# Probability box

A conspicuous problem in probability distribution elicition, for example in probabilistic modelling analysis, is that the specification is typically precise, despite hardly justified by empirical informtion in many cases.

```{attention}
epistemic uncertainy remains on the shape, parameters, and dependencies of the distributions.
```

[Probability box](https://en.wikipedia.org/wiki/Probability_box) (abbreviated as `p-box`) essentially represents bounds on the cumulative distribution function (c.d.f) of the underlying random variable. Let $[\overline{F}, \underline{F}]$ denotes the **set** of all nondecreasing functions from the reals into $[0,1]$ such that $\underline{F} \le F \le \overline{F}$. This means that, $[\overline{F}, \underline{F}]$ denotes a p-box for a random varaible $X$ whose c.d.f $F$ is unknown except that it is within the "box" circumscribed by the lower ($\underline{F}$) and upper bound ($\overline{F}$).

$$
\underline{F} \le F(x) \le \overline{F}
$$

p-box collectively reflects the variability (aleatoric uncertainty) and incertitude (epistemic uncertainty) in one structure for the uncertain quantity of interest. The horizontal span of the probabilty bounds are a function of the variability and the vertical breadth of the bounds is a function of ignorance.

<!-- a plot -->
```{image} ../../assets/pbox_illustration.png
:align: center
```

```{hint}
There is a storng link between p-box and Dempster-Shafer structures (which `PyUncertainNumber` also explicitly provides support :boom:). Each can be converted to the other. However, it should be noted such translation is not one-to-one.
```

`PyUncertainNumber` provides support for p-boxes ranging from [characterisation](./guides/uc.md), aggregation, [propagation](./guides/up.md). Go check out these links for details as to the computation with p-boxes. Meanwhile, quick examples are show below:

```python
from pyuncertainnumber import pba

p = pba.normal([0, 12], [1,4])
p.display()
```

<!-- a plot -->
```{image} ../../assets/myAnimation.gif
:align: center
:width: 600px
```

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