that is given by

$$p\big{(}\gamma_{1:K},z_{0:K}\big{|}y_{0:K}\big{)}=\frac{1}{p\big{(}y_{0:K}\big{)}}p\big{(}y_{0:K}\big{|}\gamma_{1:K},z_{0:K}\big{)}p\big{(}\gamma_{1:K},z_{0:K}\big{)}\,.$$

a quantity, that is proportional to the product of the likelihood term p( yo:K 11:K, zo:K ) and the prior term p(71:K; zo:K)
In rhythm transcription and tempo tracking, the prior encodes our background knowledge about the nature of musical scores and tempo deviations. For example, we can construct a prior that prefers "simple" scores and smooth tempo variations.

The likelihood term relates the tempo and the score to actual observed onset times. In this respect, the likelihood is a model for short time expressive timing deviations and motor errors that are introduced by the performer.

![0_image_0.png](0_image_0.png)

## 2.1 Score Prior

To define a score γι:κ, we first introduce a sequence of quantization locations co:κ.  A
quantization location ck specifies the score time of the k ' th onset. We let γk denote the interval between quantization locations of two consecutive onsets

$$\begin{array}{r c l}{{\gamma_{k}}}&{{=}}&{{c_{k}\,-\,c_{k}-1}}\end{array}$$
$\left(1\right)$. 
For example consider the conventional music notation ¶ ¶ which encodes the score ¶:3 =
[1 0.5 0.5]. Corresponding quantization locations are co:3 = [0 1 1.5 2].

One simple way of defining a prior distribution on quantization locations p ( ck ) is specifying a table of probabilities for c k  mod 1 (the fraction of c k ). For example if we wish to