Where does the bias come from in LogLog and HyperLoglog?
In the original LogLog paper from P. Flajolet, the cardinality estimation function is presented as
$$E := alpha_m m2^{1 over m} {^{sum_{j} M^{(j)}}}$$
Where $alpha$ is needed to correct the systematic bias in the asymptotic limit which cause overestimates.
However I am having trouble identifying where does this bias come from.
My intuition tells me this is related to the fact we retain the $max$ count of leading zeros, which is probably skewed to higher values.
Wikipedia states this is due to the risk of hash collisions, but that seems wrong as hash collision would underestimate not overestimates (this is actually an improvement in the HyperLogLog paper)
Has anyone a more explicit definition (ideally in layman terms) or reason explaining this bias?
Thank you!
probability-theory probability-limit-theorems
asked 2 days ago
Pierre LacavePierre Lacave
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In the original LogLog paper from P. Flajolet, the cardinality estimation function is presented as
$$E := alpha_m m2^{1 over m} {^{sum_{j} M^{(j)}}}$$
Where $alpha$ is needed to correct the systematic bias in the asymptotic limit which cause overestimates.
However I am having trouble identifying where does this bias come from.
My intuition tells me this is related to the fact we retain the $max$ count of leading zeros, which is probably skewed to higher values.
Wikipedia states this is due to the risk of hash collisions, but that seems wrong as hash collision would underestimate not overestimates (this is actually an improvement in the HyperLogLog paper)
Has anyone a more explicit definition (ideally in layman terms) or reason explaining this bias?
Thank you!
probability-theory probability-limit-theorems
asked 2 days ago
Pierre LacavePierre Lacave
114
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Pierre Lacave is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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This question has an open bounty worth +100
reputation from Pierre Lacave ending in 7 days.
Looking for an answer drawing from credible and/or official sources.
The two papers discussed here contrast LL and HLL; note in particular their respective uses of arithmetic and harmonic means. By the AM-HM inequality, you'd expect at least one - probably both - to be biased.
– J.G.
2 days ago
add a comment |
In the original LogLog paper from P. Flajolet, the cardinality estimation function is presented as
$$E := alpha_m m2^{1 over m} {^{sum_{j} M^{(j)}}}$$
Where $alpha$ is needed to correct the systematic bias in the asymptotic limit which cause overestimates.
However I am having trouble identifying where does this bias come from.
My intuition tells me this is related to the fact we retain the $max$ count of leading zeros, which is probably skewed to higher values.
Wikipedia states this is due to the risk of hash collisions, but that seems wrong as hash collision would underestimate not overestimates (this is actually an improvement in the HyperLogLog paper)
Has anyone a more explicit definition (ideally in layman terms) or reason explaining this bias?
Thank you!
probability-theory probability-limit-theorems
asked 2 days ago
Pierre LacavePierre Lacave
114
New contributor
Pierre Lacave is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
In the original LogLog paper from P. Flajolet, the cardinality estimation function is presented as
$$E := alpha_m m2^{1 over m} {^{sum_{j} M^{(j)}}}$$
Where $alpha$ is needed to correct the systematic bias in the asymptotic limit which cause overestimates.
However I am having trouble identifying where does this bias come from.
My intuition tells me this is related to the fact we retain the $max$ count of leading zeros, which is probably skewed to higher values.
Wikipedia states this is due to the risk of hash collisions, but that seems wrong as hash collision would underestimate not overestimates (this is actually an improvement in the HyperLogLog paper)
Has anyone a more explicit definition (ideally in layman terms) or reason explaining this bias?
Thank you!
probability-theory probability-limit-theorems
probability-theory probability-limit-theorems
asked 2 days ago
Pierre LacavePierre Lacave
114
New contributor
Pierre Lacave is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
Pierre LacavePierre Lacave
114
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Pierre Lacave is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 days ago
Pierre Lacave
asked 2 days ago
Pierre LacavePierre Lacave
114
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Pierre Lacave is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 2 days ago
Pierre LacavePierre Lacave
114
asked 2 days ago
Pierre LacavePierre Lacave
114
114
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Pierre Lacave is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
Pierre Lacave is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Pierre Lacave is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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This question has an open bounty worth +100
reputation from Pierre Lacave ending in 7 days.
Looking for an answer drawing from credible and/or official sources.
This question has an open bounty worth +100
reputation from Pierre Lacave ending in 7 days.
Looking for an answer drawing from credible and/or official sources.
The two papers discussed here contrast LL and HLL; note in particular their respective uses of arithmetic and harmonic means. By the AM-HM inequality, you'd expect at least one - probably both - to be biased.
– J.G.
2 days ago
add a comment |
The two papers discussed here contrast LL and HLL; note in particular their respective uses of arithmetic and harmonic means. By the AM-HM inequality, you'd expect at least one - probably both - to be biased.
– J.G.
2 days ago
The two papers discussed here contrast LL and HLL; note in particular their respective uses of arithmetic and harmonic means. By the AM-HM inequality, you'd expect at least one - probably both - to be biased.
– J.G.
2 days ago
The two papers discussed here contrast LL and HLL; note in particular their respective uses of arithmetic and harmonic means. By the AM-HM inequality, you'd expect at least one - probably both - to be biased.
– J.G.
2 days ago
add a comment |
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The two papers discussed here contrast LL and HLL; note in particular their respective uses of arithmetic and harmonic means. By the AM-HM inequality, you'd expect at least one - probably both - to be biased.
– J.G.
2 days ago