- Is Statistics something you do to data? Is it
*procedural*?

- Ideally, it's a way of
*thinking*to avoiding fooling yourself & others.

- In many disciplines, "Statistics" is calculation, not thinking.

- Consequence of how statistics is taught and of peverse incentives:
*Cargo-Cult Statistics*.

- Want to use data $Y \in \Re^n$ to learn about the (unknown) state of the world, $\theta$ in a mathematical model of physical system.

- Often $\theta$ is a function of position and/or time: infinite-dimensional

- "Know" that $\theta \in \Theta$.

*measurement model*: If $\theta = \eta$, $Y \sim P_\eta$.

- Known measure $\mu$ that dominates all $P_\eta$ s.t. $\eta \in \Theta$.

- Density of $P_\eta$ at $y$ w.r.t. $\mu$ is

*likelihood of $\eta$ given $Y = y$*is $p_\eta(y)$, viewed as function of $\eta$.

- Typically impossible to estimate $\theta$ with any useful level of accuracy (maybe not even identifiable).

- Generally possible and scientifically interesting to estimate some
*parameter*$\lambda = \lambda[\theta]$

- Uses
*prior probability distribution*$\pi$ on $\Theta$ and likelihood $p_\eta(y)$.

- $\pi$ and $p_\eta$ imply a joint distribution of $\theta$ and $Y$.

*Marginal distribution*or*predictive distribution*of $Y$ is

*Posterior distribution of $\theta$ given*$Y=y$:

- All the information in the prior and the data is in the posterior distribution.

- Posterior distribution $\pi_\lambda(d \ell | Y = y)$ of $\lambda[\theta]$ is induced by posterior distribution of $\theta$:

*Descriptive*: people*are*Bayesian.

*Normative*: people*should be*Bayesian.

*Practical*: "data swamp the prior" so doesn't matter

- My guess: popular because gives a general recipe and smaller error bars than frequentist methods.

- But error bars don't have the same meaning.

- To use Bayesian framework
*must*quantify beliefs and constraints as a prior $\pi$.

- Constraint $\theta \in \Theta$ captured as $\pi(\Theta) = 1$.

- But infinitely many probability distributions assign probability 1 to $\Theta$.

Priors selected or justified in ~5 ways:

- to make the calculations simple
- because the particular prior is conventional
- so that the prior satisfies some invariance
- with the assertion that the prior is "uninformative" (e.g., Laplace's principle)
- because the prior roughly matches the relative frequencies of values in some population.

Main difference:

Frequentists treat $\theta$ as an unknown element of $\Theta$.

Bayesians treat $\theta$ as if drawn at random from $\Theta$ using $\pi$.

Bayesian approach requires *much* stronger assumptions.

Consider Bayesian and frequentist versions of 2 summaries:

mean squared error (frequentist) and posterior mean squared error (Bayesian)

confidence sets (frequentist) and credible regions (Bayesian)

MSE is an expectation with respect to the distribution of the data $Y$, holding the parameter $\theta = \eta$ fixed.

PMSE is an expectation with respect to the posterior distribution of $\theta$, holding the data $Y = y$ fixed.

A random set $I(Y)$ of possible values of $\lambda$ is
a $1-\alpha$ *confidence set* for $\lambda[\theta]$ if
$$
P_\eta \{ I(Y) \ni \lambda[\eta] \} \ge 1 - \alpha, \;\; \forall \eta \in \Theta.
$$

Probability w.r.t. distribution of the data $Y$, holding $\eta$ fixed.

*posterior credible region* for $\lambda[\theta]$ if
$$
P_{\pi( d\theta | Y=y)} (\lambda[\theta] \in I(y))
\equiv \int_{I(y)} \pi_\lambda(d \ell | Y = y) \ge 1-\alpha.
$$

Probability w.r.t. marginal posterior distribution of $\lambda[\theta]$, holding the data fixed.

credible level: probability that by drawing from prior, nature generates an element of the set, given the data

confidence level: probability that procedure gives a set that contains the truth

Frequentist: hold parameter constant, characterize behavior under repeated measurement

Bayesian: hold measurement constant, characterize behavior under repeatedly drawing parameter at random from the prior

- Formal Bayesian uncertainty can be made as small as desired by choosing prior appropriately.

Posterior uncertainty measures meaningful only if you believe prior

Changes the subject

Is the truth unknown? Is it a realization of a known probability distribution?

Where does prior come from?

Usually chosen for computational convenience or habit, not "physics"

Priors get their own literature

Eliciting priors deeply problemmatic

Why should I care about your posterior, if I don't share your prior?

How much does prior matter?

Slogan "the data swamp the prior." Theorem has conditions that aren't always met.

- Is all uncertainty random?

Aleatory

- Canonical examples: coin toss, die roll, lotto, roulette
- under some circumstances, behave "as if" random (but not perfectly)

- Epistemic: stuff we don't know

Standard way to combine aleatory variability epistemic uncertainty puts beliefs on a par with an unbiased physical measurement w/ known uncertainty.

Claims by introspection, can estimate without bias, with known accuracy, just as if one's brain were unbiased instrument with known accuracy

Bacon put this to rest, but empirically:

- people are bad at making even rough quantitative estimates
- quantitative estimates are usually biased
- bias can be manipulated by anchoring, priming, etc.
- people are bad at judging weights
*in their hands*: biased by shape & density - people are bad at judging when something is random
- people are overconfident in their estimates and predictions
- confidence unconnected to actual accuracy.
- anchoring affects entire disciplines (e.g., Millikan, c, Fe in spinach)

what if I don't trust your internal scale, or your assessment of its accuracy?

same observations that are factored in as "data" are also used to form beliefs: the "measurements" made by introspection are not independent of the data

- Can grade Bayesian methods using frequentist criteria

- E.g., what is the coverage probability of a credible region?

Bounded normal mean

Election audits

Observe $Y \sim N(\theta, 1)$.

Know

*a priori*that $\theta \in [-\tau, \tau]$

- Bayes "uninformative" prior: $\theta \sim U[-\tau, \tau]$

Check whether reported winner(s) really won by looking at random sample of ballots.

Absent convincing evidence that reported winners really won, keep looking.

Risk: probability that the audit does not correct the reported outcome.

Constraint: vote shares are non-negative, sum of shares $\le 1$.

Keep auditing until the (sequential) $P$-value of the hypothesis that the outcome is wrong is sufficiently small.

Known maximum risk, regardless of correct result.

Keep auditing until the conditional probability that the outcome is wrong, given the data, is sufficiently small.

Requires a prior.

"Nonpartisan" prior is invariant under permutations of the candidate names: "fair."

Includes "flat" or "uninformative" prior.

Among all procedures for constructing a valid $1-\alpha$ confidence set for a parameter, which has the smallest worst-case expected size?

Exploit duality between Bayesian and frequentist methods: least-favorable prior.

It is inappropriate to be concerned about mice when there are tigers abroad.

—George Box

Commonly ignored sources of uncertainty:

Coding errors (ex: Hubble)

Stability of optimization algorithms (ex: GONG)

"upstream" data reduction steps

Quality of PRNGs (RANDU, but still an issue)

Stark, P.B., R.L. Parker, G. Masters, and J.A. Orcutt, 1986. Strict bounds on seismic velocity in the spherical Earth, Journal of Geophysical Research, 91, 13,892–13,902.

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Stark, P.B., 1993. Uncertainty of the COBE quadrupole detection, Astrophysical Journal Letters, 408, L73–L76.

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Stark, P.B. and L. Tenorio, 2010. A Primer of Frequentist and Bayesian Inference in Inverse Problems. In Large Scale Inverse Problems and Quantification of Uncertainty, Biegler, L., G. Biros, O. Ghattas, M. Heinkenschloss, D. Keyes, B. Mallick, L. Tenorio, B. van Bloemen Waanders and K. Willcox, eds. John Wiley and Sons, NY. Preprint: https://www.stat.berkeley.edu/~stark/Preprints/freqBayes09.pdf

Stark, P.B., 2015. Constraints versus priors. SIAM/ASA Journal on Uncertainty Quantification, 3(1), 586–598. doi:10.1137/130920721, Reprint: http://epubs.siam.org/doi/10.1137/130920721, Preprint: https://www.stat.berkeley.edu/~stark/Preprints/constraintsPriors15.pdf

Stark, P.B., 2016. Pay no attention to the model behind the curtain. https://www.stat.berkeley.edu/~stark/Preprints/eucCurtain15.pdf

Kuusela, M., and P.B. Stark, 2017. Shape-constrained uncertainty quantification in unfolding steeply falling elementary particle spectra, Annals of Applied Statistics, 11, 1671–1710. Preprint: http://arxiv.org/abs/1512.00905

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