# Get Confidence Intervals for Predictions

Whittaker supports three types of uncertainty bands via the [interval](../reference/PosteriorPredictResult.md#whittaker.PosteriorPredictResult.interval) argument to `predict()`. Pointwise confidence intervals cover each individual prediction point separately. Simultaneous bands correct for the fact that you are evaluating the entire curve at once. Prediction intervals are wider still, because they must cover a new observation (not just the mean) at each point.


# Fit

Fit the `mcycle` GAM and build the prediction grid used throughout this recipe.


``` python
import numpy as np
import whittaker as wk

# Load data and fit the GAM
data = wk.load_dataset("mcycle")
model = wk.GAM("accel ~ s(times)").fit(data)

# Build prediction grid
grid = {"times": np.linspace(0, 57.6, 200)}
```


# Pointwise confidence interval

Pass `interval="confidence"` to get `.lower` and `.upper` alongside `.values`. These bands have 95% coverage at each individual x value by default.


``` python
preds = model.predict(grid, interval="confidence")
preds.values[:5].round(2)
```


    array([3.27, 4.18, 5.08, 5.97, 6.83])


The fitted values show the smooth running through the data. The lower bound reveals how far the interval extends below the point estimate at each location.


``` python
preds.lower[:5].round(2)
```


    array([-16.62, -14.63, -12.69, -10.84,  -9.09])


The upper bound is symmetric around the fitted value for Gaussian families.


``` python
preds.upper[:5].round(2)
```


    array([23.17, 22.98, 22.85, 22.77, 22.75])


Pointwise intervals are the right choice when you are reporting a single prediction or comparing the smooth at a specific x value against a reference.


# Simultaneous bands

Using `interval="simultaneous"` widens the bands so that the entire fitted curve is covered with 95% probability, not just each point in isolation. Use these whenever you want to make a statement about the shape of the curve as a whole (e.g., claiming that the smooth is everywhere above zero).


``` python
preds_sim = model.predict(grid, interval="simultaneous", level=0.95)
```


Comparing the mean band widths makes the difference concrete. The tuple below shows pointwise width first, simultaneous width second.


``` python
# Compare mean widths: pointwise vs simultaneous
(preds.upper - preds.lower).mean(), (preds_sim.upper - preds_sim.lower).mean()
```


    (np.float64(23.446407209246317), np.float64(35.70322751765385))


Simultaneous bands are always at least as wide as pointwise intervals, often noticeably so.


# Prediction interval

`interval="prediction"` adds observation-level noise to the uncertainty, producing intervals that should contain a new individual measurement rather than the conditional mean.


``` python
preds_pred = model.predict(grid, interval="prediction")
(preds_pred.upper - preds_pred.lower).mean()
```


    np.float64(87.25342679688102)


The prediction interval is wider than both confidence band types, as it must account for the irreducible scatter around the smooth.


# Summary

| CI type | Coverage guarantee | Width | When to use |
|----|----|----|----|
| `"confidence"` | Per-point (95% at each x) | Narrowest | Reporting a single x value |
| `"simultaneous"` | Whole curve (95% jointly) | Wider | Inference about curve shape |
| `"prediction"` | New observation at each x | Widest | Forecasting individual values |