A Bayesian statistician understands probabilities as numerical description of 'subjective uncertainty'. In order to make this a little bit more concrete, Bruno de Finetti considered betting odds and order books (see e.g. this paper [pdf!] for a more detailed discussion).
Meanwhile this approach is usually introduced and shortened as follows (*):
"we say one has assigned a probability p(A) to an event A if,
before knowing the value of A, one is willing to either buy or sell a lottery ticket of the form 'Worth $1 if A' for an amount $p(A).
The personalist Bayesian position adds only that this is the full meaning of
probability; it is nothing more and nothing less than this definition."
But there is a problem with that.
As every professional gambler knows, if one places a series of bets one needs to follow the Kelly criterion to avoid certain ruin in the long run. But the Kelly fraction and thus the amount one should be willing to bet is strictly zero for the case of a fair bet (when one would be willing to take both the buy and sell side).
In other words, it is wrong to pay the price suggested in the above definition, if one wants to survive in the long run.
Now, it would seem that this is one of those 'technicalities' that one can easily sweep under a rug or into a footnote. But I think it would be difficult to somehow incorporate the Kelly criterion in the above definition of probability, because in order to derive the Kelly fraction one needs to know already quite a bit about probabilities in the first place.
It may make more sense to emphasize that, of course, an order book is really a process and Bayesian probabilities are found in the limit when such order books approach the fair price and bet sizes tend towards zero. But unfortunately, in general we don't know much about the convergence of this process and real world examples of order books do not exhibit such a limit [x].
There is one additional problem, because the Kelly criterion is actually the limiting case of a more general rule. Indeed, if the estimated probabilities contain just a small amount of noise, bankruptcy still looms even if the Kelly criterion is used. Therefore professional gamblers know that one must bet 'less than Kelly'; In general a rational agent will bet 'less than Kelly' due to risk aversion and this means that the direct relationship between bet sizes and probability, as proposed in the above definition, is lost completely [x].
Therefore, I suggest that Bayesians simply refrain from using order books etc. (free markets are becoming quite unpopular anyways) to define probability and simply state that 'subjective uncertainty' is a self evident concept. After all, we know that self evidence is a very reliable kind of rug.
(*) I do not want to pick on this particular paper, since similar 'definitions' are
nowadays used in many cases. But since the paper is about the foundations of quantum theory I would like to point out that I have it on good authority that God does not play dice.
[x] If risk aversion and probability estimates of the participating agents are correlated, this would be already one reason why the order book would not converge towards the 'fair price'.