interval estimation - university of illinois · interval estimation rarely are we interested in...
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Interval Estimation
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Interval Estimation
● Rarely are we interested in just a single point estimate for a parameter
● Confidence intervals are used to– Express uncertainty in an estimate
– Determine whether a hypothesized value fallswithin the interval
● Interval estimates on predicted values
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● Def'n: The fraction of intervals calculated from a large number of data sets generated by the same process that would include the true parameter value
● Def'n: Posterior probability that the parameter lies within the interval
Frequentist Confidence Interval
Bayesian Credible Interval
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Credible Intervals
● Analytically estimated from posterior CDF● Numerically estimated from quantiles of sample● NOT estimated based on standard deviation● Not necessarily symmetric● Equal tail interval: both tails have the same
probability● Highest posterior density: narrowest possible
interval
∫−∞
A
p∣Y d=∫B
∞
p∣Y d=/2
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Normal
0.025
0.975
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Beta-Binomial
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Analytical CI in R
> ## Normal 95% CI> mu = 5> sigma = 3> qnorm(c(0.025,0.975),mu,sigma)[1] -0.879892 10.879892> qnorm(c(0.925,0,975),1,0)[1] -1.959964 1.959964>> ## Beta 95% CI> p = 3> n = 13> qbeta(c(0.025,0.975),p,n-p)[1] 0.05486064 0.48413775
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Numerical Credible Intervals
250 9750
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Numerical CI in R
> quantile(beta,c(0.025,0.975)) 2.5% 97.5% 9.362793 10.779553
● Why numerical estimates of quantiles take longer to converge
● e.g. from 10000 steps, most extreme 250 used
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Model Credible Interval
● Is a transformation of random variables, f(y'|)
– f(x) is our process model
– We are interested in the PDF of some new point y'
– the model parameters are random, p(|Y)
● Formally this transformation is
● Easier to understand/solve numerically
f y ' =p∣Y Posterior
det ddy ' Jacobian
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Example: Regression
● Plotting y = b0 + b1 * x
● For each [b0,b1] in the MCMC
● Interested in distribution of E[y|x] for each x
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Regression Credible Interval
● Constructed as CI at each x
● Accounts for parameter uncertainty
● Does not account for variability of the data model
● Jensen's Inequality f x∣≠f x∣
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Numerical CI in R
xpred <- 1:20ycred <- matrix(NA,nrow=10000,ncol=2)
for(g in 1:10000){ ycred[g,] <- b0[g] + b1[g] * xpred}
ci <- apply(ycred,2,quantile,c(0.025,0.975))
lines(xpred,ci[1,],col=3,lty=2)lines(xpred,ci[2,],col=3,lty=2)
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Bayesian Prediction
p y '∣y =∫ p y '∣Likelihood of newdata
p ∣Y Posterior
d
● Consider an observed data set Y and a model with parameters
● Want to calculate the posterior PDF of some new data point y'
● Need to integrate over all values can take on for ALL the model parameters (including variances)
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Bayesian Prediction Intervals
● CI of p(y'|y) for each x
● Includes both data and parameter uncertainty
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Numerical PI in R
xpred <- 1:20ypred <- matrix(NA,nrow=10000,ncol=2)
for(g in 1:10000){ Ey = b0[g] + b1[g] * xpred ypred[g,] <- rnorm(1,Ey,sig[g])}
pi <- apply(ypred,2,quantile,c(0.025,0.975))
lines(xpred,pi[1,],col=3,lty=2)lines(xpred,pi[2,],col=3,lty=2)
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Frequentist Confidence Interval
● Will consider four approaches to estimating confidence interval– Standard Error
– Likelihood Profile
– Fisher Information
– Bootstrap
● All require additional assumptions
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● Frequentist CI does not assume a density centered on the MLE
● Cannot integrate likelihood profile
● Assumes density centered on upper/lower bound and calc. tail probabilities
● Equivalent if symmetric(e.g. Normal)
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Normal CI● Goal:
– Find u and
l the
locations where the distributions should be centered so that they have the desired tail probability
● As we know at = 0.05 (95% CI) these are located at+/- 1.96 2
● Approx 1.96 SE
∫−∞
MLE
N x∣l ,2=/2
∫MLE
∞
N x∣u ,2=/2
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Std Error Estimator
● Only approximate if not Normal● Always symmetric● Can lead to non-sensible estimates for other
distributions● Choice of likelihood not equivalent to
distribution of parameter estimator● Other methods of estimating frequentist CI
based on likelihood surface