But in the special IID case,

$$

p(W_t | W_{t-1}, \ldots, W_0) = p(W_t)

p(W_t | W_{t-1}, \ldots, W_0) = p(W_t) ,

$$

and partial history $W_{t-1}, \ldots, W_0$ contains no information about the probability of $W_t$.

so that the partial history $W_{t-1}, \ldots, W_0$ contains no information about the probability of $W_t$.

So in the IID case, there is **nothing to learn** about the densities of future random variables from past random variables.

We could say that *objectively*, meaning *after* nature has chosen either $F$ or $G$, the probability that the data are generated as draws from $F$ is either $0$

or $1$.

We now drop into this setting a partially informed decision maker who knows

We now drop into this setting a partially informed decision maker who

- both $F$ and $G$, but

- knows both $F$ and $G$, but

- not the $F$ or $G$ that nature drew once-and-for-all at $t = -1$

- does not know whether at $t = -1$ nature had drawn $F$ or whether nature had drawn $G$ once-and-for-all

So our decision maker does not know which of the two distributions nature selected.

Thus, although our decision maker knows $F$ and knows $G$, he does not know which of these two known distributions nature had selected to draw from.

The decision maker describes his ignorance with a **subjective probability**

$\tilde \pi$ and reasons as if nature had selected $F$ with probability

So there is something to learn from the past about the future.

But what and how?

## Exchangeability

While the sequence $W_0, W_1, \ldots$ is not IID, it can be verified that it is

**exchangeable**, which means that the ``re-ordered'' joint distributions $h(W_0, W_1)$ and $h(W_1, W_0)$

**exchangeable**, which means that the joint distributions $h(W_0, W_1)$ and $h(W_1, W_0)$ of the ''re-ordered'' sequences

satisfy

$$

Equation {eq}`eq_definetti` represents our instance of an exchangeable joint density over a sequence of random

variables as a **mixture** of two IID joint densities over a sequence of random variables.

For a Bayesian statistician, the mixing parameter $\tilde \pi \in (0,1)$ has a special interpretation

as a subjective **prior probability** that nature selected probability distribution $F$.

A Bayesian statistician interprets the mixing parameter $\tilde \pi \in (0,1)$ as a decision maker's subjective belief -- the decision maker's **prior probability** -- that nature had selected probability distribution $F$.

```{note}

DeFinetti {cite}`definetti` established a related representation of an exchangeable process created by mixing

sequences of IID Bernoulli random variables with parameter $\theta \in (0,1)$ and mixing probability density $\pi(\theta)$

that a Bayesian statistician would interpret as a prior over the unknown

Bernoulli parameter $\theta$.

```

## Bayes' Law