## Exercise 1: Pseudocounts

The main text states that you can think of the Dirichlet parameter $\alpha = [\alpha_1, \alpha_2, ..., \alpha_n]$ “as a kind of prior” over categories $[A_1, A_2, ..., A_n]$. α is not a prior in the usual sense, since it is not a probability distribution. What α represents instead is a virtual observation. Thus if $\alpha = [2, 2, 1]$, that is the equivalent of having already observed the first category and second category twice each, and the third category one time only.

Complete the code below to prove that setting $\alpha = [2, 3, 1, 1, 1]$ is equivalent to setting $\alpha = [1, 1, 1, 1, 1]$ and then observing the first category once and the second category twice:

var colors = ['black', 'blue', 'green', 'orange', 'red'];

var observedData = [
{bag: 'bag1', draw: 'blue'},
{bag: 'bag1', draw: 'blue'},
{bag: 'bag1', draw: 'black'}]

// first model: set alpha = [1, 1, 1, 1, 1] and observe observedData
var observed = Infer({method: 'MCMC', samples: 20000}, function(){
var makeBag = mem(function(bag){
var colorProbs = T.toScalars(dirichlet(ones([colors.length, 1])))
return Categorical({vs: colors, ps: colorProbs})
})

var obsFn = function(datum){
observe(makeBag(datum.bag), datum.draw)
}

mapData({data: observedData}, obsFn)

return {bag1: sample(makeBag('bag1'))}
})

viz.marginals(observed)

// second model. Set alpha = [2, 3, 1, 1, 1]
var usealpha = Infer(

)

viz.marginals(usealpha) // should roughly match first figure


## Exercise 2: Rotten apples

On any given day, a given grocery store has some number of apples for sale. Some of these apples may be mushy or even rotten. The probability that each apple is rotten is not independent: a ripening fruit emits chemicals that encourages other fruit to ripen as well. As they say, one rotten apple spoils the whole barrel.

For each apple in a barrel, assume the probability that the apple is rotten is flip(p) where p is drawn from some prior. An appropriate prior distribution is Beta. Recall that the Beta distribution is just a Dirichlet that returns a vector of length one. So it, too, is defined based on pseudocounts [a, b]. Thus Beta({a: 10, b: 2}) returns the equivalent of a Beta distribution conditioned on having previously seen 10 heads and 2 tails.

To get a sense of the Beta distribution, run the following code:

viz(Beta({a: 1, b: 1})
viz(Beta({a: 10, b: 1})
viz(Beta({a: 1, b: 10})
viz(Beta({a: .1, b: .2})


Note that the final example gives a very nice prior for our apples: most of the time, the probability of a rotten apple is quite low. The rest of the time, the probability is very high. Middling probabilities are rare.

#### a)

Write a function makeBarrel that returns a function (a ‘barrel’) that takes a single argument N and returns a vector representing the rottenness of N apples from that barrel (where true is rotten and false is not rotten). That is, the following code:

var abarrel = makeBarrel('b')
abarrel(5)


should return something like [true, true, true, false, true].

Complete the following codebox:



var post = Infer({method: 'forward'}, function(){
//helper function to inspect your code. Do not edit.
var abarrel = makeBarrel('b')
return Math.sum(abarrel(10))
})
viz(post)


#### b)

Some grocery stores have fresher produce than others. So let’s create a function makeStore that returns a makeBarrel function, which works as it did in (a). Importantly, each store has its own Beta parameters [a, b] drawn from some prior.

HINT: In order to maintain the likelihood that in a given barrel, either most of the apples are rotten or few are, you need to ensure that a < 1 and b < 1. However, if a is much larger than b (or vice versa), you will get extreme results with every apple being rotten or every apple being good.

NOTE: No need to be overly fancy with this prior. Pick something simple that you know will give you what you want: stores that tend to have bad barrels and stores that tend to have good barrels.

var makeStore = // your code here

// Following code inspects your functions
viz(Infer({method: 'forward', samples:10000}, function(){
var S = makeStore('S')
var B1 = S('B1')
var B2 = S('B2')
return Math.abs(Math.sum(B1(10))-Math.sum(B2(10)))
})) // should generally be little difference

viz(Infer({method: 'forward', samples:10000}, function(){
var S1 = makeStore('S1')
var S2 = makeStore('S2')
var B1 = S1('B1')
var B2 = S2('B2')
return Math.abs(Math.sum(B1(10))-Math.sum(B2(10)))
})) // difference should be larger on average


#### c)

We can keep going. Some cities are located in apple country and thus have more access to fresh apples. Most stores in those cities are going to mostly have good barrels with good apples. Other cities have less access to fresh apples, and so more of their stores will have bad barrels with rotten apples.

In the code block below, create a makeCity function, which returns a makeStore function, which works as in (b). In (b), each store had a prior on [a, b]. Let’s put a prior on that prior, such that cities either tend to have good stores or tend to have bad stores.

HINT: Again, it is not necesary to have an overly fancy prior here. If you are spending hours trying to find just the right prior distribution, you are over-thinking it.

var makeCity = // your code here

//Make sure the following code runs:
var C1 = makeCity("C1")
var S1 = C1("S1")
var B1 = S1("B1")

viz(Infer({method: 'forward'}, function(){
return Math.sum(B1(10))
}))
//repeat to see different kinds of cities


#### d)

Suppose you go to a store in a city. The store has a barrel of 10 apples, 7 of which are rotten. You leave and go to another store in the same city. It also has has a barrel with 10 apples. Using your code above, how many of these apples are likely to be rotten?

// your code here