Do we need central banks to be independent?

In the developed world, the central banks are independent and target inflation and economic stability. However, the independence in some ways is questionable. In most cases, the government's fiscal policies and the central bank's monetary policies tend to achieve a same goal; thus despite they make policies independently the goals of these two channels are the same. Moreover, sometimes fiscal policies need the support of monetary policies. The aim of independence of the central banks is to make central banks more credible and maximize the effects of the signals sent to the markets. If we could maintain the credibility of central banks, there is no need to make central banks independent as they could never be fully independent as the monetary system has to be backed a sovereign.