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Posted to dev@commons.apache.org by Phil Steitz <ph...@gmail.com> on 2011/01/02 23:05:23 UTC
[math] meaning of "support" in distributions classes
We don't precisely define what we mean by the support of a distribution
anywhere. I have been assuming that we mean the smallest closed set such
that its complement has probability 0. This would make, for example, the
support of the Beta distribution [0, 1] independent of the parameters. But
then isSupportLowerBoundInclusive currently returns false for Beta. I must
have one of the concepts wrong. Could it be that the
upper/lowerboundInclusive attributes are only meaningful in the discrete
case?
Phil
Re: [math] meaning of "support" in distributions classes
Posted by Mikkel Meyer Andersen <mi...@mikl.dk>.
2011/1/3 Phil Steitz <ph...@gmail.com>:
> On Mon, Jan 3, 2011 at 2:04 PM, Ted Dunning <te...@gmail.com> wrote:
>
>> I don't think you are missing anything. Moreover, I think that wikipedia
>> just has an error in this regard.
>>
>> Following their chain of definitions leads to this example:
>>
>>
>> http://en.wikipedia.org/wiki/Support_(measure_theory)#A_uniform_distribution<http://en.wikipedia.org/wiki/Support_%28measure_theory%29#A_uniform_distribution>
>>
>> If the uniform distribution on the open interval (0,1) has the closed set
>> [0,1] as its support then the beta distribution
>> obviously does as well. In fact, the definition they use starts with "The
>> largest closed set ...".
>>
>> Yes. We should probably state somewhere in the javadoc that we are using
> that definition. A possible modification that would make the Wikipedia Beta
> example make sense (but make the Uniform example wrong ;) would be to
> consider whether or not the endpoints are in the domain of the density
> function. I don't see that info as adding a lot of value, so am +1 for just
> dropping the isXxxIncluded properties, but leaving isSupportConnected in
> place.
+1
>
> Phil
>
>
>> On Mon, Jan 3, 2011 at 11:00 AM, Mikkel Meyer Andersen <mi...@mikl.dk>
>> wrote:
>>
>> > > I am happy to keep them if I can get a clear understanding of what they
>> > > mean. As I said in the original post, I think I must be missing
>> > something
>> > > that makes them meaningful. If you use the definition that I gave of
>> > > support, other than infinities, the endpoints are always going to be
>> > > included. Could well be I am missing something.
>> > No, I don't think that you've missed anything. I probably haven't
>> > given it a decent thought when I included them to begin with. So the
>> > right think is to remove those functions following the de facto
>> > definition of support.
>> >
>>
>
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Re: [math] meaning of "support" in distributions classes
Posted by Phil Steitz <ph...@gmail.com>.
On Mon, Jan 3, 2011 at 2:04 PM, Ted Dunning <te...@gmail.com> wrote:
> I don't think you are missing anything. Moreover, I think that wikipedia
> just has an error in this regard.
>
> Following their chain of definitions leads to this example:
>
>
> http://en.wikipedia.org/wiki/Support_(measure_theory)#A_uniform_distribution<http://en.wikipedia.org/wiki/Support_%28measure_theory%29#A_uniform_distribution>
>
> If the uniform distribution on the open interval (0,1) has the closed set
> [0,1] as its support then the beta distribution
> obviously does as well. In fact, the definition they use starts with "The
> largest closed set ...".
>
> Yes. We should probably state somewhere in the javadoc that we are using
that definition. A possible modification that would make the Wikipedia Beta
example make sense (but make the Uniform example wrong ;) would be to
consider whether or not the endpoints are in the domain of the density
function. I don't see that info as adding a lot of value, so am +1 for just
dropping the isXxxIncluded properties, but leaving isSupportConnected in
place.
Phil
> On Mon, Jan 3, 2011 at 11:00 AM, Mikkel Meyer Andersen <mi...@mikl.dk>
> wrote:
>
> > > I am happy to keep them if I can get a clear understanding of what they
> > > mean. As I said in the original post, I think I must be missing
> > something
> > > that makes them meaningful. If you use the definition that I gave of
> > > support, other than infinities, the endpoints are always going to be
> > > included. Could well be I am missing something.
> > No, I don't think that you've missed anything. I probably haven't
> > given it a decent thought when I included them to begin with. So the
> > right think is to remove those functions following the de facto
> > definition of support.
> >
>
Re: [math] meaning of "support" in distributions classes
Posted by Ted Dunning <te...@gmail.com>.
I don't think you are missing anything. Moreover, I think that wikipedia
just has an error in this regard.
Following their chain of definitions leads to this example:
http://en.wikipedia.org/wiki/Support_(measure_theory)#A_uniform_distribution
If the uniform distribution on the open interval (0,1) has the closed set
[0,1] as its support then the beta distribution
obviously does as well. In fact, the definition they use starts with "The
largest closed set ...".
On Mon, Jan 3, 2011 at 11:00 AM, Mikkel Meyer Andersen <mi...@mikl.dk> wrote:
> > I am happy to keep them if I can get a clear understanding of what they
> > mean. As I said in the original post, I think I must be missing
> something
> > that makes them meaningful. If you use the definition that I gave of
> > support, other than infinities, the endpoints are always going to be
> > included. Could well be I am missing something.
> No, I don't think that you've missed anything. I probably haven't
> given it a decent thought when I included them to begin with. So the
> right think is to remove those functions following the de facto
> definition of support.
>
Re: [math] meaning of "support" in distributions classes
Posted by Mikkel Meyer Andersen <mi...@mikl.dk>.
2011/1/3 Phil Steitz <ph...@gmail.com>:
> On Mon, Jan 3, 2011 at 1:41 PM, Mikkel Meyer Andersen <mi...@mikl.dk> wrote:
>
>> 2011/1/3 Phil Steitz <ph...@gmail.com>:
>> > On Mon, Jan 3, 2011 at 2:23 AM, Mikkel Meyer Andersen <mi...@mikl.dk>
>> wrote:
>> >
>> >> Hi,
>> >>
>> >> You're right, Phil. Support for beta is [0, 1] and not (0, 1) as stated
>> on
>> >> Wikipedia. As you mention, support for continuous distributions is
>> closed,
>> >> hence the corresponding isInclusive-functions can be discussed. I
>> thought
>> >> about it being useful for infinity, but we could let users deal with
>> this
>> >> themselves?
>> >>
>> >
>> > Yeah, sorry I missed this before. It hit me when I was working on the
>> 2_X
>> > retrofit and it looked like Beta was wrong (I see now Wikipedia seems to
>> be
>> > using some other definition - or is just wrong). I dropped the
>> > inclusive/exclusive functions there. I think in the discrete case, this
>> can
>> > be handled by convention and the only issue there is the same as the
>> > continuous one - infinities - but these are all the same. So I propose
>> that
>> > we drop these functions in 3.0 as well. The isSupportConnected property
>> > still logically makes sense; though it is always true for the 2_X
>> > distributions, so I dropped it there.
>> +1
>> I agree. For now we don't need them, neither in 2.2 nor 3.0. Is there
>> any sense in keeping them if we later on includes distributions where
>> it would be beneficial to have such functions, or should we simply
>> just add them then?
>> >
>>
> I am happy to keep them if I can get a clear understanding of what they
> mean. As I said in the original post, I think I must be missing something
> that makes them meaningful. If you use the definition that I gave of
> support, other than infinities, the endpoints are always going to be
> included. Could well be I am missing something.
No, I don't think that you've missed anything. I probably haven't
given it a decent thought when I included them to begin with. So the
right think is to remove those functions following the de facto
definition of support.
Cheers, Mikkel.
>
> Phil
>
>
>> > Phil
>> >
>> >>
>> >> Cheers, Mikkel.
>> >> Den 02/01/2011 23.05 skrev "Phil Steitz" <ph...@gmail.com>:
>> >> > We don't precisely define what we mean by the support of a
>> distribution
>> >> > anywhere. I have been assuming that we mean the smallest closed set
>> such
>> >> > that its complement has probability 0. This would make, for example,
>> the
>> >> > support of the Beta distribution [0, 1] independent of the parameters.
>> >> But
>> >> > then isSupportLowerBoundInclusive currently returns false for Beta. I
>> >> must
>> >> > have one of the concepts wrong. Could it be that the
>> >> > upper/lowerboundInclusive attributes are only meaningful in the
>> discrete
>> >> > case?
>> >> >
>> >> > Phil
>> >>
>> >
>>
>> ---------------------------------------------------------------------
>> To unsubscribe, e-mail: dev-unsubscribe@commons.apache.org
>> For additional commands, e-mail: dev-help@commons.apache.org
>>
>>
>
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Re: [math] meaning of "support" in distributions classes
Posted by Phil Steitz <ph...@gmail.com>.
On Mon, Jan 3, 2011 at 1:41 PM, Mikkel Meyer Andersen <mi...@mikl.dk> wrote:
> 2011/1/3 Phil Steitz <ph...@gmail.com>:
> > On Mon, Jan 3, 2011 at 2:23 AM, Mikkel Meyer Andersen <mi...@mikl.dk>
> wrote:
> >
> >> Hi,
> >>
> >> You're right, Phil. Support for beta is [0, 1] and not (0, 1) as stated
> on
> >> Wikipedia. As you mention, support for continuous distributions is
> closed,
> >> hence the corresponding isInclusive-functions can be discussed. I
> thought
> >> about it being useful for infinity, but we could let users deal with
> this
> >> themselves?
> >>
> >
> > Yeah, sorry I missed this before. It hit me when I was working on the
> 2_X
> > retrofit and it looked like Beta was wrong (I see now Wikipedia seems to
> be
> > using some other definition - or is just wrong). I dropped the
> > inclusive/exclusive functions there. I think in the discrete case, this
> can
> > be handled by convention and the only issue there is the same as the
> > continuous one - infinities - but these are all the same. So I propose
> that
> > we drop these functions in 3.0 as well. The isSupportConnected property
> > still logically makes sense; though it is always true for the 2_X
> > distributions, so I dropped it there.
> +1
> I agree. For now we don't need them, neither in 2.2 nor 3.0. Is there
> any sense in keeping them if we later on includes distributions where
> it would be beneficial to have such functions, or should we simply
> just add them then?
> >
>
I am happy to keep them if I can get a clear understanding of what they
mean. As I said in the original post, I think I must be missing something
that makes them meaningful. If you use the definition that I gave of
support, other than infinities, the endpoints are always going to be
included. Could well be I am missing something.
Phil
> > Phil
> >
> >>
> >> Cheers, Mikkel.
> >> Den 02/01/2011 23.05 skrev "Phil Steitz" <ph...@gmail.com>:
> >> > We don't precisely define what we mean by the support of a
> distribution
> >> > anywhere. I have been assuming that we mean the smallest closed set
> such
> >> > that its complement has probability 0. This would make, for example,
> the
> >> > support of the Beta distribution [0, 1] independent of the parameters.
> >> But
> >> > then isSupportLowerBoundInclusive currently returns false for Beta. I
> >> must
> >> > have one of the concepts wrong. Could it be that the
> >> > upper/lowerboundInclusive attributes are only meaningful in the
> discrete
> >> > case?
> >> >
> >> > Phil
> >>
> >
>
> ---------------------------------------------------------------------
> To unsubscribe, e-mail: dev-unsubscribe@commons.apache.org
> For additional commands, e-mail: dev-help@commons.apache.org
>
>
Re: [math] meaning of "support" in distributions classes
Posted by Mikkel Meyer Andersen <mi...@mikl.dk>.
2011/1/3 Phil Steitz <ph...@gmail.com>:
> On Mon, Jan 3, 2011 at 2:23 AM, Mikkel Meyer Andersen <mi...@mikl.dk> wrote:
>
>> Hi,
>>
>> You're right, Phil. Support for beta is [0, 1] and not (0, 1) as stated on
>> Wikipedia. As you mention, support for continuous distributions is closed,
>> hence the corresponding isInclusive-functions can be discussed. I thought
>> about it being useful for infinity, but we could let users deal with this
>> themselves?
>>
>
> Yeah, sorry I missed this before. It hit me when I was working on the 2_X
> retrofit and it looked like Beta was wrong (I see now Wikipedia seems to be
> using some other definition - or is just wrong). I dropped the
> inclusive/exclusive functions there. I think in the discrete case, this can
> be handled by convention and the only issue there is the same as the
> continuous one - infinities - but these are all the same. So I propose that
> we drop these functions in 3.0 as well. The isSupportConnected property
> still logically makes sense; though it is always true for the 2_X
> distributions, so I dropped it there.
+1
I agree. For now we don't need them, neither in 2.2 nor 3.0. Is there
any sense in keeping them if we later on includes distributions where
it would be beneficial to have such functions, or should we simply
just add them then?
>
> Phil
>
>>
>> Cheers, Mikkel.
>> Den 02/01/2011 23.05 skrev "Phil Steitz" <ph...@gmail.com>:
>> > We don't precisely define what we mean by the support of a distribution
>> > anywhere. I have been assuming that we mean the smallest closed set such
>> > that its complement has probability 0. This would make, for example, the
>> > support of the Beta distribution [0, 1] independent of the parameters.
>> But
>> > then isSupportLowerBoundInclusive currently returns false for Beta. I
>> must
>> > have one of the concepts wrong. Could it be that the
>> > upper/lowerboundInclusive attributes are only meaningful in the discrete
>> > case?
>> >
>> > Phil
>>
>
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Re: [math] meaning of "support" in distributions classes
Posted by Phil Steitz <ph...@gmail.com>.
On Mon, Jan 3, 2011 at 2:23 AM, Mikkel Meyer Andersen <mi...@mikl.dk> wrote:
> Hi,
>
> You're right, Phil. Support for beta is [0, 1] and not (0, 1) as stated on
> Wikipedia. As you mention, support for continuous distributions is closed,
> hence the corresponding isInclusive-functions can be discussed. I thought
> about it being useful for infinity, but we could let users deal with this
> themselves?
>
Yeah, sorry I missed this before. It hit me when I was working on the 2_X
retrofit and it looked like Beta was wrong (I see now Wikipedia seems to be
using some other definition - or is just wrong). I dropped the
inclusive/exclusive functions there. I think in the discrete case, this can
be handled by convention and the only issue there is the same as the
continuous one - infinities - but these are all the same. So I propose that
we drop these functions in 3.0 as well. The isSupportConnected property
still logically makes sense; though it is always true for the 2_X
distributions, so I dropped it there.
Phil
>
> Cheers, Mikkel.
> Den 02/01/2011 23.05 skrev "Phil Steitz" <ph...@gmail.com>:
> > We don't precisely define what we mean by the support of a distribution
> > anywhere. I have been assuming that we mean the smallest closed set such
> > that its complement has probability 0. This would make, for example, the
> > support of the Beta distribution [0, 1] independent of the parameters.
> But
> > then isSupportLowerBoundInclusive currently returns false for Beta. I
> must
> > have one of the concepts wrong. Could it be that the
> > upper/lowerboundInclusive attributes are only meaningful in the discrete
> > case?
> >
> > Phil
>
Re: [math] meaning of "support" in distributions classes
Posted by Mikkel Meyer Andersen <mi...@mikl.dk>.
Hi,
You're right, Phil. Support for beta is [0, 1] and not (0, 1) as stated on
Wikipedia. As you mention, support for continuous distributions is closed,
hence the corresponding isInclusive-functions can be discussed. I thought
about it being useful for infinity, but we could let users deal with this
themselves?
Cheers, Mikkel.
Den 02/01/2011 23.05 skrev "Phil Steitz" <ph...@gmail.com>:
> We don't precisely define what we mean by the support of a distribution
> anywhere. I have been assuming that we mean the smallest closed set such
> that its complement has probability 0. This would make, for example, the
> support of the Beta distribution [0, 1] independent of the parameters. But
> then isSupportLowerBoundInclusive currently returns false for Beta. I must
> have one of the concepts wrong. Could it be that the
> upper/lowerboundInclusive attributes are only meaningful in the discrete
> case?
>
> Phil
Re: [math] meaning of "support" in distributions classes
Posted by Ted Dunning <te...@gmail.com>.
Ok. That makes sense.
Sent from my iPhone
On Jan 2, 2011, at 9:26 PM, Phil Steitz <ph...@gmail.com> wrote:
> On Sun, Jan 2, 2011 at 11:46 PM, Ted Dunning <te...@gmail.com> wrote:
>
>> On Sun, Jan 2, 2011 at 2:05 PM, Phil Steitz <ph...@gmail.com> wrote:
>>
>>> We don't precisely define what we mean by the support of a distribution
>>> anywhere. I have been assuming that we mean the smallest closed set such
>>> that its complement has probability 0.
>>
>>
>> Why closed?
>>
>> Why not just the smallest set such that the complement has probability 0?
>>
>
> Because that in general will not be well-defined. Consider, for example,
> the support of the Beta distribution. The smallest closed set whose
> complement has probability 0 is [0, 1] (independently of the parameters).
> If the definition does not require that the set be closed, then when you
> consider (0, 1], [0, 1), [0, 1] - {x} for any x in [0,1], or [0, 1] minus
> any finite number of points...you see that there will be no unique smallest
> (in terms of inclusion) set whose complement has probability 0.
>
> Phil
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Re: [math] meaning of "support" in distributions classes
Posted by Phil Steitz <ph...@gmail.com>.
On Sun, Jan 2, 2011 at 11:46 PM, Ted Dunning <te...@gmail.com> wrote:
> On Sun, Jan 2, 2011 at 2:05 PM, Phil Steitz <ph...@gmail.com> wrote:
>
> > We don't precisely define what we mean by the support of a distribution
> > anywhere. I have been assuming that we mean the smallest closed set such
> > that its complement has probability 0.
>
>
> Why closed?
>
> Why not just the smallest set such that the complement has probability 0?
>
Because that in general will not be well-defined. Consider, for example,
the support of the Beta distribution. The smallest closed set whose
complement has probability 0 is [0, 1] (independently of the parameters).
If the definition does not require that the set be closed, then when you
consider (0, 1], [0, 1), [0, 1] - {x} for any x in [0,1], or [0, 1] minus
any finite number of points...you see that there will be no unique smallest
(in terms of inclusion) set whose complement has probability 0.
Phil
Re: [math] meaning of "support" in distributions classes
Posted by Ted Dunning <te...@gmail.com>.
On Sun, Jan 2, 2011 at 2:05 PM, Phil Steitz <ph...@gmail.com> wrote:
> We don't precisely define what we mean by the support of a distribution
> anywhere. I have been assuming that we mean the smallest closed set such
> that its complement has probability 0.
Why closed?
Why not just the smallest set such that the complement has probability 0?