Mutual Information Calculator
Analyze joint states, marginals, entropy, and information transfer. Built for experiments, detectors, channels, and measurements. See results above the form with intuitive visual summaries.
Enter Joint Data
Use counts from a paired physics experiment or directly enter joint probabilities for two binary states.
Example Data Table
This example represents paired detector observations where each device reports a low or high state during the same measurement interval.
| Detector A state | Detector B state | Observed count | Joint probability |
|---|---|---|---|
| Low | Low | 40 | 0.4000 |
| Low | High | 10 | 0.1000 |
| High | Low | 15 | 0.1500 |
| High | High | 35 | 0.3500 |
Formula Used
Mutual Information
I(X;Y) = Σ p(x,y) × logb [ p(x,y) / ( p(x)p(y) ) ]
Marginal Probabilities
p(x) = Σ p(x,y) and p(y) = Σ p(x,y)
Entropy Terms
H(X) = -Σ p(x) logb p(x), H(Y) = -Σ p(y) logb p(y), H(X,Y) = -Σ p(x,y) logb p(x,y)
From Counts to Probabilities
p(x,y) = n(x,y) / N, where N is the total number of paired observations.
In physics, mutual information helps quantify shared structure between two observed states, channels, detectors, spin configurations, or sampled signal conditions.
How to Use This Calculator
- Enter a label for your experiment or paired measurement set.
- Choose whether you are entering raw counts or already normalized probabilities.
- Rename the X and Y state labels to match your detector, channel, or signal states.
- Fill in the four joint values for the binary state combinations.
- Select the log base for bits, nats, or hartleys.
- Press the calculate button to show results above the form.
- Review entropy values, normalized mutual information, and uncertainty reduction percentages.
- Use the CSV or PDF buttons to export the current result summary.
Frequently Asked Questions
1. What does mutual information measure?
It measures how much knowing one variable reduces uncertainty about the other. Higher values mean stronger statistical dependence between paired states or observed outcomes.
2. Why is mutual information useful in physics?
It helps evaluate relationships between detector outputs, channel states, spin measurements, sensor readings, and correlated experimental observations without assuming a purely linear relationship.
3. Can I enter probabilities instead of counts?
Yes. Choose the probabilities mode and enter the four joint probabilities. If they do not sum exactly to one, the calculator normalizes them automatically.
4. What is the difference between bits and nats?
The difference is the logarithm base. Base 2 reports bits, while base e reports nats. The dependence structure is the same, but the numerical units change.
5. What does normalized mutual information show?
It rescales mutual information so you can compare dependence strength more easily across different datasets. Values closer to one indicate stronger shared structure.
6. What happens when a joint cell is zero?
Zero-probability cells contribute nothing to the mutual information sum. This follows the standard limiting behavior used in entropy and information calculations.
7. Is this calculator limited to binary variables?
This version is designed for two-state X and Y variables, which fits many detector, signal, and event classification problems. Larger state spaces need expanded matrices.
8. How should I interpret a low mutual information value?
A low value suggests weak dependence. Knowing one state gives little help predicting the other, so the paired observations are closer to statistical independence.