Is there a Bitcoin bubble? Jason Kuznicki thinks so and believes that he has conclusive proof. He blogs three graphs that show more or less that there is a lot of speculation in Bitcoin. But does speculation prove that there’s a bubble? Let’s use Bayes’s rule to think about this carefully.

Bayes’s rule is a mathematical tool for thinking about the incorporation of new evidence into subjective probabilities. Let’s suppose that there is some proposition A for which you have a prior belief. Somebody offers evidence B for or against A. How much should you change your belief in A based on evidence B?

Bayes’s rule boils the answer down to a simple mathematical form:

$$P(A|B) = P(B|A)\dfrac{P(A)}{P(B)}$$

In English, the probability of A given B equals the probability of B given A, times the probability of A and divided by the probability of B.

So to evaluate Jason’s argument and see how much we should change our estimate of a Bitcoin bubble based on the evidence that there’s speculation, we can simply assign the proposition and the evidence to A and B. In this case, A is the proposition that there’s a bubble, and B is the evidence that there’s speculation in Bitcoin. If we figure out our subjective probabilities for B|A and B, we can use those to determine how different P(A|B) should be from P(A).

So what is B|A? Since B is the evidence that there is speculation in Bitcoin and A is the proposition that there is a bubble, B|A simply states the proposition that given that there is a bubble, there is speculation. It seems pretty much impossible to have a bubble without speculation, so I’ll go with a subjective probability of 1. Picking a different value here will only work against Jason’s argument.

So what is the probability of B, the fact that there is speculation in Bitcoin? The Bitcoin ecosystem isn’t built out yet. Most of the protocol’s most exciting uses haven’t even seen the light of day yet. As I blogged last week, multisignature transactions are barely in use yet, but they form the foundation for a decentralized architecture of arbitration. Ed Felten at Princeton is working on decentralized prediction markets. Jerry Brito points to microtransactions, or even nearly-continuous transactions, as another exciting future use scenario.

Given that we don’t know whether this ecosystem will ever materialize, holders of bitcoin are necessarily speculating. If the ecosystem matures and is useful, bitcoins will be worth something. If none of these innovations come about, or if we decide they’re not that useful after all, then bitcoins will probably be worth nothing. There’s no way out of speculating, because we simply don’t know for sure if the ecosystem will come along. Almost the entire “fundamental value” of Bitcoin rests on future events.

So the probability of B, I think, is 1. When P(B|A) is 1, and P(B) is 1, what does Bayes’s rule reduce to?

$$P(A|B) = P(A)$$

B simply offers no information as to whether A is true.

A similar argument can be made when Bitcoin’s volatility is offered as evidence of a bubble. Bitcoin is a thinly-traded asset where supply does not adjust to accommodate demand. It is going to be volatile. So the fact that Bitcoin is volatile adds no new information to the question of whether it’s a bubble.

What does provide information? I think the most reliable evidence is on the maturation (or not) of the Bitcoin ecosystem. If Bitcoin seemed static right now, I would interpret that as evidence of a bubble. But it doesn’t. Every day, people are working to build businesses that leverage some of the unique features of Bitcoin’s protocol. As long as that continues, I think it’s most reasonable to be highly agnostic about the correct price of Bitcoin.