Does incorporation of prior expert opinions with Bayesian analysis actually work in practise or is it too much to ask of non-statisticians? Suppose we have a sample from some population of people and we want to perform Bayesian regression of height vs weight using this sample. Suppose the true relationship between height $y$ and weight $x$ is:
$$
y = \beta_0 + \beta_1 x^2 = 3 + 2x^2.
$$
The plot of the sample data is:

To implement a Bayesian linear regression model on this data I assume Gaussian priors $\beta_0 \sim \mathcal{N}(\mu_0,\sigma_0)$ and $\beta_1 \sim \mathcal{N}(\mu_1,\sigma_1)$ and that the error is normally distributed with mean zero and constant variance.
Now it remains to choose the parameters of the priors. Suppose I know a doctor who is an expert on the height/weight relationship so I visit him to obtain his expert knowledge.
Here is where I am having issues:

*

*The parameter $\mu_1$ represents the expected value of a variable that indicates how much a person's height increases as their weight squared goes up by one unit. Is a doctor really going to be able to give a good answer to this equation? We are dealing with a medical professional here, not a statistician/mathematician. I am taking a simplified example for the sake of convenience here, in practise we could have a multiple regression model with many predictors and be asking a field expert for their opinion on the mean of the several coefficients.

*The variable $\sigma_1$ is worse again. Are we really expected to believe that a medical professional will not just be able to provide a good guess for $\mu_1$ but that they can also give a good guess for the standard deviation of how much a person's height increases as their weight squared goes up by one unit? It seems way too complicated to ask this of statistician/mathematician?

*We also require the doctor to give an estimate for the covariance of $\beta_0$ and $\beta_1$. A doctor will know this?

*Suppose we extend our model to multiple regression of height $y$ vs weight $x$ and age $w$ (and assume a linear relationship for age):
$$
y = \beta_0 + \beta_1 x^2 + \beta_2 w.
$$
To incorporate the expert opinion of the doctor we now require him to to give us entries for the covariance matrix of $\beta_0,\beta_1$, and $\beta_2$. This seems to be a completely ridiculous proposition.

So how does it work in practise when a Bayesian statistican wants to generate priors for the regression coefficients based the opinion of some expert in a particular field? It seems to me that an expert could only be realistically be expected to give a decent estimate of $\mu_0$ and $\sigma_0$. Asking for the other information seems like a lost cause.
Yet incorporating expert information is often highlighted as one of the advantages of Bayesian over Frequentist statistics. Is it truly only a theoretical advantage, or can we actually make use of it in the real world?
If the difficulty is because I have somehow overcomplicated matters, can you explain to me what questions I should ask the doctor to get good parameters for the priors in the model for height vs weight? How exactly should I phrase the questions?
 A: This can be a challenging problem and one has to think quite carefully about it. There's multiple frameworks for how to elicit expert judgments such as SHELF or Delphi.
Asking non-statistical experts (and for that matter statisticians) about some quantity of interest in a way that is not a natural way of thinking for them is not a good idea. And, people mostly don't think in terms of regression equations (in fact it's a very unnatural way for most people). Instead, you could ask doctors for their judgements on the plausible heights a person that weighs $x$ kilograms and is $z$ years old. You can ask about a set of $(x, z)$ combinations that span the space of plausible values (i.e. you probably don't want to ask about totally unrealistic values) and identify the terms in your regression equation well - and perhaps even let you double-check whether your experts a-priori really believe in the functional form that you are proposing. The width of their distributions of course reflects both the uncertainty on coefficients, as well as $\sigma$, so it's important to ask about judgements when $y$ is low and high to clearly get at $\sigma$. It gets a bit trickier, if heteroskedascity is possible.
Doing it this way also ensures that you get a joint prior that avoids completely absurd prior beliefs (e.g. independent priors on $\beta_0$, $\beta_1$ and $\beta_2$ could possibly put a decent amount of prior weight on totally implausible predictions like heights <0 m or >3 m for humans).
