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population 100k
- sample 20k? then problem comes from the population
uncertainty in the estimation of the ESS itself?
- ess itself is an estimator
- can provide an investigation into the accuracy of the ESS itself?
- the more ESS is more averaged? Is that because of the inaccuracy of estimating the ESS? or should it deviate from the ESS?
--- if population is different from the expected population, then could be different
------ ex: draws could be more homogeneous than the superpopulation, so should expect to see more effective sample size, but because the population itself is an easier population to learn from/work with?
--------- ex: plot the ESS versus the population variance? would explain to some extent if the population variance drawn is smaller.
maybe can't talk about effective sample size with one fixed dataset?
-
-
ESS should be compared to the population beta, and not the superpopulation beta?
- depending on what you want to generalize to
- if superpopulation beta, usually have larger MSEs
Notion of effective sample size with respect to what? [probably good to add a paragraph.]
- indication of information measure +1
-- information cannot be defined without replication, with respect to which replications you care about/which information you care about?
----- this will separate between the finite population (fixed) and superpopulation.
----- not saying one is better than the other -- they are different, but depends on what is the level of uncertainty you care about?
- can add a histogram adding uncertainty of the ESS itself.
brad efron and hinkley [???] {Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information} -> observed versus expected fisher information
--- observed one is much more relevant: known that score and observed FI can be highly correlated, but if you standardize the observed FI by the expected FI (difference) --> known to be independent of the MLE. Not as surprising because if not dependent on beta, then is the same.
--- LLN kicks in for the sum on the right, so on the right, the expected fisher information is un-correlated with the score, while the observed fisher information somewhat is.
------ can reproduce the right-side one, if the score itself you plug in the population beta.
--- score itself, because you put in the average population beta, should be zero
--- the marginal distribution of the score is the same
---- score function evaluated at population beta, expanded around the sample beta is just (beta - beta hat) times derivative at sample beta.
----- so the difference is proportional to beta hat minus beta
----- if population information matrix,
fisher information -- measures variability -- usually free of the estimate itself?
x^2 ( 1 - P(Y) ) (P(Y))
P(Y) = expit(beta * x)
-- at least check via integral calculator if expit function has nice integral. otherwise, simulate.
FI is meant to be a residual?
^ in linear regression is clear
- so, should be stable to perturbations in beta...?
- beta hat would drive ps more towards 1 or 0?
--- can formally taylor expand wrt beta, and plug in beta or beta hat, and see what happens?
in normal distribution,
In practice, can't do either of the score vs fi plots?
- for score function, can always plug in beta
-- for fi, can also always plug in beta
^ and can give a curve as well?
^ can also be quite practical! and can do it probably directly
for non-linear problem, can probably learn something about the ddc and jn without looking at the truth, maybe even the bound.
approximate elasticity by the derivative??
-- maybe some approximation that could help understand the Jn, but not as much for the linear case?
---- sometimes things are inconsistent, but could still have the relationship between Jn and ddc.
1) you could have a biased frame, but 2) if you only want to study the effect of the bias, then doesn't matter if you have a biased frame or biased response?
-- note that there is an impact of the sample size!
----- impact of biased response is ONLY on the intended sample
----- suppose that my intended sample is very small.
----- sampled vs intended feels unfair?
----- one R is single R, other R is the product of the two?
----- even if in expectation, R are the same, impact is different?
------ even if same ddc, the stages are different because N matters and does diff damage to the error.
- sample 20k? then problem comes from the population
uncertainty in the estimation of the ESS itself?
- ess itself is an estimator
- can provide an investigation into the accuracy of the ESS itself?
- the more ESS is more averaged? Is that because of the inaccuracy of estimating the ESS? or should it deviate from the ESS?
--- if population is different from the expected population, then could be different
------ ex: draws could be more homogeneous than the superpopulation, so should expect to see more effective sample size, but because the population itself is an easier population to learn from/work with?
--------- ex: plot the ESS versus the population variance? would explain to some extent if the population variance drawn is smaller.
maybe can't talk about effective sample size with one fixed dataset?
-
-
ESS should be compared to the population beta, and not the superpopulation beta?
- depending on what you want to generalize to
- if superpopulation beta, usually have larger MSEs
Notion of effective sample size with respect to what? [probably good to add a paragraph.]
- indication of information measure +1
-- information cannot be defined without replication, with respect to which replications you care about/which information you care about?
----- this will separate between the finite population (fixed) and superpopulation.
----- not saying one is better than the other -- they are different, but depends on what is the level of uncertainty you care about?
- can add a histogram adding uncertainty of the ESS itself.
brad efron and hinkley [???] {Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information} -> observed versus expected fisher information
--- observed one is much more relevant: known that score and observed FI can be highly correlated, but if you standardize the observed FI by the expected FI (difference) --> known to be independent of the MLE. Not as surprising because if not dependent on beta, then is the same.
--- LLN kicks in for the sum on the right, so on the right, the expected fisher information is un-correlated with the score, while the observed fisher information somewhat is.
------ can reproduce the right-side one, if the score itself you plug in the population beta.
--- score itself, because you put in the average population beta, should be zero
--- the marginal distribution of the score is the same
---- score function evaluated at population beta, expanded around the sample beta is just (beta - beta hat) times derivative at sample beta.
----- so the difference is proportional to beta hat minus beta
----- if population information matrix,
fisher information -- measures variability -- usually free of the estimate itself?
x^2 ( 1 - P(Y) ) (P(Y))
P(Y) = expit(beta * x)
-- at least check via integral calculator if expit function has nice integral. otherwise, simulate.
FI is meant to be a residual?
^ in linear regression is clear
- so, should be stable to perturbations in beta...?
- beta hat would drive ps more towards 1 or 0?
--- can formally taylor expand wrt beta, and plug in beta or beta hat, and see what happens?
in normal distribution,
In practice, can't do either of the score vs fi plots?
- for score function, can always plug in beta
-- for fi, can also always plug in beta
^ and can give a curve as well?
^ can also be quite practical! and can do it probably directly
for non-linear problem, can probably learn something about the ddc and jn without looking at the truth, maybe even the bound.
approximate elasticity by the derivative??
-- maybe some approximation that could help understand the Jn, but not as much for the linear case?
---- sometimes things are inconsistent, but could still have the relationship between Jn and ddc.
1) you could have a biased frame, but 2) if you only want to study the effect of the bias, then doesn't matter if you have a biased frame or biased response?
-- note that there is an impact of the sample size!
----- impact of biased response is ONLY on the intended sample
----- suppose that my intended sample is very small.
----- sampled vs intended feels unfair?
----- one R is single R, other R is the product of the two?
----- even if in expectation, R are the same, impact is different?
------ even if same ddc, the stages are different because N matters and does diff damage to the error.
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