Theorems like the Lovász Local Lemma?analog of principle of inclusion-exclusionprobabilities of increasing events under different product measures.Can you explain the description of the Lovasz Local Lemma by Moser+Tardos?To what extent can the following zero-one laws be relaxed?A question on independenceHelp with derivation of probability density of event generation & event detectionWhat is the early history of the concepts of probabilistic independence and conditional probability/expectation?Fixing (non)-independency of a the subfamilies of finitely many events.Negative association in a “k out of n” processWhat's the probability of two independent events in time domain?
Theorems like the Lovász Local Lemma?
analog of principle of inclusion-exclusionprobabilities of increasing events under different product measures.Can you explain the description of the Lovasz Local Lemma by Moser+Tardos?To what extent can the following zero-one laws be relaxed?A question on independenceHelp with derivation of probability density of event generation & event detectionWhat is the early history of the concepts of probabilistic independence and conditional probability/expectation?Fixing (non)-independency of a the subfamilies of finitely many events.Negative association in a “k out of n” processWhat's the probability of two independent events in time domain?
$begingroup$
The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have a hypothesis of the form "Let events E_1, E_2, ... satisfy [relaxed form of independence]" and a conclusion of the form "Then the probability of [compound event] satisfies [inequality]"?
(I hope this question isn't too broad. I frequently encounter problems with events that are "almost independent", either in the sense that most subsets are independent or in the sense that the probabilities of compound events are well-approximated by assuming independence, and I am looking for general tools that may be helpful when these situations come up.)
pr.probability probabilistic-method
asked 3 hours ago
AustinAustin
1763
$endgroup$
add a comment |
$begingroup$
The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have a hypothesis of the form "Let events E_1, E_2, ... satisfy [relaxed form of independence]" and a conclusion of the form "Then the probability of [compound event] satisfies [inequality]"?
(I hope this question isn't too broad. I frequently encounter problems with events that are "almost independent", either in the sense that most subsets are independent or in the sense that the probabilities of compound events are well-approximated by assuming independence, and I am looking for general tools that may be helpful when these situations come up.)
pr.probability probabilistic-method
asked 3 hours ago
AustinAustin
1763
$endgroup$
1
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
3 hours ago
2
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago
add a comment |
$begingroup$
The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have a hypothesis of the form "Let events E_1, E_2, ... satisfy [relaxed form of independence]" and a conclusion of the form "Then the probability of [compound event] satisfies [inequality]"?
(I hope this question isn't too broad. I frequently encounter problems with events that are "almost independent", either in the sense that most subsets are independent or in the sense that the probabilities of compound events are well-approximated by assuming independence, and I am looking for general tools that may be helpful when these situations come up.)
pr.probability probabilistic-method
asked 3 hours ago
AustinAustin
1763
$endgroup$
The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have a hypothesis of the form "Let events E_1, E_2, ... satisfy [relaxed form of independence]" and a conclusion of the form "Then the probability of [compound event] satisfies [inequality]"?
(I hope this question isn't too broad. I frequently encounter problems with events that are "almost independent", either in the sense that most subsets are independent or in the sense that the probabilities of compound events are well-approximated by assuming independence, and I am looking for general tools that may be helpful when these situations come up.)
pr.probability probabilistic-method
pr.probability probabilistic-method
asked 3 hours ago
AustinAustin
1763
asked 3 hours ago
AustinAustin
1763
asked 3 hours ago
AustinAustin
1763
asked 3 hours ago
AustinAustin
1763
asked 3 hours ago
AustinAustin
1763
1763
1
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
3 hours ago
2
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago
add a comment |
1
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
3 hours ago
2
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago
1
1
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
3 hours ago
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
3 hours ago
2
2
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X_i$ have been obtained by the Chen--Stein method. See e.g. Theorem 1, which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b_1,b_2,b_3$ of the strength of the dependence between the $X_i$'s (defined in formulas (4)--(6) of that paper).
answered 2 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
$endgroup$
add a comment |
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$begingroup$
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X_i$ have been obtained by the Chen--Stein method. See e.g. Theorem 1, which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b_1,b_2,b_3$ of the strength of the dependence between the $X_i$'s (defined in formulas (4)--(6) of that paper).
answered 2 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
$endgroup$
add a comment |
$begingroup$
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X_i$ have been obtained by the Chen--Stein method. See e.g. Theorem 1, which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b_1,b_2,b_3$ of the strength of the dependence between the $X_i$'s (defined in formulas (4)--(6) of that paper).
answered 2 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
$endgroup$
add a comment |
$begingroup$
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X_i$ have been obtained by the Chen--Stein method. See e.g. Theorem 1, which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b_1,b_2,b_3$ of the strength of the dependence between the $X_i$'s (defined in formulas (4)--(6) of that paper).
answered 2 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
$endgroup$
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X_i$ have been obtained by the Chen--Stein method. See e.g. Theorem 1, which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b_1,b_2,b_3$ of the strength of the dependence between the $X_i$'s (defined in formulas (4)--(6) of that paper).
answered 2 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
answered 2 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
answered 2 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
answered 2 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
19.8k22259
add a comment |
add a comment |
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1
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
3 hours ago
2
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago