This calculator evaluates statistical decisions under uncertainty. Enter states, actions, prior probabilities, and a loss matrix. It then computes each action's expected loss, identifies the minimum value, and reports the Bayes risk with a decision ranking.
Use 2 to 8 states and 2 to 8 actions for readable tables.
| Action | Low Demand | Moderate Demand | High Demand | Expected Loss |
|---|---|---|---|---|
| Small Inventory | 8 | 14 | 30 | 16.20 |
| Balanced Inventory | 12 | 9 | 16 | 11.70 |
| Large Inventory | 24 | 13 | 7 | 14.00 |
Example priors: Low Demand = 0.30, Moderate Demand = 0.50, High Demand = 0.20. The smallest expected loss is 11.70, so the Bayes risk is 11.70.
For each action aj, compute the expected loss using prior probabilities across all states:
R(aj) = Σ [ P(θi) × L(aj, θi) ]
Here, P(θi) is the prior probability of state θi, and L(aj, θi) is the loss from choosing action aj when that state occurs.
The Bayes risk is the minimum expected loss among all candidate actions:
Bayes Risk = min R(aj)
If the priors you enter do not add to 1.0000, this calculator normalizes them before computing expected losses.
- Choose how many states of nature and actions you want to compare.
- Enter a name and prior probability for every state.
- Enter a name for each action.
- Fill the loss matrix with the cost or penalty for every action-state pair.
- Click Calculate Bayes Risk to see the best action, Bayes risk, and ranked expected losses.
- Use the CSV button to export summary results or the PDF button to save the visible report.
1. What does Bayes risk measure?
Bayes risk measures the smallest expected loss across competing decisions when you combine prior probabilities with a specified loss function.
2. Why are prior probabilities required?
Priors represent your belief about how likely each state is before observing new data. They weight the losses in the expected loss calculation.
3. What if my priors do not sum to one?
The calculator rescales them automatically. It divides each prior by the total prior sum, then uses the normalized values for all computations.
4. Can I use costs instead of losses?
Yes. If your decision problem is framed with costs, enter them directly as losses. The best action remains the one with the lowest expected value.
5. What does the relative gap show?
The relative gap shows how much worse an action is than the best action. A zero gap means that action achieves the Bayes risk.
6. Can this help with classification decisions?
Yes. Bayes risk is widely used in classification, quality control, forecasting, and operations whenever different mistakes have different penalties.
7. Does a lower Bayes risk always mean a better rule?
Within the same model, priors, and loss structure, yes. Lower Bayes risk means smaller expected loss under the assumptions you entered.
8. What should I enter in the loss matrix?
Enter the penalty for choosing each action under each state. Use consistent units, such as money, error points, or utility loss.