Articles Cited by Public access. Title Sort Sort by citations Sort by year Sort by title. Articles 1—20 Show more. Help Privacy Terms. Why trees migrate so fast: confronting theory with dispersal biology and the paleorecord JS Clark The American Naturalist 2 , , Particle motion and the theory of charcoal analysis: source area, transport, deposition, and sampling JS Clark Quaternary research 30 1 , , BioScience 48 1 , , Climate Dynamics 30 7 , , Quantifying global soil carbon losses in response to warming Nature , , Fire and climate change during the last yr in northwestern Minnesota JS Clark Ecological Monographs 60 2 , , American Journal of Botany 86 1 , , Plant Migration and Climate Change: A more realistic portrait of plant migration is essential to predicting biological responses to global warming in a world drastically … LF Pitelka American Scientist 85 5 , , A hypothesis that hasn't been thought of automatically receives a subjective knowledge probability of zero, even though it may be the correct explanation, in reality.
For example, forest managers use an indicator of growth potential, the site index, in timber supply projections. For decades, they have considered standard methods for estimating the site index to be reliable. Previously, those estimates would have been assigned a probability of zero. Confidence probability refers to the degree of belief that a person has in a particular hypothesis. For example, she might decide to leave her jacket at home, based on her degree of confidence that the weather will be warm.
Similarly, a fisheries manager might decide to open a hatchery, based on his confidence that survival to adulthood is not density dependent.
Confidence probability is essentially subjective: internal to the observer, a characteristic that differentiates it strongly from chance probability. Confidence probability is especially important for this discussion. The central assumption of Bayesian analysis is that probability should be interpreted subjectively as confidence or degree of belief in a hypothesis Morgan and Henrion Nonetheless, Bayesian analysis uses the mathematics of chance probability probability theory to manipulate these subjective probabilities.
It is reasonable to question whether quantities derived from the subjective concept, confidence, can reasonably be processed by the same calculus as quantities based on chance probability. Indeed, experiments show that probability theory is poor at predicting people's confidence in their own judgments under uncertainty Gigerenzer et al. Control probability describes the degree to which the assessed probability of an event depends not only on its own stochastic characteristics, but also on what the assessor does to influence the event, regardless of the efficacy of his actions.
For example, many people feel safer i. In conservation ecology and resource management, this concept may cause problems when managers, who exert control over a resource, estimate probabilities on hypotheses concerning that resource.
Plausibility probability ought, perhaps, to have little relevance in scientific contexts, but even a scientific presentation is more likely to be believed i.
Lively discussion, description of personal experience, clever catch-phrases, and skillful marshaling of supporting information are all elements of story-telling that may play into the hypothesized algorithms for plausibility in scientific communication.
For example, the tactics of talented writers such as Richard Dawkins help to convince readers of the plausibility of their arguments. These variants of subjective probability present a challenge to the ecologist who elicits probability estimates from experts or wishes his audience to follow an analysis of single-event probabilities. The same pattern of information can influence subjective estimates of "probability" in opposite directions when the data are processed by different cognitive algorithms.
For example, it is possible to assign a high knowledge probability to a hypothesis because all other known alternatives have been ruled out, but still assign it a low confidence probability because there is little evidence in its favor. Specific cognitive pitfalls of concern to ecological Bayesian analysts In the next sections, I will examine three well-known cognitive illusions and their relevance to Bayesian analyses in ecology.
In each case, I will show how appropriate input format and activation of the intended algorithm chance can improve people's intuitive ability to process probabilities. Overestimation of single-event probabilities. Probability theory demands that an exhaustive set of mutually exclusive single-event probabilities must sum to 1. This apparently obvious distributional constraint, however, is often violated in intuitive reasoning about single-event probabilities. For example, when asked to estimate single-event probabilities of individual stochastic outcomes e.
In other experiments, subjects tend not to revise the probabilities that they assign to a set of hypotheses when the set is enlarged. For example, when subjects assign probabilities of guilt to a list of suspects for a fictional murder, suspects who could "easily" have committed the murder are assigned the same high probabilities, regardless of the number of other suspects that are introduced Teigen , Robinson and Hastie , Teigen In an ecological context, overestimation can be especially troublesome where probability estimates from different experts must be combined.
For example, a manager might ask a fire expert, an entomologist, and a meteorologist to estimate the probability that a stand of trees will be destroyed in the next 50 years by fire, insects, or windthrow, respectively. If each of these single estimates is overestimated, the total probability for the stand's destruction in the time period could be seriously biased upward.
Teigen attributes this cognitive bias to activation of the algorithm for tendency, under which assessed probability is an attribute of an individual outcome or hypothesis. As such, it is not constrained by the set of outcomes under consideration. Subjectively, probabilities in this context seem to be treated as though on an unbounded ordinal scale; the total can increase without limit as more outcomes are added to the set Robinson and Hastie In contrast, probability theory assumes that single-event probabilities occupy a bounded ratio scale 0 - 1.
The lack of intuition about distributional constraints and the apparent mismatch of scales should be a concern for ecologists using expert elicitation.
It affects both accuracy of individual estimates and coherence of the set of estimates i. Expert elicitation should involve discussion and ad hoc correction of coherence among the estimates Morgan and Henrion , Ferrell , but possible distortions produced by this practice have not been studied Ferrell Neither exposure to basic probability distributions nor training at specific aspects of probability estimation has generally improved either coherence or accuracy in expert elicitation Teigen b, Robinson and Hastie , Ferrell Single-event probabilities differ crucially from most other variables in ecology because of distributional constraints.
Ordinary variables can be considered attributes of the entity to which they apply, e. In contrast, although they superficially resemble other scientific data, probabilities from Bayesian analyses cannot be used in another context.
A single-event probability estimate adjusted to fit coherently into one set of hypotheses or range of parameter values is not an attribute of the hypothesis to which it is attached.
You cannot say that a hypothesis or outcome Y has single-event probability X, because the probability will change in any other context involving even a slightly different set of hypotheses or range of parameter values.
For example, suppose a Bayesian analyst considered two population models, one assuming two age classes and another with three age classes. He might assign them probabilities of 0. If he added a third model including four age classes to the analysis, the original probabilities would no longer be valid. The probabilities do not "belong" to the models. These critical constraints may be obscured if a chance probability is interpreted subjectively as tendency.
Correcting overestimation. Fortunately, experimental evidence suggests that overestimation of probability estimates is reduced by reporting or eliciting them as frequencies. The frequency format apparently causes people to process the information more like chance probabilities, resulting in a dramatic improvement in both accuracy and coherence Teigen a, Gigerenzer and Hoffrage Surprisingly, although many elaborate, ad hoc strategies for improving probability estimates exist Morgan and Henrion , Ferrell , Chaloner , they do not usually include the use of frequency format.
Frequency format in expert elicitation deserves evaluation as a simple means of combating biases. For example, compare the following questions that might be addressed to an expert. Probability format: "What is the probability that the rate of population increase r for this Spectacled Eider population is less than What is the probability that it is between What is the probability that it is greater than 0. How many would exhibit r between How many would exhibit r greater than 0.
Activating mental imagery seems to be more effective than abstract presentations in mobilizing an expert's experience Brunner et al. The frequency format increases people's intuitive ease with probabilistic information because it converts the mathematics into simple operations of set theory Gigerenzer and Hoffrage , Cosmides and Tooby In the previous example, the expert who provides "probabilities" for ranges of r must develop a mental probability density function and then integrate it over each range.
In contrast, when he estimates the "number of populations out of " exhibiting r within each range, he need only divide the imagined set of similar populations into three subsets and estimate the number of populations in each. Furthermore, reporting the probability of a hypothesis as a frequency may help analysts to avoid the temptation to apply estimates inappropriately to new contexts.
This is because we define a frequency only with respect to a specified reference group, e. Unlike a single-event probability misinterpreted as tendency, the frequency the size of the subset within the reference class applies to the subset, rather than to any individual case within the subset.
Finally, Bayesian analysts stress the advantage of being able to calculate probability density functions over continuous variables. However, the analyst must divide continuous variables into discrete ranges if he is to discuss the results in ordinary language, express them as hypotheses, or use them as "states of nature" in a decision analysis. For example, Taylor et al. Because it requires definition of a reference class and its subsets, the frequency format automatically encourages analysts to find meaningful divisions for continuous variables over which Bayesian probabilities have been calculated.
Conjunction fallacy. The idea of "conjunction" is central to probability theory and seems intuitively obvious. Two hypotheses are "in conjunction" or "conjoint" if they are true at the same time. Because probabilities are never greater than 1. In other words, it is harder for two things to be true at the same time than for one thing to be true. Despite the simplicity of the mathematics involved, the "conjunction fallacy" is perhaps the most familiar and pervasive of cognitive illusions.
It occurs when people estimate a higher probability for two hypotheses in conjunction than for either hypothesis alone. The "Linda problem" described previously is a good example. Teigen suggests that the conjunction fallacy results from the operation of the cognitive algorithm that processes subjective probabilities in terms of plausibility Table 3. A story generally becomes more believable and thus is assigned a higher subjective probability as more details are added, even though each detail would constitute a separate hypothesis to be treated in conjunction if the story were an exercise in chance probability.
In ecology, the conjunction fallacy is a concern whenever people must estimate or understand the probability of an event that depends on a series of preceding uncertain events Ferrell For example, the occurrence of a forest fire depends upon the forest reaching a threshold humidity level, a source of ignition, and a minimum wind speed within a specified area and time.
If an expert estimates a probability of 0. However, if the expert were to imagine all three factors together in circumstances that made the subjective plausibility algorithm active, he might judge a fire to be highly probable. It is necessary to structure elicitation interviews carefully to avoid this sort of confusion. Intuitive difficulties with conjunction are also a consideration when expert elicitation or a Bayesian analysis involves a hypothesis or model plus one or more uncertain parameters.
The probability of a model is the product of 1 the probability of the model's structure and 2 the probabilities of all of its assumptions. The product of many probabilities generally will be smaller than the product of fewer probabilities. Nonetheless, where several models are being compared, the conjunction fallacy may cause a model or hypothesis that specifies much detail assumed parameter values to be judged more probable than either a more general hypothesis within which it is nested, or a structurally different model with fewer assumptions.
For example, the Chief Forester of British Columbia compared a series of models that predict long-term harvest level projections Pedersen The models included progressively more silvicultural interventions and ecosystem management activities.
Pedersen presented a long list of factors assumed to influence harvest projections, as evidence of the validity of the most complex model. He concluded that its projections would be a good basis for management decisions. However, audience members might be drawn into the conjunction fallacy if they were reassured by the thought "That model must be good.
It looks like they have thought of everything. The model is true only if all of its assumptions are simultaneously true. Furthermore, because conjunction violations are difficult to detect intuitively, Bayesian analyses calculating probabilities of complex models can be hard for analysts to set up correctly and for readers to understand and evaluate. For example, Walters and Ludwig presented a Bayesian analysis of population parameters for a harvested fish population.
Conjoint relationships between the model and its assumed parameters are so complex that their analysis is difficult for even sophisticated readers to grasp.
Similarly, Sainsbury , used Bayesian analysis to estimate probabilities for four structurally different models of population interactions proposed to explain the dynamics of a multispecies fish community in Australia. He analyzed each model in conjunction with one set of parameter values Sainsbury et al.
He constrained the posterior probabilities for the four model-parameter set conjunctions to add to 1. If the other combinations had been admitted to the analysis, the posterior probabilities for the four chosen model-parameter set conjunctions would have been much smaller than those reported.
An unwary reader might place high confidence in the "model" with a reported posterior probability of 0. Correcting conjunction violations. As with the overestimation problem, the conjunction fallacy disappears when placed in a context of frequencies Gigerenzer , Gigerenzer and Hoffrage Most subjects, when asked "Out of people like Linda, how many are bank tellers?
Frequency formats apparently help subjects to avoid the plausibility interpretation and generate estimates consistent with chance probability calculus G.
Gigerenzer and R. Hertwig, unpublished manuscript. This strategy should be easy to apply in expert elicitation. A series of questions like the following is vulnerable to conjunction violations: "What is the probability that the population of mountain goats in Cathedral Park, British Columbia, is declining?
What is the probability that it is declining and disappearing from marginal habitats? How many are declining? Of those, how many are declining and disappearing from marginal habitat? It would probably prove helpful for authors to articulate their results in frequency terms, because the specification of the reference class automatically exposes the conjunctions between parameters and models.
For example, a comparison of Tables 2 and 4 shows how Pascual and Hilborn's posterior probabilities could be translated into frequency format.
The concrete presentation in Table 4 confers several advantages. First, the choice of a reference class size that people can visualize usually between 10 and sets a reasonable limit on the inclusion of infinitesimal probabilities. The choice of as the reference class size in Table 4 gives the frequency estimates two significant figures, a reasonable degree of resolution for most ecological analyses.
The original thirteen rows of data reported to five significant figures by Pascual and Hilborn boil down to just seven rows containing non-zero frequencies. Second, frequencies encourage critical examination of the conjoint results.
Consider the left column of Table 4: cases with a "low" slope for the recruitment function. In all cases but one in this group, the intercept beta is between 0. Does this degree of uniformity correspond with the reader's own intuition? It is much more difficult to perceive and evaluate this result in the original table.
Finally, the exercise of reinterpreting results in frequency format could indicate to authors whether or not their results will be successful "memes. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email. Posted on July 30, by Sara Stoudt Leave a comment.
Before we get going, if you want a refresher about what exactly Bayesian thought entails, check out this previous post.
Anderson et al. Share this: Twitter Facebook. Like this: Like Loading Leave a Reply Cancel reply Enter your comment here Fill in your details below or click an icon to log in:.
Email required Address never made public. Name required. Follow Following. Ecology for the Masses Join 1, other followers. Sign me up. Already have a WordPress.
0コメント