Modifying priors on the parameters of the Hill function

[Aminshnn]

I trained my model for different settings, hyper-parameters and in all cases, the saturation function was concave. I just found that the default value for the slope is 1 (deterministic value) which results in a cancave Hill function. I am wondering if it is set to exactly 1 for the optimization part to get global optimum? I am wondering if the optimization routine can find the global budget allocation if we use S-shaped Hill function? Also, when changed the prior of slope_m to TruncatedNormal(1, 0.5, 0, 2), the model didn't converge. Does it mean it should that the model cannot determine if the Saturation function is concave or S-shaped?
Another issue is that when I used TruncatedNormal(1, 0.1, 1, 2) to have an S-shaped function, it seems that posterior of slope_m didn't change much from its prior.
In this case, what should we do?

[gousesb]

Hello @Aminshnn ,

Thank you for contacting us!

Meridian uses a Deterministic slope by default to ensure model stability and avoid overparameterization. By fixing this value, the model produces a concave Hill function, which reflects the standard marketing assumption that every additional dollar spent results in diminishing returns. While the optimization routine can support S-shaped curves (slope > 1), they are harder to estimate reliably and make the optimization less stable. Concave functions are generally preferred because they reduce complexity and lead to more consistent optimization behavior.

The convergence issues you experienced when using a flexible prior like TruncatedNormal(1, 0.5, 0, 2) are likely due to parameter non-identifiability. This typically occurs when the data does not have sufficient variation or a wide enough spend range to distinguish between concave and S-shaped responses. In such cases, multiple curve shapes can explain the data equally well, leading to poor convergence . Similarly, when you used TruncatedNormal(1, 0.1, 1, 2) and the Posterior did not change much from your Prior, it indicates that your data was uninformative for that parameter. In Meridian’s Bayesian framework, the model updates the prior based on the Likelihood (the signal from your data).

If the data does not exhibit a clear threshold, the model may struggle to confidently learn an S-shaped response. In such uninformative cases, rely on prior knowledge or external results (e.g., lift studies or geo-experiments) to select priors. If you lack outside evidence for an S-shape (threshold spend) for a channel, it's recommended to keep the slope Deterministic. This maintains model identifiability, ensuring stable and reliable budget optimization results for decision-making.

Feel free to reach out for any further queries or feedback regarding Meridian.

Google Meridian Support Team