[SciPy-User] Confidence interval for bounded minimization

josef.pktd@gmai... josef.pktd@gmai...
Thu Feb 23 00:23:44 CST 2012


On Thu, Feb 23, 2012 at 1:10 AM,  <josef.pktd@gmail.com> wrote:
> On Thu, Feb 23, 2012 at 12:09 AM, Christopher Jordan-Squire
> <cjordan1@uw.edu> wrote:
>> On Wed, Feb 22, 2012 at 2:02 PM, Nathaniel Smith <njs@pobox.com> wrote:
>>> On Wed, Feb 22, 2012 at 8:48 PM,  <josef.pktd@gmail.com> wrote:
>>>> On Wed, Feb 22, 2012 at 3:26 PM, Greg Friedland
>>>> <greg.friedland@gmail.com> wrote:
>>>>> Hi,
>>>>>
>>>>> Is it possible to calculate asymptotic confidence intervals for any of
>>>>> the bounded minimization algorithms? As far as I can tell they don't
>>>>> return the Hessian; that's including the new 'minimize' function which
>>>>> seemed like it might.
>>>>
>>>> If the parameter ends up at the bounds, then the standard statistics
>>>> doesn't apply. The Hessian is based on a local quadratic
>>>> approximation, which doesn't work if part of the local neigborhood is
>>>> out of bounds.
>>>> There is some special statistics for this, but so far I have seen only
>>>> the description how GAUSS handles it.
>>>>
>>>> In statsmodels we use in some cases the bounds, or a transformation,
>>>> just to keep the optimizer in the required range, and we assume we get
>>>> an interior solution. In this case, it is possible to use the standard
>>>> calculations, the easiest is to use the local minimum that the
>>>> constraint or transformed optimizer found and use it as starting value
>>>> for an unconstrained optimization where we can get the Hessian (or
>>>> just calculate the Hessian based on the original objective function).
>>>
>>> Some optimizers compute the Hessian internally. In those cases, it
>>> would be nice to have a way to ask them to somehow return that value
>>> instead of throwing it away. I haven't used Matlab in a while, but I
>>> remember running into this as a standard feature at some point, and it
>>> was quite nice. Especially when working with a problem where each
>>> computation of the Hessian requires an hour or so of computing time.
>>>
>>
>> Are you talking about analytic or finite-difference gradients and
>> hessians? I'd assumed that anything derived from finite difference
>> estimations wouldn't give particularly good confidence intervals, but
>> I've never needed them so I've never looked into it in detail.
>
> statsmodels has both, all discrete models for example have analytical
> gradients and hessians.
>
> But for models with a complicated log-likelihood function, there isn't
> much choice, second derivatives with centered finite differences are
> ok, scipy.optimize.leastsq is not very good. statsmodels also has
> complex derivatives which are numerically pretty good but they cannot
> always be used.
>
> I think in most cases numerical derivatives will have a precision of a
> few decimals, which is more precise than all the other statistical
> assumptions, normality, law of large numbers, local definition of
> covariance matrix to calculate "large" confidence intervals, and so
> on.
>
> One problem is that choosing the step size depends on the data and
> model. numdifftools has adaptive calculations for the derivatives, but
> we are not using it anymore.
>
> Also, if the model is not well specified, then the lower precision of
> finite difference derivatives can hurt. For example, in ARMA models I
> had problems when there are too many lags specified, so that some
> roots should almost cancel. Skipper's implementation works better
> because he used a reparameterization that forces some nicer behavior.
>
> The only case in the econometrics literature that I know is that early
> GARCH models were criticized for using numerical derivatives even
> though analytical derivatives were available, some parameters were not
> well estimated, although different estimates produced essentially the
> same predictions (parameters are barely identified)
>
> Last defense: everyone else does it, maybe a few models more or less,
> and if the same statistical method is used, then the results usually
> agree pretty well.
> (But if different methods are used, for example initial conditions are
> treated differently in time series analysis, then the differences are
> usually much larger. Something like: I don't worry about numerical
> problems at the 5th or 6th decimal if I cannot figure out what these
> guys are doing with their first and second decimal.)
>
> (maybe more than anyone wants to know.)

In case it wasn't clear: analytical derivatives are of course much
better, and I would be glad if the scipy.stats.distributions or sympy
had the formulas for the derivatives of the log-likelihood functions
for the main distributions. (but it's work)

Josef


>
> Josef
> .
>
>>
>> -Chris
>>
>>
>>> -- Nathaniel
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