Poisson Log Likelihood Count Increment Tool

Description

Return increment=ln(rate)-ln(count+1) for positive rate and an integer count from zero through Number.MAX_SAFE_INTEGER-1.

Event probabilities and probability masses lie between zero and one. Marginals, intersections, conditioning events, and endpoint atoms describe different quantities. A probability bound or local consistency check can constrain a model without specifying its complete distribution or dependence structure.

Return increment=ln(rate)-ln(count+1) for positive rate and an integer count from zero through Number.MAX_SAFE_INTEGER-1.

When to use this tool

Use Poisson Log Likelihood Count Increment when you need the stated event identity, boundary policy, distribution component, or local feasibility check.

How to use the tool and how it works

Enter count, rate. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read increment. Use the output convention below.

Return increment=ln(rate)-ln(count+1) for positive rate and an integer count from zero through Number.MAX_SAFE_INTEGER-1.^[1]

Input field	Meaning and units	Accepted input
count	Count under the stated event or record definition.	Required. Safe integer; minimum 0.
rate	Rate	Required. Finite number; strictly above 0.

Output field	Interpretation	Returned shape
increment	Increment	Required. Finite number.

json

{
  "input": {
    "count": 3,
    "rate": 8
  },
  "output": {
    "increment": 0.6931471805599452
  }
}

Limitations and assumptions

This is the change in Poisson log probability from count to count+1 at fixed rate, not a derivative with respect to rate or an entire likelihood.
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. Signed zero is exported as zero.

Alternative or Complementary methods

Use a full probability table or distribution when a local mass, bound, or identity is insufficient. Specify dependence, conditioning, support order, and endpoint policy before combining terms.

References

Calculation and input contract: probability/poisson-log-likelihood-count-increment — tool.io

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