One of the problems with the English language is that words such as "mistake" and "error" connote shortfalls from a feasible ideal when in fact, in some important cases, errors are inevitable and unavoidable.
I'm prompted to say this by Yian Mui's piece which points out that "the Fed has continually underestimated how much support the economy needs" in the last seven years. A similar thing is true in the UK. For example, the Bank of England forecast two years ago that inflation would now be around 2% but in fact even the core rate is just 1%.
These "errors", though, are not corrigible. Macroeconomic forecasts are inevitably subject to a margin of "error". And this margin is especially big just when we need forecasts most, to warn us of recession; as Prakash Loungani pointed out in 2000 economists have a perfect record of failing to foresee recessions, a fact corroborated by subsequent experience. These "errors" aren't due merely or even mainly to economists' incompetence but to the fact that the economy is a complex process whose outcomes depend upon inherently unpredictable network effects. As GLS Shackle said years ago, accurate forecasting is pretty much impossible*.
This in turn means that monetary policy "errors" are common and inevitable, simply because policy affects the real economy and inflation with a lag - remember the title of Tony Yates' blog - and so the right policy now requires foresight as to future economic conditions.
In the absence of such foresight, though, policy will often be "wrong". The recession of 2009 implies that policy was too tight in 2007-08 and the fact that inflation now is well below its (symmetric, remember) target implies that policy was too tight some months ago.
These "errors", though, are not eliminable bugs but features; forecast-based policies will often be wrong.
It's for this reason that, like Tony, I am lukewarm about money GDP targets: I can't get excited about whether it's better for the Bank to miss a money GDP target than an inflation target.
Instead, in thinking about monetary policy we must give great weight to the inevitability that policy will be wrong.
This has, of course, long been a feature of the literature. Back in the 1960s, William Brainard said (pdf) that it meant that central banks should (pdf) be slow to change rates - a principle which the Fed followed for years. And textbooks say that central banks should (pdf) minimize a loss function. The precise form of this function is, though, debatable. Simon says that at low inflation, the Bank should put more weight upon output losses than inflation overshoots, but JP Koning says this isn't necessarily true if interest rates can become negative.
One reason why is that the gilt market could interpret even the honest policy errors of a rate-setting Chancellor as being in fact politically motivated, by a desire to stoke up booms. This would raise inflation expectations and hence nominal interest rates. Independence removes this interpretation and hence allows rates to be lower: this, or something like it, was the argument upon which independence was founded in the 90s.
There is, though, another reason for independence. If Chancellors set interest rates the media, in its imbecile failure to see that "errors" are inevitable, would blame them for even honest and unavoidable "mistakes": imagine how much worse the criticism would be of Gordon Brown if he had been responsible for interest rates in the mid-00s. The political case for central bank independence is to ensure that somebody else is the fall guy for inevitable policy "errors".
* The fact that one or two people foresaw the crisis of 2008 doesn't refute my point. Successful monetary policy requires not that forecasts be right sometimes, but that they be right all the time. AFAIK, nobody has achieved this, even roughly.
** I mean operational independence: Richard is right that its independence is circumscribed.