Bimodal prior

Hi,
I’m new to Bayesian analysis and BRM. After a couple of days, I haven’t found how to address this issue, so thank you for your advice:

I have an ordinal variable (e.g., 1,2,3,4,5) which the result of a Likert scale (We asked participants to tell us from 1 to 5 how much they liked the stimuli heard over headphones). We want to predict that outcome from several continuous variables measured from this stimuli. For example, some of the stimuli was produced with a low frequency (in Hz) and some was produced with a high frequency. All the stimuli was produced by a single speaker. Naturally, this variable has a bimodal distribution. I want to include this information in the prior for this variable, but I just can’t find how to do it.

Can somebody point me in the right direction, please?

Could your variable (low/high frequency) be coded as 1 and 2?

No, it’s a continuous predictor

Between which values?

Hi Richard,

The values are between 80 Hz to 400 Hz. I’d like to find how increasing frequency affects the outcome.
I also have a categorical variable with ‘low’ and ‘high,’ as you previously suggested, but we prefer the continuous variable for our study.

Cheers,

Can’t you just standardize the predictor, i.e., (x_i - \bar{x}) /\sigma_x, and then set a Normal(0,whatever) on it?

I think there is some confusion here. If a predictor variable is bimodal, then you don’t need to do anything special with the prior. The "prior for a predictor variable* is more accurately a prior for the coefficient that multiplies the predictor variable. Is there any reason that your domain knowledge suggests a bimodal prior distribution for the estimated coefficient? If not, you don’t need a bimodal prior here, regardless of whether the predictor is bimodally distributed or not.

On the other hand, if your outcome variable is bimodal, note that the “prior” for this variable is encapsulated by the entire model and the priors on all of its parameters. The prior on the outcome is what we call the prior predictive distribution. If you are hoping to ensure that this prior predictive distribution is consistent with your observed bimodal outcome data, then reply to this message and we can move forward with that slightly more focused question.

6 Likes

Hi Jacob,

“Is there any reason that your domain knowledge suggests a bimodal prior distribution for the estimated coefficient?”

No there isn’t.

“If not, you don’t need a bimodal prior here, regardless of whether the predictor is bimodally distributed or not.”

I think you’re answer is very clarifying. I was confusing the distribution of the predictor with the distribution for the coefficients, if I understood correctly. Then, I think in my case, should I use, say, weakly informative priors based on my beliefs about the effect of the predictor? e.g., if I believe that increasing frequency will have a positive effect on the ratings of the stimuli (i.e., participants would be more likely to rate high pitch stimuli with higher scores) I should specify, for example, a prior with a normal distribution a la
normal(4,2)?
Does this make sense or I’m still lost in the bushes? Thanks again, and if you recommend me where to read about this, I’d appreciate it.

Cheers,

2 Likes

How strong is your belief that it should be positive? You need to defend your priors later… Also, do prior predictive checks to see what the priors imply! :)

How strong is your belief that it should be positive? You need to defend your priors later…

not very strong, TBH

You need to defend your priors later… Also, do prior predictive checks to see what the priors imply! :)

Would do, thanks for the suggestion and have nice weekend!

If it’s not very strong then I would be careful and center it on ‘0’.

If it’s not very strong then I would be careful and center it on ‘0’.

Even though the output is an ordinal variable?

…So, what if we were so sure—cocksure, even—that the true effect of being unemployed is to lower estimated pro-immigration sentiment by 2.12 points? Here, we believe this is the true effect and our regression model is not picking this up even though the t -test is. If we were so sure about this, we could specify this as a prior distribution on the estimated coefficient for unemployment with a normal distribution plugging in that difference in means as the mean of the distribution and the standard error of the t -test as the standard deviation of that distribution

I’m not deadly sure of the effect of the predictor in my case, but I thought that I should center a normal distribution around the presumed effect of pitch on likeness, i.e., I believe that increasing pitch would increase the probability of choosing a higher rate (4 in my case), is this wrong?

1 Like

Well the outcome is not that important in this case.