Sunday, April 22, 2012

Frequentists, Bayesians, and HBO's "Girls"

Yielding to pressure, I watched the first episode of HBO's "Girls" on YouTube — well, the first ten minutes or so. The experience wasn't a total waste: I got an example of the difference between frequentists and Bayesians from it.

The protagonist, whose name I can't remember (henceforth "she"), has an unpaid internship that she took on the expectation of a job. She doesn't get the job and it's implied that there never was a job.*

Given that there's only that one data point, frequentists would have to decline any conclusion regarding the existence of the potential job. (The point estimate would be irrelevant without a variance for that estimate.)

Bayesians have a different view of things.

There are two possible states of the world: boss told the truth ($T$) or boss lied ($L$). There are two possible events: she gets a job ($J$) or she doesn't ($N$).

Without any information about the boss, we'll assume that the probability of truth or lie before any event was observed (that is the apriori probability) is

$\Pr(T) = p_0 = 1/2$,
$\Pr(L) = 1-p_0 = 1/2$.

The $1/2$ is the maximum entropy assumption for $p_0$, meaning we are the most uncertain about the truthfulness of the boss.

If the boss lied, then we can never observe event $J$,

$\Pr(J|L) = 0$,
$\Pr(N|L) = 1$.

If the boss told the truth, and there was in fact a potential job, she might still not get the job, as she might be a bad match. Given no other information, we can assume the same high-entropy case, here for the conditional probabilities:

$\Pr(J|T) = 1/2$,
$\Pr(N|T) = 1/2$.

We can now determine the probability that the the boss was telling the truth:

$\Pr(T|J) = 1$,
$\Pr(T|N)  = \frac{\Pr(N|T)  \Pr(T)}{\Pr(N|T) \Pr(T)+\Pr(N|L) \Pr(L)}= \frac{1/4}{1/4 + 1/2} = 1/3.$

Since she didn't get the job, there's a $2/3$ chance that there was never any job.

Note that there's really no magic on the Bayesian side; we bring a lot of baggage to the problem with the apriori and conditional probabilities. But in doing so we make our assumptions and ignorance explicit, which allows us to make inferences.

It's not magic, it's Bayes.

-- -- -- --
* I was going to write "SPOILER ALERT," but then I realized there's no way to spoil the show more than it already is...

Counterintuitive solution for being a late chronotype

Hi. I'm Joe and I'm a late chronotype.

A late chronotype is someone whose energy level, after waking up, increases more slowly than than the average person's; also known as "not a morning person." Typically this slow start is balanced by high levels of energy in the evening, when other people are crashing. (Panel I below depicts this for illustration.)

Mismatching Chronotypes

Many late chronotypes believe that the solution to their problem is to sleep late. That is exactly the wrong approach. The problem of being a late chronotype is that our level of energy doesn't match everyone else's. Starting the day later only increases the problem (as illustrated in panel II).

The solution, which may sound counter-intuitive, is to get up much earlier than everyone else, therefore reaching peak energy at the same time as everyone else (as shown in panel III).

I have used a number of approaches to manage being a late chronotype (caffeine, no breakfast, exercise, ice-cold morning shower), but none was ever as effective as being on Boston time while living in California.

Monday, April 2, 2012

Bundling for a reason

There's much to dislike about the current monetization of television shows, but bundling isn't necessarily a bad idea for the channels.

On a recent episode of The Ihnatko Almanac podcast, Andy Ihnatko, talking about HBO pricing and release schedule for Game Of Thrones (which he had blogged about before), said that a rule of commerce is "when customers have money to give you for your product, you take it" (paraphrased). I don't like to defend HBO, but that rule is incomplete: it should read "...you take it as long as it doesn't change your ability to get more money from other customers."

An example (simplistic for clarity, but the reason why HBO bundles content):

In this example HBO has three shows: Game of Thrones, Sopranos, Sex and the City; and there are only three customers in the world, Andy, Ben, and Charles. Each of the customers values each of the shows differently. What they're willing to pay for one season of each show is:

$ \begin{array}{lccc}
 & \mathrm{GoT} & \mathrm{Sopranos}  & \mathrm{SatC} \\
\mathrm{Andy} &100 & 40 &10\\
\mathrm{Ben}  & 40 & 10 & 100 \\
\mathrm{Charles}   & 10 &100 & 40\\
\end{array}$

HBO can sell each of them a subscription for $\$150$/yr. Or it can price each show at $\$100$ and get a total of $\$100$ from each customer (any other price is even worse). This is the standard rationale for all bundling: take advantage of uncorrelated preferences.

By keeping the shows exclusively on their channel for a year, they get to realize those $\$150$ from the "high value" customers. After that, HBO sells the individual shows to make money off of people who don't value the HBO channel enough to subscribe (people other than Andy, Ben, or Charles above). This is standard time-based price segmentation.

This is not to say that HBO and other content providers won't have to adapt; but their release schedule is not just because they're old-fashioned.