alternative to KS that integrates squared differences between CDFs.

Mathematical Definition:

ω² = ∫[Fₙ(x) - F(x)]² dF(x)

Usage:

# Cramér-von Mises test requires numeric columns
drift = th.compare(baseline, current, method="cvm", columns=["age", "salary", "score"])

Note: Cramér-von Mises test only works with numeric columns and requires at least 2 samples in each dataset.

Characteristics:

Aspect	Description
Column Type	Numeric only
Best For	Detecting differences in entire distribution shape
Sensitivity	More sensitive to tail differences than KS
Output	ω² statistic, p-value
Threshold	p-value < 0.05 indicates drift

2.8 Anderson-Darling Test¶

Anderson-Darling test gives more weight to the tails of the distribution.

Mathematical Definition:

A² = -n - (1/n) × Σ (2i-1) × [ln(F(xᵢ)) + ln(1-F(x_{n+1-i}))]

Usage:

# Anderson-Darling test requires numeric columns
drift = th.compare(baseline, current, method="anderson", columns=["age", "salary", "score"])

Note: Anderson-Darling test only works with numeric columns and requires at least 2 samples in each dataset.

Characteristics:

Aspect	Description
Column Type	Numeric only
Best For	Detecting differences in distribution tails
Sensitivity	More sensitive to tail deviations than KS or CvM
Output	A² statistic, p-value
Threshold	p-value < 0.05 indicates drift

Interpretation (based on critical values):

p-value	Interpretation
p > 0.25	No significant difference
0.05 < p <= 0.25	Weak evidence of difference
0.01 < p <= 0.05	Moderate evidence of difference
p <= 0.01	Strong evidence of difference

2.9 Hellinger Distance¶

Hellinger distance measures the similarity between two probability distributions with desirable metric properties.

Mathematical Definition:

H(P, Q) = (1/√2) × √(Σ(√pᵢ - √qᵢ)²)

Where pᵢ and qᵢ are probabilities for category/bin i.

Usage:

drift = th.compare(baseline, current, method="hellinger")

Interpretation:

Hellinger Value	Interpretation
H = 0	Distributions are identical
H < 0.1	Very similar distributions
0.1 <= H < 0.2	Moderate difference
H >= 0.2	Significant difference
H = 1	Distributions have no overlap

Characteristics:

Aspect	Description
Column Type	Numeric and Categorical
Range	Bounded [0, 1]
Symmetry	Symmetric: H(P,Q) = H(Q,P)
Metric	True metric (satisfies triangle inequality)
Relationship	H(P,Q) = √(1 - BC(P,Q)) where BC is Bhattacharyya coefficient

2.10 Bhattacharyya Distance¶

Bhattacharyya distance measures the overlap between two probability distributions, with connections to classification error bounds.

Mathematical Definition:

D_B(P, Q) = -ln(BC(P, Q))
BC(P, Q) = Σ√(pᵢ × qᵢ)  (Bhattacharyya coefficient)

Usage:

drift = th.compare(baseline, current, method="bhattacharyya")

Interpretation:

Bhattacharyya Distance	Interpretation
D_B ≈ 0	Distributions are identical
D_B < 0.1	Very similar distributions
0.1 <= D_B < 0.2	Moderate difference
D_B >= 0.2	Significant difference

Characteristics:

Aspect	Description
Column Type	Numeric and Categorical
Range	[0, +∞)
BC Coefficient	Bounded [0, 1] (reported in details)
Application	Related to Bayes classification error
Relationship	Related to Hellinger: H² = 1 - BC

2.11 Total Variation Distance¶

Total Variation (TV) distance measures the maximum difference in probability between two distributions.

Mathematical Definition:

TV(P, Q) = (1/2) × Σ|pᵢ - qᵢ| = max_A |P(A) - Q(A)|

Usage:

drift = th.compare(baseline, current, method="tv")
# or
drift = th.compare(baseline, current, method="total_variation")

Interpretation:

TV Value	Interpretation
TV = 0	Distributions are identical
TV < 0.1	Very similar distributions
0.1 <= TV < 0.2	Moderate difference
TV >= 0.2	Significant difference
TV = 1	Distributions have completely disjoint support

Characteristics:

Aspect	Description
Column Type	Numeric and Categorical
Range	Bounded [0, 1]
Symmetry	Symmetric: TV(P,Q) = TV(Q,P)
Metric	True metric (satisfies triangle inequality)
Interpretation	"Largest possible probability difference for any event"

Relationship with Hellinger:

H²(P,Q) ≤ TV(P,Q) ≤ √2 × H(P,Q)

2.12 Energy Distance¶

Energy distance is a statistical distance that characterizes the equality of distributions and has desirable metric properties.

Mathematical Definition:

E(P, Q) = 2×E[|X-Y|] - E[|X-X'|] - E[|Y-Y'|]

Where X, X' ~ P and Y, Y' ~ Q are independent samples.

Usage:

# Energy distance requires numeric columns
drift = th.compare(baseline, current, method="energy", columns=["age", "salary"])

Note: Energy distance only works with numeric columns.

Interpretation:

Normalized Energy	Interpretation
E ≈ 0	Distributions are identical
E < 0.1	Very similar distributions
0.1 <= E < 0.2	Moderate difference
E >= 0.2	Significant difference

Characteristics:

Aspect	Description
Column Type	Numeric only
Range	[0, +∞), normalized by pooled std
Metric	True metric (satisfies triangle inequality)
Consistency	E(P,Q) = 0 if and only if P = Q
Computational	O(n²) for exact, can subsample for efficiency

2.13 Maximum Mean Discrepancy (MMD)¶

Maximum Mean Discrepancy is a kernel-based distance measure that compares distributions in a reproducing kernel Hilbert space (RKHS).

Mathematical Definition:

MMD²(P, Q) = E[k(X,X')] + E[k(Y,Y')] - 2×E[k(X,Y)]

Where k is a kernel function (default: Gaussian RBF kernel).

Usage:

# MMD requires numeric columns
drift = th.compare(baseline, current, method="mmd", columns=["feature1", "feature2"])

Note: MMD only works with numeric columns.

Kernel Options (configurable via API):

Kernel	Formula	Best For
RBF (default)	k(x,y) = exp(-γ‖x-y‖²)	General purpose
Linear	k(x,y) = x·y	Linear differences
Polynomial	k(x,y) = (1 + x·y)²	Non-linear patterns

Interpretation:

MMD Value	Interpretation
MMD ≈ 0	Distributions are identical (in RKHS)
MMD < 0.1	Very similar distributions
0.1 <= MMD < 0.2	Moderate difference
MMD >= 0.2	Significant difference

Characteristics:

Aspect	Description
Column Type	Numeric only
Range	[0, +∞)
Non-parametric	No density estimation required
High-dimensional	Works well where density estimation fails
Bandwidth	Auto-selected via median heuristic or custom
Computational	O(n²), can subsample for efficiency

Currently Available Methods Summary¶

Method	`th.compare()`	ML API	Column Type
`auto`	✅	-	Any (auto-select)
`ks`	✅	-	Numeric only
`psi`	✅	✅	Numeric only
`chi2`	✅	-	Categorical
`js`	✅	✅ (`jensen_shannon`)	Any
`kl`	✅	-	Numeric only
`wasserstein`	✅	✅	Numeric only
`cvm`	✅	-	Numeric only
`anderson`	✅	-	Numeric only
`hellinger`	✅	-	Any
`bhattacharyya`	✅	-	Any
`tv`	✅	-	Any
`energy`	✅	-	Numeric only
`mmd`	✅	-	Numeric only

3. Anomaly Detection Methods¶

3.1 Z-Score Method¶

Z-score identifies outliers based on standard deviations from the mean.

Mathematical Definition:

z = (x - μ) / σ

Usage:

from truthound.ml import ZScoreAnomalyDetector

detector = ZScoreAnomalyDetector(threshold=3.0)
detector.fit(df)
result = detector.detect(df)

Characteristics:

Aspect	Description
Best For	Normally distributed data
Threshold	Typically
Assumption	Data is approximately Gaussian

3.2 Interquartile Range (IQR)¶

IQR method uses quartiles to define outlier boundaries.

Mathematical Definition:

IQR = Q3 - Q1
Lower Bound = Q1 - k × IQR
Upper Bound = Q3 + k × IQR

Usage:

from truthound.ml import IQRAnomalyDetector

detector = IQRAnomalyDetector(multiplier=1.5)

Parameters:

k Value	Description
1.5	Standard outliers
3.0	Extreme outliers

Characteristics: - Distribution-free and robust to extreme values - Works best for symmetric or approximately symmetric distributions - Resistant to extreme outliers

3.3 Modified Z-Score (MAD)¶

Modified Z-score uses Median Absolute Deviation for robustness.

Mathematical Definition:

MAD = median(|xᵢ - median(x)|)
M = 0.6745 × (xᵢ - median(x)) / MAD

Usage:

from truthound.ml import MADAnomalyDetector

detector = MADAnomalyDetector(threshold=3.5)

Characteristics: - Highly resistant to outliers - Threshold: |M| > 3.5 typically

3.4 Isolation Forest¶

Isolation Forest isolates anomalies by random recursive partitioning.

Principle: Anomalies are easier to isolate and require fewer splits.

Anomaly Score:

s(x, n) = 2^(-E(h(x)) / c(n))

Where: - h(x) is the path length for observation x - c(n) is the average path length for n samples - E(h(x)) is the expected path length

Usage:

from truthound.ml import IsolationForestDetector

detector = IsolationForestDetector(
    contamination=0.1,
    n_estimators=100
)
detector.fit(df)
result = detector.detect(df)

Interpretation:

Score	Interpretation
s ≈ 1	Anomaly
s ≈ 0.5	Normal
s < 0.5	Definitely normal

Characteristics: - Linear time complexity O(n) - Effective for high-dimensional data

3.5 Local Outlier Factor (LOF)¶

LOF identifies anomalies based on local density deviation.

Algorithm: 1. Compute k-nearest neighbors for each point 2. Calculate local reachability density (LRD) 3. Compare LRD of a point to LRDs of its neighbors

Formula:

LOF(x) = (Σ LRD(neighbor) / LRD(x)) / k

Usage:

from truthound.ml import LOFDetector

detector = LOFDetector(n_neighbors=20)

Interpretation:

LOF Value	Interpretation
LOF ≈ 1	Normal (similar density to neighbors)
LOF > 1	Lower density than neighbors (potential outlier)
LOF >> 1	Significant outlier

3.6 Mahalanobis Distance¶

Mahalanobis distance accounts for correlations between variables.

Mathematical Definition:

D = √((x - μ)ᵀ × Σ⁻¹ × (x - μ))

Where: - x is the observation vector - μ is the mean vector - Σ is the covariance matrix

Usage:

from truthound.ml import MahalanobisDetector

detector = MahalanobisDetector(threshold=3.0)

Characteristics: - Scale-invariant - Accounts for correlations between variables - D² follows a chi-square distribution with p degrees of freedom

3.7 Additional Anomaly Methods¶

Method	Approach	Best For
`grubbs`	Grubbs' test	Single outliers in univariate data
`esd`	Generalized ESD	Multiple outliers in univariate data
`dbscan`	Density-based clustering	Arbitrary-shaped clusters
`svm`	One-Class SVM	Non-linear boundaries
`autoencoder`	Reconstruction error	High-dimensional, complex patterns
`percentile`	Percentile bounds	Simple threshold-based detection
`tukey`	Tukey fences	Robust statistical bounds

3.8 Ensemble Methods¶

from truthound.ml import EnsembleAnomalyDetector

ensemble = EnsembleAnomalyDetector(
    detectors=[zscore_detector, iqr_detector, iso_detector],
    voting_strategy="majority"  # or "unanimous", "any"
)

4. Distribution Analysis¶

4.1 Normality Tests¶

Test	Method	Best For
Shapiro-Wilk	W statistic	Small samples (n < 5000)
D'Agostino-Pearson	Skewness + kurtosis	Medium samples
Kolmogorov-Smirnov	CDF comparison	Large samples
Anderson-Darling	Weighted CDF	General use

4.2 Descriptive Statistics¶

profile = th.profile(df)

# Available statistics per column:
# - count, null_count, null_ratio
# - mean, std, variance
# - min, max, range
# - q25, median (q50), q75
# - skewness, kurtosis
# - unique_count, unique_ratio

4.3 Entropy and Information¶

Shannon Entropy:

H(X) = -Σ p(xᵢ) × log₂(p(xᵢ))

Usage:

profile = th.profile(df)
# Entropy available in profile.columns[col].entropy

5. Statistical Thresholds¶

5.1 Default Thresholds¶

Method	Threshold	Interpretation
Z-Score	3.0	> 3 standard deviations
IQR	1.5	Outside 1.5 × IQR
MAD	3.5	> 3.5 modified z-scores
LOF	1.5	LOF score > 1.5
Isolation Forest	0.1	Top 10% anomaly scores
KS Test	0.05	p-value threshold
PSI	0.25	Significant drift threshold
Chi-Square	0.05	p-value threshold

5.2 Threshold Configuration¶

# Drift detection with custom thresholds
drift = th.compare(
    baseline, current,
    method="psi",
    threshold=0.1  # More sensitive threshold
)

# Anomaly detection with custom thresholds
from truthound.ml import ZScoreAnomalyDetector

detector = ZScoreAnomalyDetector(threshold=2.5)  # More sensitive

6. Method Selection Guide¶

6.1 By Data Type¶

Data Type	Drift Method	Anomaly Method
Continuous	KS, Wasserstein, Energy	Z-Score, IQR, Isolation Forest
Categorical	Chi-Square, JS, Hellinger, TV	Mode deviation, category frequency
Ordinal	KS, Wasserstein	IQR, percentile
High-dimensional	MMD, Energy	Isolation Forest, Autoencoder
Time Series	KS with windows	LOF, ARIMA residuals
Probability Distributions	Hellinger, Bhattacharyya, TV	-

6.2 By Sample Size¶

Sample Size	Recommended Methods
n < 100	Exact tests, bootstrap
100 < n < 10,000	KS, Chi-Square, Z-Score
n > 10,000	PSI, JS, Isolation Forest
n > 1,000,000	Sampled methods, streaming

6.3 By Sensitivity Requirements¶

Requirement	Methods
High sensitivity	Anderson-Darling, MAD (low threshold), Energy
Balanced	KS, PSI, IQR, Hellinger, TV
Low false positives	Mahalanobis, Ensemble voting, Bhattacharyya
True metric needed	Hellinger, TV, Energy, MMD

6.4 Decision Tree¶

Is data continuous?
├─ Yes
│   ├─ Normally distributed? → Z-Score for anomaly, KS for drift
│   └─ Skewed/Unknown? → IQR for anomaly, PSI for drift
└─ No (Categorical)
    ├─ Few categories (< 20)? → Chi-Square, mode deviation
    └─ Many categories? → Frequency analysis, JS divergence

Is data high-dimensional?
├─ Yes (> 10 features) → Isolation Forest, MMD
└─ No → Standard univariate methods

Are there existing outliers?
├─ Yes → MAD, IQR (robust methods)
└─ No → Z-Score, Mahalanobis

7. Mathematical Foundations¶

7.1 Empirical Distribution Functions¶

The empirical CDF is defined as:

F̂ₙ(x) = (1/n) × Σ 1_{Xᵢ ≤ x}

This forms the basis for KS, CvM, and Anderson-Darling tests.

7.2 Information Theory Basics¶

Entropy measures uncertainty:

H(X) = -Σ P(x) × log P(x)

Relative Entropy (KL Divergence) measures distribution difference:

D_KL(P || Q) = Σ P(x) × log(P(x) / Q(x))

7.3 Hypothesis Testing Framework¶

Statistical tests follow the framework:

Null Hypothesis (H₀): No difference between distributions
Alternative Hypothesis (H₁): Distributions differ
Test Statistic: Computed from data
p-value: Probability of observing test statistic under H₀
Decision: Reject H₀ if p-value < α (typically 0.05)

7.4 Multiple Testing Correction¶

When testing multiple columns:

Method	Formula	Use Case
Bonferroni	α' = α / n	Conservative, independent tests
Holm	Sequential adjustment	Less conservative
Benjamini-Hochberg	FDR control	Many tests, some false positives acceptable

# Truthound applies Benjamini-Hochberg by default for multiple columns
drift = th.compare(baseline, current, correction="bh")

8. References¶

Kolmogorov, A. N. (1933). "Sulla determinazione empirica di una legge di distribuzione"
Smirnov, N. (1948). "Table for Estimating the Goodness of Fit of Empirical Distributions"
Liu, F. T., Ting, K. M., & Zhou, Z. H. (2008). "Isolation Forest"
Breunig, M. M., et al. (2000). "LOF: Identifying Density-Based Local Outliers"
Iglewicz, B., & Hoaglin, D. C. (1993). "How to Detect and Handle Outliers"
Kullback, S., & Leibler, R. A. (1951). "On Information and Sufficiency"
Lin, J. (1991). "Divergence measures based on the Shannon entropy"
Mahalanobis, P. C. (1936). "On the generalized distance in statistics"
Vaserstein, L. N. (1969). "Markov processes over denumerable products of spaces"
Tukey, J. W. (1977). "Exploratory Data Analysis"
Pearson, K. (1900). "On the criterion that a given system of deviations..."
Hellinger, E. (1909). "Neue Begründung der Theorie quadratischer Formen..."
Bhattacharyya, A. (1943). "On a measure of divergence between two statistical populations"
Gretton, A., et al. (2012). "A Kernel Two-Sample Test" (MMD)
Székely, G. J., & Rizzo, M. L. (2004). "Testing for equal distributions in high dimension"

Statistical Methods¶

Table of Contents¶

1. Overview¶

2. Drift Detection Methods¶

2.1 Kolmogorov-Smirnov Test (KS)¶

2.2 Population Stability Index (PSI)¶

2.3 Chi-Square Test¶

2.4 Jensen-Shannon Divergence (JS)¶

2.5 Kullback-Leibler Divergence (KL)¶

2.6 Wasserstein Distance (Earth Mover's Distance)¶

2.7 Cramér-von Mises Test¶

2.8 Anderson-Darling Test¶

2.9 Hellinger Distance¶

2.10 Bhattacharyya Distance¶

2.11 Total Variation Distance¶

2.12 Energy Distance¶

2.13 Maximum Mean Discrepancy (MMD)¶

Currently Available Methods Summary¶

3. Anomaly Detection Methods¶

3.1 Z-Score Method¶

3.2 Interquartile Range (IQR)¶

3.3 Modified Z-Score (MAD)¶

3.4 Isolation Forest¶

3.5 Local Outlier Factor (LOF)¶

3.6 Mahalanobis Distance¶

3.7 Additional Anomaly Methods¶

3.8 Ensemble Methods¶

4. Distribution Analysis¶

4.1 Normality Tests¶

4.2 Descriptive Statistics¶

4.3 Entropy and Information¶

5. Statistical Thresholds¶

5.1 Default Thresholds¶

5.2 Threshold Configuration¶

6. Method Selection Guide¶

6.1 By Data Type¶

6.2 By Sample Size¶

6.3 By Sensitivity Requirements¶

6.4 Decision Tree¶

7. Mathematical Foundations¶

7.1 Empirical Distribution Functions¶

7.2 Information Theory Basics¶

7.3 Hypothesis Testing Framework¶

7.4 Multiple Testing Correction¶

8. References¶

See Also¶