For the number of rabbits in a closed system to increase, the system must contain at least two rabbits.
No negative rabbits.
You cannot pull a rabbit from a hat unless at least one rabbit has previously been placed in the hat.
You cannot "borrow" a rabbit from an empty hat, even with a binding promise to return the rabbit later.
Naturally, there is a strong desire to substitute intellectual capital for labor. That is why investigators often try to base causal inference on statistical models. With this approach, Pvalues play a crucial role. The technology is relatively easy to use and promises to open a wide variety of questions to the research effort. However, the appearance of methodological rigor can be deceptive. Like confidence intervals, Pvalues generally deal with the problem of sampling error not the problem of bias. Even with sampling error, artifactual results are likely if there is any kind of search over possible specifications for a model, or different definitions of exposure and disease. Models may be used in efforts to adjust for confounding and other sources of bias, but many somewhat arbitrary choices are made. Which variables to enter in the equation? What functional form to use? What assumptions to make about error terms? These choices are seldom dictated either by data or prior scientific knowledge. That is why judgment is so critical, the opportunity for error so large and the number of successful applications so limited.
 David A. Freedman, 1999
If we are uncritical we shall always find what we want: we shall look for, and find, confirmations, and we shall look away from, and not see, whatever might be dangerous to our pet theories. In this way it is only too easy to obtain what appears to be overwhelming evidence in favor of a theory which, if approached critically, would have been refuted.
 Karl Popper
Curve fitting, data summary
Clustering, classification
Exploratory data analysis, thought experiments
Estimation & inference in a known model
Prediction
Causal inference
Compact description of the data
Predict future data
Predict the effect of an interventionrequires "physics"
Parametric: functional form explicit
Nonparametric: no functional form or implicit functional form
ANNs, trees, forests, etc.: flexible classes of functions
Collection of models
Algorithms for fitting those models to various kinds/sizes of data
Folklore about how to select models/features
Domainspecific knowledge
Domainspecific hacks
Fallacies do not cease to be fallacies because they become fashions.
 G.K. Chesterton
Usually.
$$ \ln \frac{p}{1p} = X \beta.$$
Commonly used to model binary responses & for classification.
What justifies modeling the response as random?
You are one of a group of 100 people. You learn that one will die in the next year.
What's the chance it's you?
You are one of a group of 100 people. You learn that one is named "Philip."
What's the chance it's you?
underlying phenomenon is random (quantum physics)
deliberate randomization (randomized experiments, random sampling)
subjective probability
model that's supposed to describe the phenomenon
metaphor: phenomenon behaves "as if random"
Scientist creates randomness by taking a random sample, assigning subjects at random to treatment or control, etc.
Scientist invents (assumes) a probability model for data the world gives.
(1) allows sound inferences.
(2) is often "garbage in, garbage out."
Empirical support?
Plausible?
Iffy?
Absurd?
"Probability" often invoked without thinking
Reflexive way to try to represent uncertainty
Not all uncertainty can be represented by a probability
"Aleatory" versus "Epistemic"
Aleatory
Epistemic: stuff we don't know
Bayesian way to combine aleatory & 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's triumph over Aristotle should put this to rest, but empirically:
What varies?
What is constant (conditioned on)?
How do we know how the variable varies?
Are we talking about our world, or a set of possible worlds?
General approach (viz, don't even need data to make an estimate)
Guaranteed to have some good frequentist properties (if prior is proper and model dimension is finite)
Can be thought of as a kind of regularization
Elegant math from perspective of decision theory: convexify strategy space
Utterly different interpretation than frequentist uncertainties
Credible regions versus confidence regions
Observe $Y \sim N(\theta, 1)$.
Know a priori that $\theta \in [\tau, \tau]$
Bayes "uninformative" prior: $\theta \sim U[\tau, \tau]$
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 problematic
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.
—George Box
Commonly ignored sources of uncertainty:
Coding errors (ex: Hubble)
Stability of optimization algorithms (ex: GONG)
"upstream" data reduction steps
Quality of PRNGs
Expression  full  scientific notation 

$2^{32}$  4,294,967,296  4.29e9 
$2^{64}$  18,446,744,073,709,551,616  1.84e19 
$2^{128}$  3.40e38  
$2^{32 \times 624}$  9.27e6010  
$13!$  6,227,020,800  6.23e9 
$21!$  51,090,942,171,709,440,000  5.11e19 
$35!$  1.03e40  
$2084!$  3.73e6013  
${50 \choose 10}$  10,272,278,170  1.03e10 
${100 \choose 10}$  17,310,309,456,440  1.73e13 
${500 \choose 10}$  2.4581e20  
$\frac{2^{32}}{{50 \choose 10}}$  0.418  
$\frac{2^{64}}{{500 \choose 10}}$  0.075  
$\frac{2^{32}}{7000!}$  $<$ 1e54,958  
$\frac{2}{52!}$  2.48e68 
Have a collection of numbers, e.g., MME climate model predictions of warming
Take mean and standard deviation.
Report mean as the estimate; construct a confidence interval or "probability" statement from the results, generally using Gaussian critical values
IPCC does this
No random sample; no stochastic errors.
Even if there were a random sample, what justifies using normal theory?
Even if random and normal, misinterprets confidence as probability. Garbled; something like Fisher's fiducial inference
Ultimately, quantifauxcation.
Consider taking a sample of soup to tell whether it is too salty.
Tendency to treat haphazard as random
Probability, pvalue, confidence intervals, etc., apply only if the sample is random (or for some kinds of measurement errors)
Do not apply to samples of convenience, haphazard samples, etc.
Do not apply to populations.
Monte Carlo is a way to substitute computing for calculation.
Does not reveal anything that was not already an assumption in the calculation.
Distribution of the output results from the assumptions in the input.
The randomness is an assumption, not a conclusion; the distribution is an assumption, not a conclusion.
Avian / windturbine interactions
Earthquake probabilities
Climate models and climate change probabilities
Wind turbines kill birds, including raptors.
how many, and of what species?
how concerned should we be?
what design and siting features matter?
how do you build/site less lethal turbines?
Periodic ontheground surveys, subject to:
censoring
shrinkage/scavenging
background mortality
pieces of two birds, or two pieces of one bird?
how far from the point of injury does a bird land? attribution...
Is it possible to ...
make an unbiased estimate of mortality?
reliably relate the mortality to individual turbines in wind farms?
Common: Mixture of a point mass at zero and some distribution on the positive axis. E.g., "Zeroinflated Poisson"
Countless alternatives, e.g.:
observe $\max\{0, \mbox{Poisson}(\lambda_j)b_j\}$, $b_j > 0$
observe $b_j\times \mbox{Poisson}(\lambda_j)$, $b_j \in (0, 1)$.
observe true count in area $j$ with error $\epsilon_j$, where $\{\epsilon_j\}$ are dependent, not identically distributed, nonzero mean
Changes the question from bird deaths to the chance of heads for hypothetical coins.
Why random?
Why Poisson?
Why independent from site to site? From period to period? From bird to bird? From encounter to encounter?
Why doesn't chance of detection depend on size, coloration, groundcover, …?
Why do different observers miss carcasses at the same rate?
What about background mortality?
Probabilistic seismic hazard analysis (PSHA): basis for building codes in many countries & for siting nuclear power plants
Models locations & magnitudes of earthquakes as random; mags iid
Models ground motion as random, given event. Distribution depends on the location and magnitude of the event.
Claim to estimate "exceedance probabilities": chance acceleration exceeds some threshold in some number of years
In U.S.A., codes generally require design to withstand accelerations w probability ≥2% in 50y.
PSHA arose from probabilistic risk assessment (PRA) in aerospace and nuclear power.
Those are engineered systems whose inner workings are known but for some system parameters and inputs.
Inner workings of earthquakes are almost entirely unknown: PSHA is based on metaphors and heuristics, not physics.
Some assumptions are at best weakly supported by evidence; some are contradicted.

Model earthquake occurrence as a marked stochastic process with known parameters.
Model ground motion in a given place as a stochastic process, given the quake location and magnitude.
Then,
probability of a given level of ground movement in a given place is the integral (over space and magnitude) of the conditional probability of that level of movement given that there's an event of a particular magnitude in a particular place, times the probability that there's an event of a particular magnitude in that place
That earthquakes occur at random is an assumption not based in theory or observation.
involves taking rates as probabilities
Models amount to saying there's an "earthquake deck"
Turn over one card per period. If the card has a number, that's the size quake you get.
Journals full of arguments about how many "8"s in the deck, whether the deck is fully shuffled, whether cards are replaced and reshuffled after dealing, etc.
Why not say earthquakes are like terrorist bombings?
What advantage is there to the casino metaphor?
3 of the most destructive recent earthquakes were in regions seismic hazard maps showed to be relatively safe (2008 Wenchuan M7.9, 2010 Haiti M7.1, & 2011 Tohoku M9) Stein, Geller, & Liu, 2012
What good are the numbers?
https://www.ipcc.ch/pdf/supportingmaterial/uncertaintyguidancenote.pdf (at p.2)
… quantified measures of uncertainty in a finding expressed probabilistically (based on statistical analysis of observations or model results, or expert judgment).
… Depending on the nature of the evidence evaluated, teams have the option to quantify the uncertainty in the finding probabilistically. In most cases, author teams will present either a quantified measure of uncertainty or an assigned level of confidence.
… Because risk is a function of probability and consequence, information on the tails of the distribution of outcomes can be especially important. … Author teams are therefore encouraged to provide information on the tails of distributions of key variables …
As mentioned above
Result is gibberish.
Recall from above:
"Risky Business" project cochaired by Michael R. Bloomberg, Henry Paulson, Tom Steyer.
Funded by Bloomberg Philanthropies, the Paulson Institute, and TomKat Charitable Trust. Also Skoll Global Threats Fund, Rockefeller Family Fund, McKnight Foundation, Joyce Foundation.
"While our understanding of climate change has improved dramatically in recent years, predicting the severity & timing of future impacts remains a challenge. Uncertainty surrounding the level of GHG emissions going forward & the sensitivity of the climate system to those emissions makes it difficult to know exactly how much warming will occur, & when. Tipping points, beyond which abrupt & irreversible changes to the climate occur, could exist. Due to the complexity of the Earth’s climate system, we do not know exactly how changes in global average temperatures will manifest at a regional level. There is considerable uncertainty about how a given change in temperature, precipitation, or sea level will impact different sectors of the economy, & how these impacts will interact."
"In this American Climate Prospectus, we aim to provide decisionmakers in business & in government with the facts about the economic risks & opportunities climate change poses in the United States."
They estimate the effect of changes in temperature and humidity on mortality, a variety of crops, energy use, labor force, and crime, at the county level through 2099.
"evidencebased approach."
Even the notion that if you knew the temperature and humidity in every cubic centimeter of the atmosphere for the next hundred years, you could accurately predict the effect on violent crime is absurd.
Adding the uncertainty in temperature and humidity obviously doesn't make the problem easier.
Ditto for labor, crop yields, mortality, etc.
And that's even if ceteris were paribus—which they will not be.
And that's even for next year, much less for 80 years from now.
And that's even for predicting a global average, not countylevel predictions.