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I have a strategy Y which is influenced by some factors X1, ..., Xn (for example asset volatility, distribution of macroeconomic factors). At moment t0 I have historical distribution(prior) of X1, ..., Xn and Y (past strategy performance). How can I model expected distribution of strategy performance(Y_posterior) based on the most recent information about X1, ...., Xn. How can I model that? This sounds like Bayesian inference problem with Markov Chain Monte-Carlo (MCMC) used to model conditional distribution, can we model this problem with that and how?

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  • $\begingroup$ I assume Y is log return? $\endgroup$ – numerairX Dec 10 '18 at 14:45
  • $\begingroup$ It can be log return, underlying price distribution $\endgroup$ – Alexandr Proskurin Dec 11 '18 at 9:29
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There appears to be some confusion on how Bayesian methodologies work and their differences to null hypothesis methods such as Pearson and Neyman Frequentist, Fisher's Likelihoodist or Chebyshev's method of moments.

Having old data is not a prior distribution. That is the data. It is the same for all modeling methodologies. The prior distribution describes everything you know about where the parameters are at that is not contained inside the data itself.

Imagine you were doing research on the calories in a new variety of green beans. A good estimator of calorie count would be estimates of calorie counts in other varieties of bean. Your prior density may be the average calories in a sample of Phaseolus coccineus and you could estimate the variability of the estimate of the variance in the same way. This generates a far more accurate answer and controls for things such as the odd chance of getting a weird sample. It is related to the concept of shrinkage estimation in Frequentist statistics.

MCMC is used to perform numerical integration on those Bayesian problems where there cannot be an analytic solution. Many simple Bayesian problems have analytic solutions and you should not use MCMC if your regression can be restated in that way. MCMC is only used to calculate the denominator's value in Bayes theorem and may not be necessary even without analytic solutions depending on your very specific problem. This is because the posterior density is proportionate to the numerator.

Bayesian methods differ substantially in the understanding of how things work. For example, there is nothing similar to a t-test or an F-test. Indeed, the idea of an F-test for regression has no analog in Bayesian methods. The idea of testing that all parameters are equal to zero is called a sharp null hypothesis and there is no way to test a sharp null in Bayesian thinking, though there are rough analogs that do not mean the same thing.

In null hypothesis methodologies, the sample is random and the parameters are fixed. In Bayesian methods, the sample is fixed and the parameters are random. Even randomness has a different meaning. Because you are fixing the parameters with a null hypothesis, randomness is due to chance alone. With Bayesian methods, there is nothing similar to a null hypothesis. You can have as many or as few hypotheses as you need and none of them are special in any way. There is no such thing as an "alternative" hypothesis as they are all alternatives. Randomness in Bayesian thinking is uncertainty. You are uncertain as to the value of the parameter, but you are not uncertain as to the value of the data that you saw. It cannot be random, you saw it.

If you decide you would like to use Bayesian methods because of their greater accuracy, as Bayesian estimators cannot be stochastically dominated, then you need to pick up an undergraduate textbook on them first and work your way forward. They don't even calculate the estimator of the mean the same way. There is no x bar. There is no s squared. No t-tests, no F tests, no Kolmogorov-Smirnov test. There is no null. There are also no confidence intervals, they use a very different construction called a credible interval. Not even close to similar. They can even be disconnected. While an appropriate Frequentist confidence interval may be (-7,5) the same Bayesian interval may be (-6,-4) unioned with (2.5,3.5). If that is a 95% interval that would imply that there is a 95% chance that the true value of the parameter is inside one of those two bounds.

I would start with Bolstad's undergraduate book introducing Bayesian statistics and really work the examples, even on the solutions you know because they could come out very different. He also produces a computational methods book as well that follows on. Do not start with it. There are other books as well.

There is a good discussion on Bayesian and Frequentist intervals that will give you a very good feel of how different they are by modeling cookie jars at the stack exchange discussion on intervals.

Finally, predictions in Bayesian thinking revolve around the Bayesian predictive distribution. It is easier to calculate than equivalent Frequentist predictive intervals and makes more sense for most problems. However, if you are testing a model that involves prediction, you will need to learn about prediction scoring methods. After you have grounded yourself in Bayesian methods, pick up Parmigiani's book on "Decision Theory," otherwise you are in danger of improperly accepting or rejecting a predictive system.

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