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There is a publicly-available dataset generated by an experiment from a publication. I replicated the experiment to generate my own dataset. I wasn't able to perfectly replicate the original experiment because the authors did not provide enough information. Even though our experimental setup is not the same, we are capturing the same type of data.

First, I applied classification methods on the publicly-available dataset. Then, I applied classification methods on my own dataset. Would it be valid to combine both datasets and apply classification methods on the pooled datasets even though they were generated by different experiments?

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Yes, you can either do a standard meta-analysis or a Bayesian analysis. The choice between the two would primarily be with regard to what question you want to be answered. How to do either is a large subject, unfortunately.

If you wanted to be able to construct better confidence intervals, unbiased point estimates, and inference then you want a standard meta-analysis. If you distrust the results of the public set, or if you were going to use the results for gambling purposes such as setting budgets, determining inventory or investing capital then you should probably use a Bayesian method. The other principal reason to use a Bayesian method would be if you have material prior information, other than just the public set of data, that could be incorporated into your study.

In the general case, you would not pool the data but rather the results.

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