我们有N个样本 X一世
Xi ,从均匀分布 [0,θ][0,θ] 哪里 θθ 未知。估计θθ 从数据。
因此,贝叶斯法则...
F(θ|X一世)=F(X一世|θ)F(θ)F(X一世)
可能是:
F(X一世|θ)=∏ñ一世=1个1个θ
但是没有其他信息 θ
我们有N个样本 X一世
Xi ,从均匀分布 [0,θ][0,θ] 哪里 θθ 未知。估计θθ 从数据。
因此,贝叶斯法则...
F(θ|X一世)=F(X一世|θ)F(θ)F(X一世)
可能是:
F(X一世|θ)=∏ñ一世=1个1个θ
但是没有其他信息 θ
Answers:
This has generated some interesting debate, but note that it really doesn't make much difference to the question of interest. Personally I think that because θ
p(θ|I)=θ−1log(UL)∝θ−1L<θ<U
This distribution has the same form under rescaling of the problem (the likelihood also remains "invariant" under rescaling). The kernel of this prior, f(y)=y−1
p(θ|DI)=Nθ−N−1(L∗)−N−U−NL∗<θ<UwhereL∗=max(L,X(N))
But now suppose we use a more general prior, given by p(θ|cI)∝θ−c−1
E(θ|DI)=N+cN+c−1X(N)
So the uniform prior (c=−1
One argument against the use of the improper uniform prior in this case is that the posterior is improper when N=1
Since the purpose here is presumably to obtain some valid and useful estimate of θ
Now assume that this population consists of m
Denote for compactness maxi=1,...,n{Xi}≡X∗
The density function of the max
for the support [0,θ]
which may be improper if we don't specify the constant c
Then writing X={x1,..,xn}
f(θ∣X)∝θ−NNcNθ−1⇒f(θ∣X)=ANcNθ−(N+1)
for some normalizing constant A. We want ∫Sθf(θ∣X)dθ=1⇒∫∞x∗ANcNθ−(N+1)dθ=1
⇒ANcN1−Nθ−N|∞x∗=1⇒A=(cx∗)N
Inserting into the posterior f(θ∣X)=(cx∗)NNcNθ−(N+1)=N(x∗)Nθ−(N+1)
Note that the undetermined constant c
The posterior summarizes all the information that the specific sample can give us regarding the value of θ
Is there any intuition in this result? Well, as the number of X
"If the totality of Your information about θ external to the data D is captured by the single proposition B={{Possible values for θ}={the interval (a,b)},a<b}
Thus, you prior specification should correspond to the Jeffrey's prior if you truly believe in the above theorem."
Not part of the Uniform Prior Distribution Theorem:
Alternatively you could specify your prior distribution f(θ) as a Pareto distribution, which is the conjugate distribution for the uniform, knowing that you posterior distribution will have to be another uniform distribution by conjugacy. However, if you use the Pareto distribution, then you will need to specify parameters of the Pareto distribution in some sort of way.