In this part:

> We begin with a sadly familiar problem:

> Suppose you live in a town with a limited number of tolerable restaurants. Every Friday you go out for dinner. You randomly choose a restaurant according to a certain probability distribution $P$. If you go to the $i$th restaurant, you then choose a dish from the menu according to some probability distribution $Q_i$. How surprising will your choice be, on average?

[...]

> Glomming together probabilities

> But the interesting thing about this problem is that it involves an operation which I'll call 'glomming together' probability distributions. First you choose a restaurant according to some probability distribution $P$ on the set of restaurants. Then you choose a meal according to some probability distribution $Q_i$. If there are $n$ restaurants in town, you wind up eating meals in a way described by some probability distribution we'll call

> $$ P \circ (Q_1, \dots, Q_n )$$

> A bit more formally:

> Suppose $P$ is a probability distribution on the set $\{1,\dots, n\}$ and $Q_i$ are probability distributions on finite sets $X_i$, where $i = 1, \dots, n$. Suppose the probability distribution $P$ assigns a probability $p_i$ to each element $i \in \{1,\dots, n\}$, and suppose the distribution $Q_i$ assigns a probability $q_{i j}$ to each element $j \in X_i$.


I want to extend the $Q_i$ to all the $X_k$ with zeroes when $k \neq i$, so that each $Q_i$ is a distribution over the same set $U$, the disjoint union of the $X_i$, and $j$ runs over $U$ in the definition of the $q_{ij}$. Then what you call glomming is a mixture of distributions. People often use mixtures, but I've never heard of glomming!

The formula with the extra term reminds of `the law of total variance', see [http://statisticalmodeling.wordpress.com/2011/06/16/the-variance-of-a-mixture/](http://statisticalmodeling.wordpress.com/2011/06/16/the-variance-of-a-mixture/).