KL Divergence Calculator

Formula used

For discrete distributions P = (p₁…pₙ) and Q = (q₁…qₙ), the Kullback–Leibler divergence is:

KL(P‖Q) = Σᵢ pᵢ · log_b(pᵢ / qᵢ)

• Non-negativity: KL(P‖Q) ≥ 0, and equals 0 only when P = Q.
• Asymmetry: KL(P‖Q) ≠ KL(Q‖P) in general.
• Zero handling: If pᵢ = 0, the term is 0 by convention.

How to use this calculator

1. Paste your P values and Q values with equal length.
2. Enable normalization if your inputs are raw counts.
3. Choose KL direction, symmetric KL, or Jensen–Shannon.
4. Add epsilon smoothing if zeros cause infinity.
5. Click Compute to show results above the form.
6. Use Download CSV or PDF for shareable outputs.

Article

Why KL divergence is used in practice

KL divergence quantifies how one discrete distribution shifts from another. In monitoring, it flags drift when predicted class probabilities change over time. In A/B tests, it compares response mixes across cohorts. With base 2 logs, results are in bits; values near 0.00 indicate close alignment, while larger values show material mismatch across categories.

Choosing direction and interpreting asymmetry

KL(P‖Q) weights differences by P, so it penalizes cases where Q underestimates mass where P is high. KL(Q‖P) answers a different question and can be much larger. Use symmetric KL when you need a single distance-like indicator. Use Jensen–Shannon when you want a bounded measure that is always finite after smoothing.

Handling zeros and stability controls

Zeros are the main operational risk: if any qᵢ equals zero where pᵢ is positive, KL becomes infinite. Epsilon smoothing adds a small constant to every entry, then normalizes, preventing undefined ratios. The ratio floor further protects against underflow when probabilities are extremely small, keeping logs numerically stable without changing the overall structure.

Normalization for counts and weighted data

Many real datasets start as counts, frequencies, or weighted scores rather than probabilities. Normalization converts them to a valid distribution that sums to one, making KL comparable across samples of different sizes. If you already have probabilities, keep normalization enabled anyway; it also corrects rounding errors such as sums of 0.9999 or 1.0002.

Reading the term breakdown table

Each row shows log(pᵢ/qᵢ) and the contribution pᵢ·log(pᵢ/qᵢ). Contributions can be negative when qᵢ exceeds pᵢ, but the total remains non‑negative. The Plotly term chart makes drivers obvious: a few categories often explain most of the divergence, which is useful for debugging model shifts or distribution drift.

Reporting and repeatability with exports

CSV export supports audit trails by storing your selected mode, log base, and smoothing settings alongside the per‑index terms. The PDF report provides a quick shareable summary for stakeholders, including key metrics and a preview of terms. Together, these exports help you reproduce results and compare runs across time, datasets, or model versions.

FAQs

1) What does a KL value of zero mean?

It means the two distributions match exactly at every index after any smoothing and normalization settings you selected, so there is no information loss when using Q to represent P.

2) Why can KL divergence become infinite?

If Q assigns zero probability to an outcome that P considers possible (pᵢ>0, qᵢ=0), the log ratio explodes. Add epsilon smoothing to avoid this issue.

3) When should I use base 2 instead of natural logs?

Use base 2 when you want divergence expressed in bits, which is common in information theory and model monitoring dashboards. Natural logs report in nats.

4) What is the difference between symmetric KL and Jensen–Shannon?

Symmetric KL adds KL(P‖Q) and KL(Q‖P) and is still unbounded. Jensen–Shannon mixes the distributions first, producing a bounded, more stable divergence.

5) Should I normalize if my values already sum to one?

Yes, it is usually safe. Normalization corrects minor rounding drift and ensures the calculator treats the inputs as proper distributions, improving comparability across runs.

6) What epsilon value is a reasonable starting point?

Start small, such as 1e-12 to 1e-6, depending on scale. Larger epsilons reduce sensitivity to rare outcomes but can hide genuine zero‑mass differences.