Complexity of histogram scheme
$begingroup$
Due to work by Lugosi and Nobel available here I know that for consistent density estimation complexity of histogram scheme should increase sub exponentially. By complexity we mean:
Given a number of observations $n$ number of distinct ways any set of $n$ observations can be grouped in bins (number of distinct partitions). Given that maximum number of cells possible for $n$ observations is $a$, this number is always bounded by $a^n$
I do not understand the derivation given in the paper as it uses theorems of measure, and I have not gone through any measure theory. I want understand the reasoning at least intuitively. This is my guess:
We want to ensure that the scheme is consistent, For each partition which is member of given set of partitions empirical average converges to true average exponentially with respect to number of observations (as known from inequalities similar to Hoeffding's inequality). Since we take max over all partition, overall probability of the class will be grow linearly with number of distinct partitions in the class. It is known that if the growth function(number of distinct partitions) is sub-exponential it is necessarily polynomial. A polynomial growth function allows the the overall average to converge.
Please give me some insight regarding this.
estimation density-function
asked Dec 20 '18 at 9:02
CuriousCurious
889516
$endgroup$
add a comment |
$begingroup$
Due to work by Lugosi and Nobel available here I know that for consistent density estimation complexity of histogram scheme should increase sub exponentially. By complexity we mean:
Given a number of observations $n$ number of distinct ways any set of $n$ observations can be grouped in bins (number of distinct partitions). Given that maximum number of cells possible for $n$ observations is $a$, this number is always bounded by $a^n$
I do not understand the derivation given in the paper as it uses theorems of measure, and I have not gone through any measure theory. I want understand the reasoning at least intuitively. This is my guess:
We want to ensure that the scheme is consistent, For each partition which is member of given set of partitions empirical average converges to true average exponentially with respect to number of observations (as known from inequalities similar to Hoeffding's inequality). Since we take max over all partition, overall probability of the class will be grow linearly with number of distinct partitions in the class. It is known that if the growth function(number of distinct partitions) is sub-exponential it is necessarily polynomial. A polynomial growth function allows the the overall average to converge.
Please give me some insight regarding this.
estimation density-function
asked Dec 20 '18 at 9:02
CuriousCurious
889516
$endgroup$
add a comment |
$begingroup$
Due to work by Lugosi and Nobel available here I know that for consistent density estimation complexity of histogram scheme should increase sub exponentially. By complexity we mean:
Given a number of observations $n$ number of distinct ways any set of $n$ observations can be grouped in bins (number of distinct partitions). Given that maximum number of cells possible for $n$ observations is $a$, this number is always bounded by $a^n$
I do not understand the derivation given in the paper as it uses theorems of measure, and I have not gone through any measure theory. I want understand the reasoning at least intuitively. This is my guess:
We want to ensure that the scheme is consistent, For each partition which is member of given set of partitions empirical average converges to true average exponentially with respect to number of observations (as known from inequalities similar to Hoeffding's inequality). Since we take max over all partition, overall probability of the class will be grow linearly with number of distinct partitions in the class. It is known that if the growth function(number of distinct partitions) is sub-exponential it is necessarily polynomial. A polynomial growth function allows the the overall average to converge.
Please give me some insight regarding this.
estimation density-function
asked Dec 20 '18 at 9:02
CuriousCurious
889516
$endgroup$
Due to work by Lugosi and Nobel available here I know that for consistent density estimation complexity of histogram scheme should increase sub exponentially. By complexity we mean:
Given a number of observations $n$ number of distinct ways any set of $n$ observations can be grouped in bins (number of distinct partitions). Given that maximum number of cells possible for $n$ observations is $a$, this number is always bounded by $a^n$
I do not understand the derivation given in the paper as it uses theorems of measure, and I have not gone through any measure theory. I want understand the reasoning at least intuitively. This is my guess:
We want to ensure that the scheme is consistent, For each partition which is member of given set of partitions empirical average converges to true average exponentially with respect to number of observations (as known from inequalities similar to Hoeffding's inequality). Since we take max over all partition, overall probability of the class will be grow linearly with number of distinct partitions in the class. It is known that if the growth function(number of distinct partitions) is sub-exponential it is necessarily polynomial. A polynomial growth function allows the the overall average to converge.
Please give me some insight regarding this.
estimation density-function
estimation density-function
asked Dec 20 '18 at 9:02
CuriousCurious
889516
asked Dec 20 '18 at 9:02
CuriousCurious
889516
edited Dec 21 '18 at 11:58
Curious
asked Dec 20 '18 at 9:02
CuriousCurious
889516
asked Dec 20 '18 at 9:02
CuriousCurious
889516
asked Dec 20 '18 at 9:02
CuriousCurious
889516
889516
add a comment |
add a comment |
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