# How to specify nonlinear Bayesian regression model?

I'm a little confused about how to specify a regression model in Bayesian terms. I understand that for the simple case of plain linear regression, the response variable is normally distributed about the mean, which varies linearly, like so: $$Y \sim N(\beta X, \sigma)$$ where $$\beta$$ and $$\sigma$$ are drawn from priors and an intercept can also be added.

My data (sample shown below) are roughly exponentially shaped. Values can be zero, but not negative. My first impulse was to try a model with $$Y \sim N(\lambda e^{-\lambda X},\sigma)$$, but I'm not sure if this makes sense and, anyway, it would allow for negative values but shouldn't.

Ultimately, my goal is to estimate the variance along this curve, getting a distribution as a function of the variable on the horizontal axis.

Is there a standard approach?

• why are the circles filled with different colours and have different diameters? Do you perhaps have more than one feature? Nov 4, 2022 at 10:14
• Yes I do. Markers are colored by the same feature as the horizontal axis, so that's no issue. Marker sizes represent another feature, but I don't necessarily need to include it in the model. Nov 4, 2022 at 15:28
• Great. Another thing, you are asking about two different things (1) Bayesian regression, (2) regression for the variance. In (1) Iguess you want to model the mean. Do you have any prior information? Is there any motivation about (2) ? Nov 4, 2022 at 16:26
• Yes I want to model the mean and estimate the variance as a function of the same feature on the horizontal axis. I know that the response variable can never be negative. Nov 4, 2022 at 18:16

• Ok that makes sense. I'm still in the process of understanding what particular distribution Y assumes under these transformations. Nov 3, 2022 at 14:08
• Does it make sense to use a model in the form: $\log Y \sim N(\beta X + \alpha, \gamma X)$ where $\beta$, $\alpha$, and $\gamma$ are the parameters to estimate Nov 3, 2022 at 20:14