I'm currently an undergrad at a Canadian university and our finance courses has been brought up through the frequentist approach (ols, hypothesis testing, sampling theory). Only recently, through exploring methods in portfolio management do I realize a lot of the literature suggest using Bayesian methods (ex: Black-Litterman) to deal with estimation.

My question is, what are the benefits that both approaches contribute the most in finance? What other areas in finance are Bayesian methods being used as industry standards? Are there certain areas where one is favored than other? Should someone interested in Finance be gearing towards bayesian or frequentist?

  • $\begingroup$ I can't think of any popular methods in finance that use Bayesian as a go-to. Binomial trees follow Bayesian ideas. Great question $\endgroup$
    – rrg
    Commented Oct 1, 2016 at 11:55

4 Answers 4


I would distinguish between Bayesian inference versus Bayesian portfolio management techniques.

Inference includes estimating parameters and credible intervals (the Bayesian version of confidence intervals, which are actually far more intuitive) and forecast.

If you want to learn about Bayesian inference, get a solid foundation in frequentist methods first. A solid understanding of maximum likelihood estimation will help immensely when trying to make heads or tails of MCMC. It's too early in your thinking to be bounded to one camp or the other. Learn both (if you want) and use them as appropriate.

Bayesian portfolio management tends to be a broader term. If you knew nothing of the field and just followed the statistical literature, you would use Bayesian inference to get the posterior distribution for parameters, convert that to a posterior predictive distribution, and then select a portfolio based on this distribution. Black-Litterman (and further developed into Entropy Pooling) is a tool with two main purposes: 1) blending an investor's views with some other distribution, 2) an alternate approach to getting a prior on market returns (reverse optimization). The second step is optional, but I wouldn't consider either to be estimation, per se. A third technique that can get lumped in with Bayesian portfolio management is resampling (or Michaud resampling). This technique can be thought as similar to the first approach mentioned. However, when you have the posterior distribution, instead of getting the posterior predictive overall, you optimize a portfolio for each collection of paramters in the posterior, and then average the weights of each of the portfolios. However, you don't need to do Bayesian inference to do Michaud resampling, just some measure of the uncertainty of the parameters.

In terms of what approaches are more popular, I think Black-Litterman and Michaud resampling are probably the most well-known.

  • $\begingroup$ Hi John, thanks for the aside on bayesian portfolio mgmt and references. Some more questions: back to other question, which of the two types of inferences are more widely practiced and dominant in finance or is it mixed? What resources would you recommend in learning bayesian theory with appl. To finance? I find it very interesting and much more intuitive and applicable to finance. Lastly, how do bayes method such as bl, entropy pooling, michaud resampling perform in practice vs. Traditional weighting? Ex. 1/N, mkt weights, etc. $\endgroup$
    – Kevin Pei
    Commented Jan 3, 2015 at 0:43
  • 1
    $\begingroup$ Frequentist is far more common. The best way to learn is reading books and then writing programs. In terms of what is better in practice, I think out-of-sample Bayesian and Michaud have better properties than Mean-variance. $\endgroup$
    – John
    Commented Jan 3, 2015 at 4:21

First of all, the area of statistics you need in finance is time series analysis. That being say, as usual is statistics: the more data you have, the more sophisticated models you can use. Typically in intraday / high frequency finance, you can afford to use sophisticated methods. See MMP (and especially the appendix) for basics, and Quantitative Trading for more.

Again as usual, advantages of Bayesian vs. Least Square methods are:

  1. Using prior to obtain a better accuracy (if your prior is adequate) with the same number of observations (note that nevertheless you have a prior --but very simple:Bayesian-- with Least Squares).
  2. The robustness to outliers.
  3. The capability to deal with non i.i.d. datasets.

No preference for Bayesian or LS, but you have to understand how you answer to each of these points when you decide to use a method.

You present portfolio construction as an estimation. I would not agree: you need estimates to build a portfolio but the portfolio construction is not an estimation.

Typically BL uses a Bayesian approach to estimate covariance (and potentially expected returns and the risk aversion of market participants) under a general equilibrium consideration. Once it is done they use some of the estimates to build a portfolio using a given framework (can be Bayesian or not).

  • $\begingroup$ What makes Bayesian robust to oultiers? If you make normal assumption in regression it still involves minimizing the least squares. $\endgroup$
    – confused
    Commented May 27, 2020 at 9:17
  • $\begingroup$ @confused The proper choice of distribution for the coefficients (eg Student t), although you do not need a Bayesian treatment for that. Note that if you do not assume a normal distribution for the parameters, maximizing the likelihood is not equivalent to minimizing least squares. $\endgroup$
    – scities
    Commented May 27, 2020 at 13:55

Although I like the other answers to this question I think, there are some points which may be interesting to note and should get attention as well. Let me address each of your individual questions.

My question is, what are the benefits that both approaches contribute the most in finance?

First, I think , methodology in finance is not about being Bayesian or Frequentist but about searching for an answer using appropriate tools. As stated by @lehalle, its all about the data, the model and the objects you are after.

  1. Financial datasets are, in general, not very large. You may have firm-month observations of thousands of firms, but as companies are usually not surviving for hundreds of years, time-series are relatively short. This leads to the problem that your parameters are not going to be estimated with large precision. How can you handle this estimation uncertainty? Well, either you go for frequentist standard errors, which may be unnecessary inflated as they are only driven by the number of observations, or you choose some well-motivated priors in order to include additional information which decreases your parameter uncertainty. Trade-off here: You may impose priors which are not accepted by your peers or you may not be able to obtain some meaningful, significant results.

  2. Often, the objects you are interested in are hard to capture by Frequentist methods. Stating empirical evidence without being able to asses the estimation uncertainty is worthless (in most cases). Let me give you one example where Bayesian Methods provide you with ways to computed Credible Regions where you would probably have a hard time to compute confidence intervals: Risk-management often cares about the Value-at-Risk, some quantile of your predicted distribution of future returns.

You may estimate the parameters of the return distribution which then yields you an estimated VaR. Can Frequentists tell you something about the uncertainty of this estimation? Well, the distribution of a quantile of return distribution is rather hard to capture. Fascinating world of Monte-Carlo Markov Chain (MCMC) allows you to simulate VaRs by sampling from the posterior distribution of the parameters and getting draws of associated VaRs conditional on the likelihood function. Bayesian Computation is a simulation tool which helps you to analyze otherwise extremely complicated statements. Trade-off here: Numerical approximations are not as handy as analytical close-form solutions, but at least there is some solution...You may also think about time-varying parameter models such as Stochastic Volatility Factor models, GMM and the like perform rather poor in this area.

  1. It really depends on your aim what fits best. Always keep in mind that results are interpreted differently depending on using Bayesian vs. Frequentist approaches. I personally think, Bayesian thinking is more natural in the sense that it overlaps with my subjective feeling for probabilities. Keep in mind that sometimes it is quite nice to have statements such as the probability of the CAPM being correct given the data is X percent (you may have a look in Doron Avramovs An exact Bayes test for asset pricing models).

What other areas in finance are Bayesian methods being used as industry standards?

This I don't know but you may find Rachevs book 'Bayesian Methods in Finance' useful.

Are there certain areas where one is favored than other? Should someone interested in Finance be gearing towards bayesian or frequentist?

I don't think anything should be preferred. Open mindedness for every method which helps to understand what is going on is the key. But: it is important to understand that there are differences in methodologies and some useless comments such as Frequentist: "Bayesians can generate every result if they just torture the prior long enough" Bayesian: "Asymptotic Theory is useless as data is always finite"

  • $\begingroup$ Although this question is now almost 3 years old, I still enjoy the answers you guys have provided. Looking back, I realize the question I asked was quite loaded (as many of you have pointed out) and I would have a completely different answer now for myself than back when I first asked. Thanks again for taking your time to give your two cents $\endgroup$
    – Kevin Pei
    Commented Sep 5, 2017 at 18:37

We can use statistical inference with at least three types of data with applications in financial engineering (with examples from Bayesian).

  1. time series: trends, seasonality, change points, outliers, etc

  2. Bayesian networks (factor graphs): to model decision variables, both known and latent and estimate risk in complex decision frameworks

  3. text : for things like "named entity recognition", "concept extraction" and "sentiment analysis"

To compare Bayesian vs frequentist inference and learn about MCMC hands on approaches I found these resources and authors to be the best

  1. "[Jake VanderPlas] Frequentism and Bayesianism: A Python-driven Primer": https://arxiv.org/pdf/1411.5018.pdf. Jake thinks with both hats and this helps connect the two worlds and motivate deeper study

    Discussion - The Bayesian approach gives odds of 10 to 1 against Bob, while the naïve frequentist approach gives odds of 18 to 1 against Bob. So which one is correct? For a simple problem like this, we can answer this question empirically by simulating a large number of games and count the fraction of suitable games which Bob goes on to win. This can be coded in a couple dozen lines of Python (see part II of [VanderPlas2014]). The result of such a simulation confirms the Bayesian result: 10 to 1 against Bob winning. So what is the takeaway: is frequentism wrong? Not necessarily: in this case, the incorrect result is more a matter of the approach being “naïve” than it being “frequentist”. The approach above does not consider how p may vary. There exist frequentist methods that can address this by, e.g. applying a transformation and conditioning of the data to isolate dependence on p, or by performing a Bayesian like integral over the sampling distribution of the frequentist estimator ˆp. Another potential frequentist response is that the question itself is posed in a way that does not lend itself to the classical, frequentist approach ...

  2. "[Jake Vanderplas] Statistics, Data Mining, and Machine Learning in Astronomy" : https://books.google.ro/books?id=2fM8AQAAQBAJ&pg=PA123&lpg=PA123&dq=%22Classical+or+frequentist+statistics+is+based+on+these+tenets%22&source=bl&ots=bGr55pHCG&sig=MvEeTaaIBbGLhP0jETP6W5B-XI&hl=en&sa=X&ved=0ahUKEwiTucTnsPfVAhUhCZoKHZGeCnMQ6AEIJDAA#v=onepage&q=%22Classical%20or%20frequentist%20statistics%20is%20based%20on%20these%20tenets%22&f=false

    Classical or frequentist statistics is based on these tenets:

    • Probabilities refer to relative frequencies of events.They are objective properties of the real world.
    • Parameters (such as the fraction of coin flips, for a certain coin, that are heads) are fixed, unknown constants. Because they are not fluctuating, probability statements about parameters are meaningless.
    • Statistical procedures should have well-defined long-run frequency properties. For example, a 95% confidence interval should bracket the true value of the parameter with a limiting frequency of at least 95%.

    In contrast, Bayesian inference takes this stance:

    • Probability describes the degree of subjective belief, not the limiting frequency.
    • Probability statements can be made about things other than data, including model parameters and models themselves.
    • Inferences about a parameter are made by producing its probability distribution—this distribution quantifies the uncertainty of our knowledge about that parameter. Various point estimates, such as expectation value, may then be readily extracted from this distribution.
  3. "[Thomas Wiecki] Bayesian Data Analysis with PyMC3": https://vimeo.com/79518830 Thomas is a main contributor for PyMC3 which is a very good tool for discover, study and apply Bayesian inference.

  4. "MCMC in Python: random effects logistic regression in PyMC3" https://healthyalgorithms.com/2014/02/17/mcmc-in-python-random-effects-logistic-regression-in-pymc3/ MCMC is indeed fundamental to Bayesian inference but an algorithm serial in nature => hard to distribute / scale. For that we have Variational Bayes inference and Loopy belief propagation. https://metacademy.org/roadmaps/rgrosse/bayesian_machine_learning says that "Loopy BP can be interpreted as a variational inference algorithm."

  • $\begingroup$ It is always better to give more than links in an answer (otherwise it is a comment); please complement each link with two keys ideas you found there. $\endgroup$
    – lehalle
    Commented Aug 27, 2017 at 8:44

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