In one word, it is usually the model predictive mean that is used for prediction.
For instance, suppose we are building a model on the amount of money an insurance company pays for reimbursement. When we say we want to predict this quantity, we mean that we wish to know the amount of money the insurance company reimburses given the values the predictors (i.e., independent variables) take. This usually amounts to calculating the predicted mean by the model given the value of the predictors. Now that the predictive mean of Tweedie model is always larger than zero, we, in the fictitious scenario mentioned above, will claim that the amount of reimbursement is always larger than zero, since this is what the Tweedie model tells us in terms of its predictive mean. But that is definitely not true. A lot of zeros are actually observed. That is why the data is termed as zero-inflated data.
However, if the researcher is interested on the factors associated with the dependent variable rather than hoping to find out the expected mean given values of predictors, then Tweedie models can be selected. Following the scenario above, suppose we are interested in the correlation of the amount of money reimbursed and age instead of trying to figure out the amount of money the company is expected to pay given that the person's age is, say, 69. In this case, the absolute amount of the regression coefficient estimate as well as its sign (+ or -) can provide us information regarding the relationship of the amount of money reimbursed and age.
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