f*******n
发帖数: 392 | 1 在使用L(θ)=g(x_1,…x_n:θ)h(x_1,…x_n )来证明sufficient statistics时，总会
遇到h(x_1,…x_n )=1。不理解，因为任何函数都可以写成L(θ)=g(x_1,…x_n:θ)×1
的形式，那什么时候可以用1来证明是sufficient的呢？ | k*******a
发帖数: 772 | 2 sufficient本来就不是唯一的，有很多，本身(x1,..xn)就是一个sufficient
statistic, 如果要找好的sufficient statistics那就找minimum sufficient
complete statistic | a*********r
发帖数: 139 | 3 Likelihood function is a sufficient statistic just as the whole sample. (
This shows that sufficient statistic always exists.) However, the motivation
of finding a sufficient statistic is "to shrink the sample X without losing
information." Thus, likelihood function is pretty much useless in this
sense.
Hope this helps. | f*******n
发帖数: 392 | | I*****a
发帖数: 5425 | 5 The sample, not the likelihood function, is.
motivation
losing
【在 a*********r 的大作中提到】
: Likelihood function is a sufficient statistic just as the whole sample. (
: This shows that sufficient statistic always exists.) However, the motivation
: of finding a sufficient statistic is "to shrink the sample X without losing
: information." Thus, likelihood function is pretty much useless in this
: sense.
: Hope this helps.
| a*********r
发帖数: 139 | 6 I'm afraid that you are wrong. Both the sample and the likelihood function
are.
【在 I*****a 的大作中提到】
: The sample, not the likelihood function, is.
:
: motivation
: losing
| I*****a
发帖数: 5425 | 7 I dont know.
Do we say a likelihood function is sufficient ?
Is a likelihood function a function of parameters ?
【在 a*********r 的大作中提到】
: I'm afraid that you are wrong. Both the sample and the likelihood function
: are.
| l*********s
发帖数: 5409 | 8 Of course. and I don't see the practical distinction between likelyhood and
sample.
【在 I*****a 的大作中提到】
: I dont know.
: Do we say a likelihood function is sufficient ?
: Is a likelihood function a function of parameters ?
| I*****a
发帖数: 5425 | 9 OK. My point is, do we ever say a likelihood is sufficient, or "sufficient l
ikelihood" ?
and
【在 l*********s 的大作中提到】
: Of course. and I don't see the practical distinction between likelyhood and
: sample.
| B****n
发帖数: 11290 | 10 We do not say likelihood is sufficient because it is not a statistics, which
by definition can be computed by samples.
l
【在 I*****a 的大作中提到】
: OK. My point is, do we ever say a likelihood is sufficient, or "sufficient l
: ikelihood" ?
:
: and
| | | a*********r
发帖数: 139 | 11 The likelihood function is fully specificied by the sample. Considered as a function L_{X}(\cdot), it is a statistic. Note theta is a
dummy variable in the likelihood function. I know lots of people get confused about this.
【在 I*****a 的大作中提到】
: I dont know.
: Do we say a likelihood function is sufficient ?
: Is a likelihood function a function of parameters ?
| I*****a
发帖数: 5425 | 12 I don't quite think so.
Likelihood is a function of parameters, no matter you think in frequencists'
way or a Bayesian way. This has been well established in many textbooks, as
well as wikipedia.
Concept-wise, likelihood is not the same as "joint density/distribution", al
though mathematically they are equal.
Z = x^2 + y
It is a quadratic as a function of x, but not so as a function of y.
a function L_{X}(\cdot), it is a statistic. Note theta is a
confused about this.
【在 a*********r 的大作中提到】
: The likelihood function is fully specificied by the sample. Considered as a function L_{X}(\cdot), it is a statistic. Note theta is a
: dummy variable in the likelihood function. I know lots of people get confused about this.
| D******n
发帖数: 2836 | 13 什么叫做“用1来证明是sufficient”？
1
【在 f*******n 的大作中提到】
: 在使用L(θ)=g(x_1,…x_n:θ)h(x_1,…x_n )来证明sufficient statistics时，总会
: 遇到h(x_1,…x_n )=1。不理解，因为任何函数都可以写成L(θ)=g(x_1,…x_n:θ)×1
: 的形式，那什么时候可以用1来证明是sufficient的呢？
| I*****a
发帖数: 5425 | 14 h = 1
【在 D******n 的大作中提到】
: 什么叫做“用1来证明是sufficient”？
:
: 1
| D******n
发帖数: 2836 | 15 ok then second floor pretty much answered it, but this thread has gone
awry since the third floor, lol.
【在 I*****a 的大作中提到】
: h = 1
| I*****a
发帖数: 5425 | 16 ya. lou2 wai1 le.
【在 D******n 的大作中提到】
: ok then second floor pretty much answered it, but this thread has gone
: awry since the third floor, lol.
| k*******a
发帖数: 772 | 17 You guys need to check your textbook:
statistic is a function of observed values (X1, X2...Xn), which shouldn't
include parameter. So likelihood function, which is a function of parameter,
is NOT a statistic | I*****a
发帖数: 5425 | 18 thats exactly my point
parameter,
【在 k*******a 的大作中提到】
: You guys need to check your textbook:
: statistic is a function of observed values (X1, X2...Xn), which shouldn't
: include parameter. So likelihood function, which is a function of parameter,
: is NOT a statistic
| a*********r
发帖数: 139 | 19 Yes. A statistic depends only on the sample not the parameters involved.
Please read my answer carefully. Considered as a fucnction, L_X(\codt) is
completely specified by the sample. There is no parameter involved!
Here theta is a dummy variable. Just as f(x) is a function, then x is a dummy variable, you can call it theta, beta, t, s, whatever.
This answers the original question: whether we have a sufficient statistic if we allow h=1 which is always legal? Yes. We can always let h=1 in the Factorization Theomrem and get likelihood function which is a sufficient statistic.
"This thread has gone awry since the third floor..."??? This simply shows lots of people here do not even fully understand this very basic concept in statistics. LOL too!
No offense. Please do not claim others are wrong without careful thinking. If you still didn't get it, I strongly recommend that you ask your professors. Thanks.
【在 I*****a 的大作中提到】
: thats exactly my point
:
: parameter,
| I*****a
发帖数: 5425 | 20 So
1) Do we use the concept "sufficient" when we are talking about a statistic
? I personally never saw terms like "sufficient likelihood" or "a likelihood
function is sufficient", etc. I even tried to google it and you may try it,
too.
2) Is a likelihood function a function of parameters ? No one says a sample
{Xi} cannot fully specify a likelihood function. But this does not change th
e fact that "likelihood function is a function of parameters"
I think these two are the results everyone can copy and paste from any textb
ook.If not, then why do people write likelihood as L(theta) ?
You can say a sample {Xi} contains equal information as f(Xi...|theta ), but
you can never change the fact that L is not even a statistic, which people
seldom call it sufficient or not.
Again L(theta) = f(X1 ... | theta) is true, but when we use the word "
likelihood", we usually refer to L(theta), right ?
dummy variable, you can call it theta, beta, t, s, whatever.
if we allow h=1 which is always legal? Yes. We can always let h=1 in the
Factorization Theomrem and get likelihood function which is a sufficient
statistic.
lots of people here do not even fully understand this very basic concept in
statistics. LOL too!
If you still didn't get it, I strongly recommend that you ask your
professors. Thanks.
【在 a*********r 的大作中提到】
: Yes. A statistic depends only on the sample not the parameters involved.
: Please read my answer carefully. Considered as a fucnction, L_X(\codt) is
: completely specified by the sample. There is no parameter involved!
: Here theta is a dummy variable. Just as f(x) is a function, then x is a dummy variable, you can call it theta, beta, t, s, whatever.
: This answers the original question: whether we have a sufficient statistic if we allow h=1 which is always legal? Yes. We can always let h=1 in the Factorization Theomrem and get likelihood function which is a sufficient statistic.
: "This thread has gone awry since the third floor..."??? This simply shows lots of people here do not even fully understand this very basic concept in statistics. LOL too!
: No offense. Please do not claim others are wrong without careful thinking. If you still didn't get it, I strongly recommend that you ask your professors. Thanks.
| | | I*****a
发帖数: 5425 | 21 and ok, I may try to ask some professors tomorrow or the day after about
suffici
ent likelihood.
I don't believe PROFESSORS that much though. I believe textbooks a lot more,
especially on concepts. But i will try. I will learn stuffs either way.
dummy variable, you can call it theta, beta, t, s, whatever.
if we allow h=1 which is always legal? Yes. We can always let h=1 in the
Factorization Theomrem and get likelihood function which is a sufficient
statistic.
lots of people here do not even fully understand this very basic concept in
statistics. LOL too!
If you still didn't get it, I strongly recommend that you ask your
professors. Thanks.
【在 a*********r 的大作中提到】
: Yes. A statistic depends only on the sample not the parameters involved.
: Please read my answer carefully. Considered as a fucnction, L_X(\codt) is
: completely specified by the sample. There is no parameter involved!
: Here theta is a dummy variable. Just as f(x) is a function, then x is a dummy variable, you can call it theta, beta, t, s, whatever.
: This answers the original question: whether we have a sufficient statistic if we allow h=1 which is always legal? Yes. We can always let h=1 in the Factorization Theomrem and get likelihood function which is a sufficient statistic.
: "This thread has gone awry since the third floor..."??? This simply shows lots of people here do not even fully understand this very basic concept in statistics. LOL too!
: No offense. Please do not claim others are wrong without careful thinking. If you still didn't get it, I strongly recommend that you ask your professors. Thanks.
| a*********r
发帖数: 108 | 22 You are correct. Avidswimmer is conceptually wrong indeed. Likelihood is
never considered as a statistic.
statistic
likelihood
it,
sample
th
textb
【在 I*****a 的大作中提到】
: So
: 1) Do we use the concept "sufficient" when we are talking about a statistic
: ? I personally never saw terms like "sufficient likelihood" or "a likelihood
: function is sufficient", etc. I even tried to google it and you may try it,
: too.
: 2) Is a likelihood function a function of parameters ? No one says a sample
: {Xi} cannot fully specify a likelihood function. But this does not change th
: e fact that "likelihood function is a function of parameters"
: I think these two are the results everyone can copy and paste from any textb
: ook.If not, then why do people write likelihood as L(theta) ?
| a*********r
发帖数: 139 | 23 Make sure ask a professor who fully understands the theory of mathematical statistics based on measure theory and does serious research in mathematical
statistics.
more,
in
【在 I*****a 的大作中提到】
: and ok, I may try to ask some professors tomorrow or the day after about
: suffici
: ent likelihood.
: I don't believe PROFESSORS that much though. I believe textbooks a lot more,
: especially on concepts. But i will try. I will learn stuffs either way.
:
: dummy variable, you can call it theta, beta, t, s, whatever.
: if we allow h=1 which is always legal? Yes. We can always let h=1 in the
: Factorization Theomrem and get likelihood function which is a sufficient
: statistic.
| a*********r
发帖数: 139 | 24 With respect, actually you are conceptually wrong. Can you point out why I'm
wrong? Can you answer what happens when h=1 in the Factorization Theorem?
The only restriction for h is that h is a Borel-measurable function and it
does not depend on the parameters. Obviously, the constant function 1 is
always a valid choice.
Likelihood function can always be considered as a statistic. When we talk
about likelihood function, we fix x's and consider it as a function of the
parameter theta. Thus, theta is a dummy variable.
I'm going to stop replying if you guys still cannot get it. I strongly
recommend that you ask a professor who is conducting serious research in
mathematical statistics.
【在 a*********r 的大作中提到】
: You are correct. Avidswimmer is conceptually wrong indeed. Likelihood is
: never considered as a statistic.
:
: statistic
: likelihood
: it,
: sample
: th
: textb
| I*****a
发帖数: 5425 | 25 Come on. Don't get mad. We were just discussing a concept.
No offense here as before.
I think your claim some other people and I don't agree w/ is in the 2nd post
of
this thread, that is "likelihood is sufficient". That's all. And I said I
never saw this term, or alternatively the claim like "some likelihood is
suff
icient". People usually talk about "** statistics are sufficient" instead.
As I replied in the 2nd or 3rd post before this, I tried to search for this
term and failed.
I did ask two professors about sufficient likelihood. I think they are both
good. Both are doing theories. They said they never used this term but they
think the two are basically the same thing.
Well, everybody knows the likelihood function is fully specified by the samp
le, and there is no essential difference between the likelihood function and
the sample itself. (I agreed with it in my 1st or 2nd post.) This is not wh
ere the conflict is.
I am just saying that concepts are concepts, conventions are conventions, an
d rules are rules. People never say "sufficient likelihood", so it will be w
ierd if you create a term but actually the term was invented fifty years ago
. You said likelihood "can be" considered as a statistic, but people years
ago defined it as a function of parameters, and most people buy it. So it is
not convincing if you say I can apply all the terminology for A to B, ju
st because A and B can be considered as the same.
If you have found any paper, book, or even website except mitbbs, well defni
ning and using this term, please show me. That is the easiest to do. It will
be very hard if you only leave me with "study more, or ask good people". I
think examples are the most straigtforward. Indeed what you said is not hard
to understand, but I don't think you know what my concern was. My concern is
"That's something people don't do."
'm
【在 a*********r 的大作中提到】
: With respect, actually you are conceptually wrong. Can you point out why I'm
: wrong? Can you answer what happens when h=1 in the Factorization Theorem?
: The only restriction for h is that h is a Borel-measurable function and it
: does not depend on the parameters. Obviously, the constant function 1 is
: always a valid choice.
: Likelihood function can always be considered as a statistic. When we talk
: about likelihood function, we fix x's and consider it as a function of the
: parameter theta. Thus, theta is a dummy variable.
: I'm going to stop replying if you guys still cannot get it. I strongly
: recommend that you ask a professor who is conducting serious research in
| a*********r
发帖数: 139 | 26 I'm not mad. I just don't want to waste my time when people do not read your
answer carefully and jump to their wrong conclusion.
The second post is not mine. I started from the 3rd.
I never used the term "sufficient likelihood" if you go back to check all my
posts. All I said is likelihood function is a sufficient statistic when
considered as a function.
I feel you still do not fully understand the concept of a statistic. Yes. Likelihood function is always defined a function of the parameter. This is exactly why it is a statistic because the parameter is a dummy variable. This does not conflict the fact that it is a statistic because a statistic is simply a measurable function defined on the sample space and likelihood is a measurable function on this space.
post
this
【在 I*****a 的大作中提到】
: Come on. Don't get mad. We were just discussing a concept.
: No offense here as before.
: I think your claim some other people and I don't agree w/ is in the 2nd post
: of
: this thread, that is "likelihood is sufficient". That's all. And I said I
: never saw this term, or alternatively the claim like "some likelihood is
: suff
: icient". People usually talk about "** statistics are sufficient" instead.
: As I replied in the 2nd or 3rd post before this, I tried to search for this
: term and failed.
| n*****n
发帖数: 3123 | 27 严格的说，likelihood function 不是statistic.
statistic 只是关于sample的，不能含有parameter.
比如n(\mu, 1)
X_1是statistic, X_1-\mu 不是
概念问题，没啥好争的。 |
|
