Emission probability calculator
Enter matching symbol and count lists. Separate entries with commas or new lines.
Plotly graph
The chart uses the current results when available. Otherwise, it shows the example dataset.
Example data table
This sample shows how counts turn into raw and smoothed probabilities.
| Symbol | Count | Raw Probability | Smoothed Probability | Expected Count at 50 Trials |
|---|---|---|---|---|
| A | 22 | 0.440000 | 0.425926 | 21.296296 |
| B | 14 | 0.280000 | 0.277778 | 13.888889 |
| C | 9 | 0.180000 | 0.185185 | 9.259259 |
| D | 5 | 0.100000 | 0.111111 | 5.555556 |
Example settings: symbols A, B, C, D; counts 22, 14, 9, 5; alpha 1; future sequence length 50.
Formula used
Raw emission probability: P(xᵢ) = cᵢ / Σcⱼ
Smoothed emission probability: Pα(xᵢ) = (cᵢ + α) / (Σcⱼ + αK)
Expected count in a future sequence: E(xᵢ) = n × Pα(xᵢ)
Information or surprise: I(xᵢ) = -log₂(Pα(xᵢ))
Entropy of the emission distribution: H = -Σ Pα(xᵢ) log₂(Pα(xᵢ))
Perplexity: Perplexity = 2ᴴ. Here, cᵢ is the observed count, α is the smoothing value, K is the number of emission categories, and n is the future sequence length.
How to use this calculator
- Enter every emission symbol in the first box. Use commas or new lines.
- Enter the matching observed count for each symbol in the second box.
- Choose one symbol for focused analysis in the selected symbol field.
- Set the smoothing alpha value. Zero keeps raw probabilities unchanged.
- Add a future sequence length to estimate expected counts.
- Pick the display precision, then press the calculate button.
- Review the result cards, sorted table, and probability chart.
- Download the result or example table as CSV or PDF when needed.
FAQs
1. What does this tool measure?
It estimates how likely each emission symbol is, based on observed counts. It also adds smoothing, expected future counts, entropy, perplexity, and information content for deeper probability analysis.
2. Why would I use smoothing alpha?
Smoothing prevents zero probabilities when a symbol has no observed count. That helps when modeling sparse observations, comparing sequences, or preparing inputs for probabilistic systems that dislike impossible events.
3. What happens when alpha equals zero?
The tool returns unsmoothed probabilities from the raw counts only. This is useful when every category was already observed and you want direct frequency estimates without adjustment.
4. Why is the selected symbol important?
The selected symbol drives focused metrics like expected future count, surprise bits, and log sequence probability. It helps you inspect one emission outcome more closely without losing the full distribution view.
5. What does entropy mean here?
Entropy summarizes how spread out the emission distribution is. Higher entropy means more uncertainty across symbols. Lower entropy means the distribution is concentrated around fewer outcomes.
6. What is perplexity used for?
Perplexity converts entropy into an easier scale. It reflects the effective number of equally likely symbols in the distribution, which is helpful when comparing different emission datasets.
7. Can I paste symbols and counts on separate lines?
Yes. The tool accepts commas, semicolons, vertical bars, and line breaks. Just make sure the symbol list and count list have the same number of entries.
8. What do the CSV and PDF buttons export?
They export the visible result table or example table. That makes it easier to share the probability breakdown, document assumptions, or reuse the data in other workflows.