'Does tweedie_variance_power matter when log-transforming predictions?

I haven't been able to find any canonical sources on how tweedie_variance_power comes into play when predicting using an XGBoost algorithm with objective=reg:tweedie. My dependent variable is log-transformed auto insurance claim amount, so when I go to predict, in order to get units in dollars, I apply exp to the "raw" predictions from XGBoost (which look like they're on a log scale).

However (and perhaps this is due to this model not being a very good one), when I apply exp(log_predictions), the resultant and presumably-dollar-amount predictions are so much lower than expected, given the dollar amounts in the training data. Am I missing something? Does my tweedie_variance_power = 2 for this model need to also be accounted for when transforming back to dollar units?

Related question: Xgboost tweedie: Why is the formula to get the prediction from the link = exp(link)/ 2?



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