Bayes rule is an important theory in statistics. In this following, we discuss one of the presentation, which can be applied to interest rate model, see [MR05].

Proposition 1 (Bayes rule)Let be two probability measures on the probability space , satisfying Let be a filtration with . Then,

*Proof:* Note that and are also probability measures on , we denote the restricted probabilities by and respectively. Then, are martingale, i.e. , see Lemma 1 of here.

It’s enough to show,

It follows by

In particular, with in the above proposition, we have

In this below, we consider an application of Bayes rule to the interest rate theory. Suppose is the short rate, and is the unique equivalent martingale measure. Then, the price at time of the derivative with maturity and payoff measurable is written by

In particular, the price of zero bond is, with ,

The **forward measure** is the probability defined as

Then, by the Bayes rule, the derivative price of (1) can be written w.r.t. forward measure:

Proposition 2For all , the process

is an -martingale under .

*Proof:* It’s enough to show

which is indeed a consequence of (3) with .