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In quantitative finance tasks (asset pricing, portfolio optimization, option pricing, volatility forecasting, etc), there are frequentist, likelihoodist and Bayesian approaches or interpretations to solving the same financial problem. For the common finance tasks listed or others, could someone give an outline or summary-list of prevalent techniques for each of the 3 approaches? For example, your answer for each task could look like:

Task: Portfolio optimization

  1. Frequentist: mean-variance model
  2. Likelihood: N/A
  3. Bayesian: Black-litterman model

Why continue teaching and learning naive frequentist approaches when the approaches like Bayesian are introduced as being improvements over traditional (frequentist) statistics? For each finance problem, it would be good to also briefly identify in which situations the 3 approaches outperform one another and why.

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Let us ignore particular models such as the CAPM and only focus on the properties of those probability interpretations as they impact financial decision making or estimation.

All classical models are Frequentist models. They depend strictly on two things being true. First, the parameters are fixed and known to all. Second, the data includes a random shock. If you relax the first assumption, you run into the problem that $E(x_{t+1})$ does not exist in the case of $x_{t+1}=\beta{x}_t+\varepsilon_{t+1}$, where the shock is drawn from any distribution centered on zero with a finite variance and $\beta>1.$ If $\beta\le{1}$ then nobody would invest money.

However, if we move away from the models and get into estimation, what we need to do is ask what are Frequentist estimators optimizing?

Unfortunately, there isn't a unique answer to this question, but, generally, they are the unbiased estimators that minimize the maximum risk. When you think about that, it is a wonderful trait.

Imagine the case where you have no prior information. One way to think about prior information is as a context. You are facing a loss from using several possible estimators in lieu of a parameter as it is unobservable. You cannot get any context to protect yourself. So what you would prefer is for the math to guarantee you a level of protection, at least upon repeated use.

It also assures you of perfect accuracy, though not for any one sample.

Finally, if you need to use them to make a decision, they perfectly link inference with decisions and behavior. They cannot tell you what is true or false, but they can tell you how to behave in such a way that you will only be made a fool of no more than $\alpha$ percent of the time.

They do have some disadvantages. First, they can never use more, but they can use less information in them than the maximum likelihood estimator. They also generate incoherent odds so that if a bookie uses a Frequentist method, then, from time to time or possibly always, a player or set of players can create a circumstance where the bookie must lose in all states of nature. That is without any form of manipulation. They also can generate a meaningful level of statistical arbitrage against the bookie. The arbitrage will be invisible to the bookie, a mathematical form of color blindness.

The method of maximum likelihood, as designed by Fisher, was created for inference only. It does not give rise to the type of math that lends itself to making decisions. You do not see models built on it because p-values only give you the strength of the evidence against something. He didn't intend for there to be an alternative hypothesis. If your null is $\mu=0$, then the only choice is $\mu\ne{0}$ which is the entire number line and uninformative.

You do see estimation created on it. The method of maximum likelihood never uses more information than the Bayesian estimator and can use less because it cannot create a prior and because it can only consider one loss function. However, it is the best case solution for the observed data. Why would you want to use a mediocre likelihood function as would be implicit in the use of a Frequentist method? Of course, Frequentist methods do not use the likelihood function, but, implicitly, they would map to some point on the likelihood.

The disadvantage comes in two cases. First, if there is real prior information, the Bayesian estimator will dominate the MLE given the same loss function. Second, it doesn't give rise to coherent probabilities and as above, you can rig the market against the bookie or market maker.

Finally, Bayesian methods can be constructed from de Finetti's axioms. The consequence of this is that they are optimized for gambling. Provided that you use your real prior and not a convenient one, you are assured that your bets will be coherent and you cannot rig the market against a bookie or market maker. The point solutions they provide cannot be dominated by another type of estimator so they are never more risky than any other estimator.

The downside is in academic research. They are not merely biased estimators as the MLE is, but subjective estimators. At least, in theory, every human should have their own estimate for any given phenomenon having seen the same data. That is an editors nightmare.

What is your point estimate of the average? Mine is 3, but Joe's is 4, and Millicent thinks it is 3,000,000.

How should we decide? Well, Savage's axiomatization of probability begins with $x\succeq{y}$, and I have strong feelings about it, plus my lunch has put me in a state of disequilibrium, so the answer is 3.

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