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    Why do you want to do a second PhD? There is no advantage of obtaining a second PhD in a related field over simply switching your future research to that field. A second PhD is simply not an efficient use of your time and resources. The natural step, if you want to stay connected to academia, would be a post-doc. It's very common to switch to a slightly different field during one of your post-docs. Commented Mar 11, 2020 at 11:21
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    A PhD certifies you can do research. It is not an efficient way of certifying that you have knowledge of a particular field (that was usually a precondition before entering the program). So a second PhD is not going to help you. What you need is to fill the gaps in your knowledge and get some work experience. Probably the most efficient way to fill the knowledge gap is to join some MOOCs. Commented Mar 11, 2020 at 11:40
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    So, perform research on evenings and weekends. Nobody is saying you can't do that. All we are saying is that you don't need to do a second dissertation. It's quite unlikely that you'd find an advisor anyway, because they would also see it as a waste of their time. What you should look for is a cooperation. You should read the answer to this question. Commented Mar 11, 2020 at 13:14
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    @SecondPhD - you seem to be of the opinion that you need to go do another PhD if what you are working on isn't quite what you did your first PhD on. I would suggest that most PhDs are doing something fairly different within 5-10 years of their thesis, yet do not feel they need to get another PhD - they learned how to learn new stuff on the fly without courses in the first one. Commented Mar 11, 2020 at 14:11
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    I have realised based on the answers I've received that my math PhD will be viewed as essentially equivalent to a stats/ML PhD in industry so I can see now that there is no need to do a second PhD in this area. Commented Mar 12, 2020 at 8:45