{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# P-values, Probability, Priors, Rabbits, Quantifauxcation, and Cargo-Cult Statistics\n", "\n", "## Philip B. Stark, www.stat.berkeley.edu/~stark, @philipbstark
\n", "The Rabbit Axioms
\n", "\n", "1. For the number of rabbits in a closed system to increase,\n", "the system must contain at least two rabbits.
\n", "\n", "2. No negative rabbits.\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ " \n", "\n", "
\n", "Freedman's Rabbit-Hat Theorem
\n", "\n", "You cannot pull a rabbit from a hat unless at least one\n", "rabbit has previously been placed in the hat.\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ " \n", "\n", "
\n", "Corollary
\n", "\n", "You cannot \"borrow\" a rabbit from an empty\n", "hat, even with a binding promise to return the rabbit later.\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Applications of the Rabbit-Hat Theorem\n", "\n", "* Probablility doesn't come out of a calculation unless probability went into the calculation.\n", " - Can't turn a rate into a probability without assuming the phenomenon is random in the first place.\n", "\n", "* Can't conclude that a process is random without making assumptions\n", "that amount to assuming that the process is random.\n", "(Something has to put the randomness rabbit into the hat.)\n", "\n", "* Testing whether the process appears to be random using the\n", "_assumption_ that it is random cannot prove that it is\n", "random. (You can't borrow a rabbit from an empty hat.)\n", "\n", "* Posterior distributions don't exist without prior distributions. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## What's a P-value?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "+ A probability" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "+ But of what?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## $P$-values\n", "\n", "+ Observe data $X \\sim \\mathbb{P}$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "+ Null hypothesis $\\mathbb{P} = \\mathbb{P}_0$ (or more generally, $\\mathbb{P} \\in \\mathcal{P}_0$)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "+ Nested (monotone) hypothesis tests:\n", "\n", " - $\\{A_\\alpha : \\alpha \\in (0, 1] \\}$\n", "\n", " - $\\mathbb{P}_0 \\{ X \\notin A_\\alpha \\} \\le \\alpha$ (or more generally, $\\mathbb{P} \\{ X \\notin A_\\alpha \\} \\le \\alpha, \\; \\forall \\mathbb{P} \\in \\mathcal{P}_0$)\n", "\n", " - $A_\\alpha \\subset A_\\beta$ if $\\beta < \\alpha$ (Can always re-define $A_\\alpha \\leftarrow \\cup_{\\beta \\ge \\alpha } A_\\beta$)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "+ If we observe $X = x$, $P$-value is $\\sup \\{ \\alpha: x \\in A_\\alpha \\}$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## C.f. informal definition in terms of \"extreme\" values?\n", "\n", "+ What does \"more extreme\" mean?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## It's all about the null hypothesis\n", "\n", "+ P-values measure the strength of the evidence against the null: smaller values, stronger evidence.\n", "\n", "+ If $P$-value $p$, either:\n", "\n", " 1. the null hypothesis is false\n", " 1. an event occurred that had probability no greater than $p$\n", "\n", "+ Alternative hypothesis matters for power, but not for level.\n", "\n", "+ Rejecting the null is not evidence _for_ the alternative: it's evidence _against_\n", "the null.\n", "\n", "+ If null is unreasonable, no surprise if we reject it. Null needs to make sense.\n", "\n", "+ Unreasonable null is not support for the alternative." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## When did the rabbit enter the hat?\n", "\n", "Anytime you see a $P$-value, you should ask what the null hypothesis is.\n", "\n", "E.g., $\\mu = 0$ is not the whole null hypothesis: \n", "\n", "+ null has to completely specify (a family of possible) probability distributions of the data\n", "\n", "+ otherwise, can't set acceptance regions $\\{A_\\alpha\\}$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Anytime you see a posterior probability, you should ask what the prior was.\n", "\n", "+ You can't have a posterior distribution without a prior distribution.\n", "\n", "+ The rabbit has to get in somehow.\n", "\n", "+ It usually matters, despite claims about asymptopic results." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Complaints about $P$-values\n", "\n", "+ \"Don't tell us what we want to know.\"\n", "\n", "+ Often abused.\n", "\n", "+ Confusing" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Where does probability come from?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "+ Rates are not probabilities" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "+ Not all uncertainty is probability. Haphazard/random/unknown" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "+ A coefficient in a model may not be a \"real\" probability, even if it's called \"probability\"" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### What is Probability?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "#### Axiomatic aspect and philosophical aspect.\n", "\n", "* Kolmogorov's axioms:\n", " - \"just math\"\n", " - triple \\\$$(S, \\Omega, P)\\\$$\n", " + \\\$$S\\\$$ a set\n", " + \\\$$\\Omega\\\$$ a sigma-algebra on \\\$$S\\\$$\n", " + \\\$$P\\\$$ a non-negative countably additive measure with total mass 1" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* Philosophical theory that ties the math to the world\n", " - What does probability _mean_?\n", " - Standard theories\n", " + Equally likely outcomes\n", " + Frequency theory\n", " + Subjective theory\n", " - Probability models as empirical commitments\n", " - Probability as metaphor" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### How does probability enter a scientific problem?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* underlying phenomenon is random (radioactive decay)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* deliberate randomization (randomized experiments, random sampling)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* subjective probability\n", " - Constraints versus priors\n", " - No posterior distributions without prior distributions\n", " - Prior generally matters\n", " - elicitation issues\n", " - arguments from consistency, \"Dutch book,\" ...\n", " - why should I care about your subjective probability" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* invented model that's supposed to describe the phenomenon\n", " - in what sense?\n", " - to what level of accuracy?\n", " - description v. prediction v. predicting effect of intervention\n", " - testable to desired level of accuracy?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* metaphor: phenomenon behaves \"as if random\"" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Two very different situations:\n", "\n", "1. Scientist creates randomness by taking a random sample,\n", "assigning subjects at random to treatment or control, etc.\n", "\n", "1. Scientist invents (assumes) a probability model for data the world gives." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "(1) allows sound inferences.\n", "\n", "(2) is only as good as the assumptions." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "#### Gotta check the assumptions against the world\n", "\n", "+ Empirical support? \n", "+ Plausible? \n", "+ Iffy? \n", "+ Absurd?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "
\n", "Quantifauxcation
\n", "\n", "Assign a meaningless number, then pretend that since it's quantitative, it's meaningful.\n", "
\n", "\n", "\n", "Many P-values and other \"probabilities\" and most cost-benefit analyses are quantifauxcation.\n", "\n", "### Cargo-cult statistics\n", "\n", "Usually involves some combination of\n", "data, pure invention, _ad hoc_ models, inappropriate statistics, and logical lacunae." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Example: The 2-sample problem\n", "\n", "+ Randomization model: two lists. Are they \"different\"?\n", "\n", "+ $t$-test. Assumptions?\n", "\n", "+ Permutation distribution" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "## Example: Effect of treatment in a randomized controlled experiment\n", "\n", "11 pairs of rats, each pair from the same litter. \n", "\n", "Randomly—by coin tosses—put one of each pair into\n", "\"enriched\" environment; other sib gets \"normal\" environment.\n", "\n", "After 65 days, measure cortical mass (mg).\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "
 enriched 689 656 668 660 679 663 664 647 694 633 653 impoverished 657 623 652 654 658 646 600 640 605 635 642 difference 32 33 16 6 21 17 64 7 89 -2 11\n", "
\n", "\n", "**How should we analyze the data?**\n", "\n", "Cartoon of Rosenzweig, M.R., E.L. Bennet, and M.C. Diamond, 1972. Brain changes in response to experience, _Scientific American_, _226_, 22–29 report an experiment in which 11 triples of male rats, each triple from the same litter, were\n", "assigned at random to three different environments, \"enriched\" (E), standard, and \"impoverished.\"\n", "See also Bennett et al., 1969. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Informal Hypotheses\n", "\n", "Null hypothesis: treatment has \"no effect.\"\n", "\n", "Alternative hypothesis: treatment increases cortical mass.\n", "\n", "Suggests 1-sided test for an increase." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Test contenders\n", "\n", "+ 2-sample Student $t$-test: \n", "\n", "$$\n", " \\frac{\\mbox{mean(treatment) - mean(control)}}\n", " {\\mbox{pooled estimate of SD of difference of means}}\n", "$$\n", "\n", "+ 1-sample Student $t$-test on the differences: \n", "\n", "$$\n", "\t \\frac{\\mbox{mean(differences)}}{\\mbox{SD(differences)}/\\sqrt{11}}\n", "$$ \n", "Better, since littermates are presumably more homogeneous.\n", " \n", "+ Permutation test using $t$-statistic of differences:\n", "same statistic, different way to calculate $P$-value." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Assumptions of the tests\n", "\n", "1. 2-sample $t$-test: \n", " + masses are iid sample from normal distribution, same unknown variance, same unknown mean.\n", "\t+ Tests weak null hypothesis (plus normality, independence, non-interference, etc.).\n", "1. 1-sample $t$-test on the differences: \n", "\t+ mass differences are iid sample from normal distribution, unknown variance, zero mean.\n", "\t+ Tests weak null hypothesis (plus normality, independence, non-interference, etc.)\n", "1. Permutation test: \n", "\t+ Randomization fair, independent across pairs.\n", "\t+ Tests strong null hypothesis.\n", "\n", "Assumptions of the permutation test are true by design: That's how treatment\n", "was assigned." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Making sense of probabilities in applied problems\n", "\n", "* Probability often applied without thinking\n", "\n", "* Reflexive way to try to represent uncertainty\n", "\n", "* Not all uncertainty can be represented by a probability\n", "\n", "* \"Aleatory\" versus \"Epistemic\"" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- Aleatory\n", " + Canonical examples: coin toss, die roll, lotto, roulette\n", " + under some circumstances, behave \"as if\" random (but not perfectly)\n", "\n", "- Epistemic: stuff we don't know" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "- Standard way to combining aleatory variability epistemic uncertainty puts\n", "beliefs on a par with an unbiased physical measurement w/ known uncertainty.\n", " + Claims by introspection, can estimate without bias, with known accuracy,\n", "just as if one's brain were unbiased instrument with known accuracy\n", " + Bacon's triumph over Aristotle should put this to rest, but empirically:\n", " - people are bad at making even rough quantitative estimates\n", " - quantitative estimates are usually biased\n", " - bias can be manipulated by anchoring, priming, etc.\n", " - people are bad at judging weights _in their hands_: biased by shape & density\n", " - people are bad at judging when something is random\n", " - people are overconfident in their estimates and predictions\n", " - confidence unconnected to actual accuracy.\n", " - anchoring effects entire disciplines (e.g., Millikan, c, Fe in spinach)\n", "\n", "- what if I don't trust your internal scale, or your assessment of its accuracy?\n", "- same observations that are factored in as \"data\" are also used to form\n", "beliefs: the \"measurements\" made by introspection are not independent of the data" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Rates versus probabilities\n", "\n", "* In a series of trials, if each trial has the same probability p of success,\n", "and if the trials are independent, then the rate of successes converges (in probability)\n", "to p. Law of Large Numbers\n", "\n", "* If a finite series of trials has an empirical rate p of success, that says\n", "nothing about whether the trials are random.\n", "\n", "* If the trials are random _and_ have the same chance of success, the empirical rate\n", "is an estimate of the chance of success.\n", "\n", "* If the trials are random _and_ have the same chance of success _and_ the dependence\n", "of the trials is known (e.g., the trials are independent), can quantify the uncertainty\n", "of the estimate." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Thought experiments\n", "\n", " \n", "\n", "
\n", "You are one of a group of 100 people, of whom one will die in the next year.
\n", "\n", "What's the chance it is you?\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ " \n", "\n", "
You are one of a group of 100 people,\n", "of whom one is named \"Philip.\"
\n", "\n", "What's the chance it is you?\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Why does the first invite an answer, and the second not?\n", "\n", "Ignorance ≠ Randomness" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Cargo Cult Confidence Intervals\n", "\n", "\n", "+ Have a collection of numbers, e.g., MME climate model predictions of\n", "warming\n", "\n", "+ Take mean and standard deviation.\n", "\n", "+ Report mean as the estimate; construct a confidence interval or \"probability\" statement from the results, generally using Gaussian critical values\n", "\n", "+ IPCC does this, as do many others." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "#### What's wrong with it?\n", "\n", "+ No random sample; no stochastic errors.\n", "\n", "+ Even if there were a random sample, what justifies using normal theory?\n", "\n", "+ Even if random and normal, misinterprets confidence as probability. Garbled;\n", "something like Fisher's fiducial inference\n", "\n", "+ Ignores known errors in physical approximations\n", "\n", "+ Ultimately, quantifauxcation." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Random/haphazard/unpredictable/unknown\n", "\n", "* Consider taking a sample of soup to tell whether it is too salty.\n", " - Stirring the soup, then taking a tablespoon, gives a random sample\n", " - Sticking in a tablespoon without looking gives a haphazard sample\n", "\n", "* Tendency to treat haphazard as random\n", " - random requires deliberate, precise action\n", " - haphazard is sloppy\n", "\n", "\n", "* Notions like probability, p-value, confidence intervals, etc.,\n", "_apply only if the sample is random_ (or for some kinds of measurement errors)\n", "\n", " - Do not apply to samples of convenience, haphazard samples, etc.\n", "\n", " - Do not apply to populations." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Two brief examples\n", "\n", "* Avian / wind-turbine interactions\n", "\n", "* Earthquake probabilities" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Wind power: \"avian / wind-turbine interactions\"\n", "\n", "Wind turbines kill birds, notably raptors.\n", "\n", "+ how many, and of what species?\n", "\n", "+ how concerned should we be?\n", "\n", "+ what design and siting features matter?\n", "\n", "+ how do you build/site less lethal turbines?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Measurements\n", "\n", "Periodic on-the-ground surveys, subject to:\n", "\n", "+ censoring\n", "\n", "+ shrinkage/scavenging\n", "\n", "+ background mortality\n", "\n", "+ is this pieces of two birds, or two pieces of one bird?\n", "\n", "+ how far from the point of injury does a bird land? attribution..." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Is it possible to ...\n", "\n", "+ make an unbiased estimate of mortality?\n", "\n", "+ reliably relate the mortality to individual turbines in wind farms?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Stochastic model\n", "\n", "Common: Mixture of a point mass at zero and some distribution on the positive axis.\n", "E.g., \"Zero-inflated Poisson\"\n", "\n", "Countless alternatives, e.g.:\n", "\n", "+ observe $\\max\\{0, \\mbox{Poisson}(\\lambda_j)-b_j\\}$, $b_j > 0$\n", "\n", "+ observe $b_j\\times \\mbox{Poisson}(\\lambda_j)$, $b_j \\in (0, 1)$.\n", "\n", "+ observe true count in area $j$ with error $\\epsilon_j$,\n", "where $\\{\\epsilon_j\\}$ are dependent, not identically distributed,\n", "nonzero mean" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Consultant\n", "\n", "* bird collisions random, Poisson distributed\n", "* same for all birds\n", "* independent across birds\n", "* rates follow hierarchical Bayesian model that depends on\n", "covariates: properties of site and turbine design" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "#### What does this mean?\n", "\n", "* when a bird approaches a turbine, it tosses a coin to decide\n", "whether to throw itself on the blades\n", "* chance coin lands heads depends on site and turbine design\n", "* all birds use the same coin for each site/design\n", "* birds toss their coins independently" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Where do the models come from?\n", "\n", "+ Why random?\n", "\n", "+ Why Poisson?\n", "\n", "+ Why independent from site to site? From period to period? From bird to bird? From encounter to encounter?\n", "\n", "+ Why doesn't chance of detection depend on size, coloration, groundcover, …?\n", "\n", "+ Why do different observers miss carcasses at the same rate?\n", "\n", "+ What about background mortality?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Complications at Altamont\n", "\n", "+ Why is randomness a good model? Random is not the same as haphazard or unpredictable.\n", "+ Why is Poisson in particular reasonable?\n", "Do birds in effect toss coins, independently, with\n", "same chance of heads, every encounter with a turbine?\n", "Is #encounters $\\times P(\\mbox{heads})$ constant?\n", "+ Why estimate the parameter of a contrived model rather than actual mortality?\n", "+ Do we want to know how many birds die, or the value of $\\lambda$ in an implausible stochastic model?\n", "+ Background mortality—varies by time, species, etc.\n", "+ Are all birds equally likely to be missed? Smaller more likely than larger? Does coloration matter?\n", "+ Nonstationarity (seasonal effects—migration, nesting, etc.; weather; variations in\n", "bird populations)\n", "+ Spatial and seasonal variation in shrinkage due to groundcover, coloration, illumination, etc.\n", "+ Interactions and dependence.\n", "+ Variations in scavenging. (Dependence on kill rates? Satiation? Food preferences? Groundcover?)\n", "+ Birds killed earlier in the monitoring interval have longer time on trial for scavengers.\n", "+ Differences or absolute numbers? (Often easier to estimate differences accurately.)\n", "+ Same-site comparisons across time, or comparisons across sites?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Earthquake probabilities\n", "\n", "* Probabilistic seismic hazard analysis (PSHA): basis for\n", "building codes in many countries & for siting nuclear power plants\n", "\n", "* Models locations & magnitudes of earthquakes as random; mags iid\n", "\n", "* Models ground motion as random, given event. Distribution\n", "depends on the location and magnitude of the event.\n", "\n", "* Claim to estimate \"exceedance probabilities\": chance acceleration exceeds some\n", "threshold in some number of years\n", "\n", "* In U.S.A., codes generally require design to withstand accelerations w probability ≥2% in 50y.\n", "\n", "* PSHA arose from probabilistic risk assessment (PRA) in aerospace and nuclear power.\n", "
Those are engineered systems whose inner workings are known but for some system parameters and inputs.\n", "\n", "* Inner workings of earthquakes are almost entirely unknown: PSHA is based on metaphors and heuristics, not physics.\n", "\n", "* Some assumptions are at best weakly supported by evidence; some are contradicted." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### The PSHA equation\n", "\n", "Model earthquake occurrence as a marked stochastic process with known parameters.\n", "\n", "Model ground motion in a given place as a stochastic process, given the quake location and magnitude.\n", "\n", "Then,\n", "\n", "> probability of a given level of ground movement in a given place is the integral (over space and magnitude)\n", "of the conditional probability\n", "of that level of movement given that there's an event of a particular magnitude in a particular place,\n", "times the probability that there's an event of a particular magnitude in that place\n", "\n", "* That earthquakes occur at random is an _assumption_ not based in theory or observation.\n", "\n", "* involves taking rates as probabilities\n", " - Standard argument:\n", " - M = 8 events happen about once a century.\n", " - Therefore, the chance is about 1% per year." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Earthquake casinos\n", "\n", "* Models amount to saying there's an \"earthquake deck\"\n", "\n", "* Turn over one card per period. If the card has a number, that's the\n", "size quake you get.\n", "\n", "* Journals and journals full of arguments about how many \"8\"s in the deck,\n", "whether the deck is fully shuffled, whether cards are replaced and re-shuffled\n", "after dealing, etc." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "
\n", "But this is just a metaphor!\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Earthquake terrorism\n", "\n", "* Why not say earthquakes are like terrorist bombings?\n", " - don't know where or when\n", " - know they will be large enough to kill\n", " - know some places are \"likely targets\"\n", " - but no probabilities\n", "\n", "* What advantage is there to the casino metaphor?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Rabbits and Earthquake Casinos\n", "\n", "#### What would make the casino metaphor apt?\n", "\n", "1. The physics of earthquakes might be stochastic. But it isn't.\n", "\n", "2. A stochastic model might provide a compact, accurate description\n", "of earthquake phenomenology. But it doesn't.\n", "\n", "3. A stochastic model might be useful for predicting future seismicity.\n", "But it isn't (Poisson, Gamma renewal, ETAS)\n", "\n", "3 of the most destructive recent earthquakes were in regions seismic hazard maps showed to be relatively safe\n", "(2008 Wenchuan M7.9, 2010 Haiti M7.1, & 2011 Tohoku M9)\n", "[Stein, Geller, & Liu, 2012](http://web.missouri.edu/~lium/pdfs/Papers/seth2012-tecto-hazardmap.pdf)\n", "\n", "See also [Mulargia, Geller, & Stark, 2017](http://www.sciencedirect.com/science/article/pii/S0031920116303016)\n", "\n", "#### What good are the numbers?" ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.1" } }, "nbformat": 4, "nbformat_minor": 2 }