Measure information transfer between variables in experiments accurately. Choose log base, units, and smoothing controls. Upload joint counts, compute MI, and export tables easily.
Mutual information measures how much knowing one discrete variable reduces uncertainty about another. Using a joint probability table p(x,y) with marginals p(x)=Σy p(x,y) and p(y)=Σx p(x,y), the mutual information is:
Entropy is computed as H(X)=−Σx p(x)·log_b p(x) and similarly for H(Y). The calculator also reports H(X|Y)=H(X,Y)−H(Y) and H(Y|X)=H(X,Y)−H(X).
In physics, mutual information is used in signal analysis, detector correlations, and studies of coupled systems.
Practical tip: when using experimental counts, keep the binning strategy consistent across runs.
Example joint counts for two outcomes (2×2). This example yields mutual information of about 0.2564 bits.
| Counts | Y1 | Y2 |
|---|---|---|
| X1 | 30 | 10 |
| X2 | 10 | 50 |
Mutual information (MI) reports how strongly two discrete outcomes co-vary beyond chance. If two sensors respond independently, the joint distribution factors into marginals and MI approaches zero. If outcomes are tightly linked, MI increases because observing one variable reduces uncertainty about the other.
The log base sets the information unit: base 2 gives bits, base e gives nats, and base 10 gives hartleys. Many lab reports use bits because it connects directly to binary decisions and digital acquisition chains. Keep the base consistent when comparing runs or instrument configurations.
MI is always non‑negative and is bounded by the smaller marginal entropy: 0 ≤ I(X;Y) ≤ min(H(X), H(Y)). A value near the bound suggests one variable almost determines the other. A small value can still be meaningful when the entropies are also small.
For time-series experiments, the joint matrix is commonly produced by binning signals and counting co-occurrences. For example, two-level detectors create a 2×2 table, while multi-level quantizers form larger tables. Use consistent bin edges and sampling alignment to avoid artificial dependencies.
With limited samples, MI can be biased upward because rare bins appear more structured than they are. As a practical guide, ensure the total count greatly exceeds the number of joint bins. If you expand from 2×2 to 10×10 bins, you need far more observations to stabilize estimates.
Zero-probability cells are common when events are rare or binning is fine. The smoothing parameter ε adds a small value to every cell before normalization, preventing log(0). Use very small ε (such as 1e-12) when you only want numerical stability, not heavy regularization.
Raw MI depends on marginal entropies, which can change with operating point and noise. Normalized MI (such as I / √(H(X)H(Y))) helps compare different datasets on a 0–1 scale. This is useful when signal power changes across temperature, bias, or gain settings.
MI is widely used to test coupling hypotheses: if a control variable should influence an observable, MI should increase compared with a shuffled baseline. In communication-style setups, MI estimates shared information between input modulation and measured output, providing a compact metric for channel quality and nonlinear distortions.
It approaches zero because the joint probability equals the product of marginals. Small nonzero values can appear from finite sampling noise and binning choices.
Theoretical MI is never negative. If you see negative values, it usually indicates numerical issues, inconsistent probabilities, or rounding errors in the input matrix.
Enter counts when you have raw co-occurrence totals. Enter probabilities when your matrix is already normalized. If probabilities do not sum to one, enable auto-normalization.
Use base 2 for bits in most measurement reports, base e for analytical work with natural logs, and base 10 when matching older instrumentation or documentation conventions.
Start with ε = 0. If zeros cause instability, use a tiny ε like 1e-12. Larger ε can reduce variance but may bias MI downward by flattening the distribution.
Normalization helps compare datasets with different entropies. If noise or operating conditions change the marginal distributions, normalized MI provides a more comparable coupling score.
Compute MI for the real data, then compare to a shuffled baseline where one variable is permuted. A clear separation indicates genuine dependence rather than sampling artifacts.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.