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I have a basic question about Bayesian statistics.

Lets say that I want to make forecasts of a certain response variable, based on explanatory variables and lagged responses variables, while I have no knowledge apart from the data. This means that under a Bayesian approach, I would specify a diffuse prior to obtain the posterior distribution of my parameters.

In this topic (Why I should use Bayesian inference with uninformative prior?) I read the following sentence: ''if you are interested only in point estimates, then it (frequentist and bayesian) is basically the same''.

I would now like to ask: Does forecasting in a 'Bayesian way', when using a diffuse prior, has any advantages over just using MLE to obtain estimates for your parameters and making forecasts in 'frequentist way'? I think that for obtaining a point estimate of your parameter, a Bayesian approach does not really have any advantages over a frequentist approach, but I was wondering if this also was the case from a forecasting perspective.

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  • $\begingroup$ Yes, of course. The frequentist approach means that you choose a model and make forecasts from it. The Bayesian approach means that you also take into account some of your uncertainty about whether your model is correct, so your forecasts should have more uncertainty. $\endgroup$
    – Flounderer
    Jan 24, 2018 at 15:11
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    $\begingroup$ @Flounderer Bayesian forecasts are not more uncertain because of using priors! $\endgroup$
    – Tim ♦
    Jan 24, 2018 at 16:29
  • $\begingroup$ @Flounderer: plenty of Frequentist methods take into account uncertainty in the model in predictions, however they often use asymptotic normal approximations rather than MCMC $\endgroup$
    – Cliff AB
    Jan 16, 2020 at 22:05
  • $\begingroup$ Here's a related question on the width of interval estimates in Bayesian and frequentist forecasting. Bottom line: frequentist prediction interval and Bayesian posterior predictive (credible) interval coincide under flat priors. $\endgroup$
    – Durden
    Jul 9 at 19:51

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I think that for obtaining a point estimate of your parameter, a Bayesian approach does not really have any advantages over a frequentist approach

I would strongly oppose to that. Take the case of a coin flipped three times and we are looking for a point estimate of the probability $\theta_{head}$ of a "Heads" result. Let there be three flips with the result of 2 Heads and 1 Tails. A Frequentist approach with no prior information will result in a point estimate of $\theta_{heads} = 2/3$ - which in a case of coin flips is just nuts. No reasonable person would deduct from this experiment, that the coin will likely show heads in 2 out of three flips to come. Prior information is worth a lot when making point estimates in case of scarce data. Missing the chance of adding prior information is a big downside of Frequentist statistics. (Obviously, not having headaches to define a prior and not having to defend you choice of prior is a big advantage of Frequentism, but that'S not the point here.)

I think that for obtaining a point estimate of your parameter, a Bayesian approach does not really have any advantages over a frequentist approach, but I was wondering if this also was the case from a forecasting perspective

I think that this question is missing an important point. You should ask: "Is it reasonable, to do forecasting/prediction based on point estimates?" A Bayesian would strongly argue, that you should do prediction based on the whole posterior distribution of possible coefficients because forecasting/prediction based on point estimates disregards all information about how imprecise the point estimate is.

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  • $\begingroup$ Thank you. In my case I cannot any prior information whatsoever, and I think this is often the cases when applying statistics to real life (business) data. If I understand correctly, the forecasting advantage of Bayesian statistics is that you make predictions based on the posterior rather than a point estimate. Does this in general also imply increased out-of-sample accuracy? I could not really find any papers adressing this specific question. $\endgroup$
    – dubvice
    Jan 25, 2018 at 18:21
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    $\begingroup$ I do not know about business data. I guess you must sometimes know, that certain values are never negative and that your stock value is not going to climb 500% in a day and that a month's wage is not going to be $10 but as I expressed in parenthesis, there are certainly good reasons to stay with uninformative or wage priors instead of weakly informative priors. As for out-of-sample accuracy I have no research to cite. I guess, it is mostly about whether the true data generation process is close to your model or not, rather then school of computation. Just a guess, though. $\endgroup$
    – Bernhard
    Jan 25, 2018 at 20:54

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