Determining the number of burn-in (warmup-steps): fixed rate in tfp-CausalImpact versus dynamic in the R implementation

Hello `tfp-causalimpact` team,

Thank you for maintaining this great package! I have some questions regarding how burn-in (or warm-up) steps are handled in tfp-causalimpact, particularly in comparison to the original R implementation.

From my understanding, in tfp-causalimpact, the number of burn-in steps is fixed at 1/9 of num_results, meaning the first 1/9th of the samples are discarded. In contrast, the R implementation adaptively determines the burn-in by analyzing the log-likelihood trajectory, using the following process:

- Compute the log-likelihood for each sample.
- Consider the last fraction of samples (e.g., the final 10%) and find a high quantile (e.g., the 90th percentile) of log-likelihood in this tail portion.
- Identify the earliest point in the chain where the log-likelihood exceeds this quantile, and drop all samples before this point.

This adaptive approach can, in some cases, lead to dropping the first 90% of samples, depending on the log-likelihood trajectory. I have a few questions about this:

1. Could you confirm if my understanding of the burn-in behavior in tfp-causalimpact and the R implementation is correct? If not, I’d appreciate any corrections or clarifications.
2. In tfp-causalimpact, is there a way to dynamically determine the number of warm-up steps, similar to the R implementation? If not, is this something that could be considered for future updates?
3. Even in the R implementation, I’m curious about the logic of looking at the last fraction of samples to determine the burn-in threshold. Wouldn’t it make more sense to use the first fraction (e.g., the first 20%) to determine the threshold? This might prevent discarding a large proportion of samples and ensure the point estimates and credible intervals are based on a sufficiently large set of samples.

Thank you for your time and for considering these questions. I look forward to hearing your insights!

No files, tests, or entry points are named. Start by locating tfp-causalimpact's warmup-step handling and comparing it with the R implementation's adaptive logic. Done would require an agreed behavior for dynamic warmup handling, with its expected results documented and validated.

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