Calculator Input Form
Formula Used
For conditional independence inside each state of Z, P(Event A ∩ Event B|Z = zi) = P(Event A|Z = zi) × P(Event B|Z = zi).
The unconditional joint probability is the weighted sum across states: P(Event A ∩ Event B) = Σ P(Z = zi) × P(Event A|zi) × P(Event B|zi).
The calculator also finds P(Event A), P(Event B), the union, state contributions, and posterior state weights after the joint event occurs.
How to Use This Calculator
- Enter labels for the two target events and the conditioning variable.
- Add each hidden state of the conditioning variable.
- For every state, enter its prior probability and both conditional event probabilities.
- Keep priors summing to 1, or enable automatic normalization.
- Optionally enter a sample size to estimate expected joint counts.
- Submit the form to view summary metrics, state breakdowns, and the graph.
- Use the export buttons to save the output as CSV or PDF.
Example Data Table
| Z state | P(Z) | P(Event A|Z) | P(Event B|Z) | P(Event A ∩ Event B|Z) | Joint contribution |
|---|---|---|---|---|---|
| Low exposure | 0.500000 | 0.120000 | 0.100000 | 0.012000 | 0.006000 |
| Moderate exposure | 0.350000 | 0.280000 | 0.240000 | 0.067200 | 0.023520 |
| High exposure | 0.150000 | 0.620000 | 0.560000 | 0.347200 | 0.052080 |
| Total P(Event A ∩ Event B) | 0.081600 | ||||
Why This Model Matters
Conditional independence appears when two event indicators behave independently only after a relevant background state is known. Medical screening, credit analysis, reliability studies, survey response modeling, and mixture models often work this way. Ignoring the hidden state can distort the overall joint probability because state composition changes the final weighted average.
This calculator lets you separate within-state behavior from between-state weighting. That distinction is useful when you want transparent assumptions, reusable scenarios, and a clear audit trail. Instead of entering a single joint rate, you can show why the joint rate changes across groups, then combine those groups into one total estimate.
The summary output helps you compare marginals against the final joint event, while the posterior column shows which states dominate the observed joint outcome. The covariance and correlation values are also useful because unconditional dependence can appear even when the variables are independent inside each conditioning state.
FAQs
1. What does conditional independence mean here?
It means the two events act independently after the conditioning state is fixed. Within each state, multiply the two conditional probabilities to get the conditional joint probability.
2. Why can the overall variables still look dependent?
Mixing different states can create unconditional dependence. The weighted average can differ from simply multiplying the final marginals, even when each state satisfies conditional independence.
3. What should the prior probabilities add up to?
They should add up to 1 because they represent all possible states of the conditioning variable. You can also let the calculator normalize them automatically.
4. What if one conditional probability is zero?
That state contributes zero to the joint event for that pair of outcomes. The total result still includes contributions from every other state.
5. What does the posterior state column show?
It shows how much each state contributes after the joint event occurs. This is the probability of each state given that both events happened.
6. When is the correlation undefined?
Correlation becomes undefined when one event has zero variance, such as an event with probability 0 or 1. In that case, the indicator never changes.
7. Why include an optional sample size?
Sample size converts the final joint probability into an expected count. That helps with planning, forecasting, simulations, and quick communication of results.
8. Can I use more than three conditioning states?
Yes. The form lets you add states for richer mixture models. Each extra state needs a prior and the two conditional probabilities.