Description
Compute both Shannon entropies, entropy of their weighted mixture, and mixture entropy minus weighted component entropies.
Entropy summarizes uncertainty under a specified probability model and logarithm convention. Support indicators, one-symbol terms, and exponentiated entropies are components with different meanings.
Compute both Shannon entropies, entropy of their weighted mixture, and mixture entropy minus weighted component entropies.
When to use this tool
Use Information: Entropy Concavity Single Mixture Gap when you need one explicitly scoped entropy or support calculation.
How to use the tool and how it works
- Enter first, second, firstWeight, base. Keep parallel arrays aligned. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read result. Expand result to inspect the returned fields.
Require aligned nonempty probability vectors summing to one within 1e-12, a mixture weight in [0, 1], and base > 1. Compute both Shannon entropies, entropy of their weighted mixture, and mixture entropy minus weighted component entropies.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| first | First value or boundary under the stated formula. | Required. Array of finite number. Implementation limit: 128 items. |
| second | Second value or boundary under the stated formula. | Required. Array of finite number. Implementation limit: 128 items. |
| firstWeight | First Weight | Required. Finite number. |
| base | Numeric base or source quantity interpreted under the specific calculation above. | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Result | Required. Object. |
| result.firstEntropy | First Entropy | Calculated result member: number. |
| result.secondEntropy | Second Entropy | Calculated result member: number. |
| result.mixtureEntropy | Mixture Entropy | Calculated result member: number. |
| result.concavityGap | Concavity Gap | Calculated result member: number. |
{
"input": {
"first": [
1,
0
],
"second": [
0,
1
],
"firstWeight": 0.5,
"base": 2
},
"output": {
"result": {
"firstEntropy": 0,
"secondEntropy": 0,
"mixtureEntropy": 1,
"concavityGap": 1
}
}
}Limitations and assumptions
- Matching positions must represent the same symbols; finite precision can leave a tiny negative gap and no clamping or statistical estimation is applied.
- Outputs are nested inside result. Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Validate and align complete probability distributions before aggregating terms or interpreting an uncertainty estimate.
References
Similar or alternative tools
- Information: Symbol Probability Merge Entropy Delta
Compute the merged Shannon term minus the two separate terms, defining zero-mass terms as zero.
- Jensen Shannon One Bin Contribution
Validate first and second in [0, 1], use their mean as reference, and return half-weighted base-two relative-entropy terms, treating zero terms as zero.