Unconditional Probability Research Articles

Invariant probabilistic sensitivity analysis for building energy models

One of the key challenges in the field of building-performance analysis is to identify the key source of input parameter/s that has/have largest effect on the energy efficiency of the buildings. Critical input parameters can be identified through the use of sensitivity analysis models. In this paper, we outline an invariant probabilistic sensitivity analysis technique to enable screening of the most important parameters in the building energy models. The proposed probabilistic sensitivity analysis method measures the average distance between unconditional probability distribution and conditional (on an input) probability distribution of an output. The distribution's description is based on the scale-invariant heat kernel signature (SI-HKS). In this paper, SI-HKS method is extended for invariant probabilistic sensitivity analysis. We demonstrate the application of the proposed method through the use of Energy Plus building-performance modelling tool.

Subjunctive Conditional Probability

There seem to be two ways of supposing a proposition: supposing “indicatively” that Shakespeare didn’t write Hamlet, it is likely that someone else did; supposing “subjunctively” that Shakespeare hadn’t written Hamlet, it is likely that nobody would have written the play. Let P(B//A) be the probability of B on the subjunctive supposition that A. Is P(B//A) equal to the probability of the corresponding counterfactual, A □→B? I review recent triviality arguments against this hypothesis and argue that they do not succeed. On the other hand, I argue that even if we can equate P(B//A) with P(A □→B), we still need an account of how subjunctive conditional probabilities are related to unconditional probabilities. The triviality arguments reveal that the connection is not as straightforward as one might have hoped.

The qualitative paradox of non-conglomerability

A probability function is non-conglomerable just in case there is some proposition E and partition \(\pi \) of the space of possible outcomes such that the probability of E conditional on any member of \(\pi \) is bounded by two values yet the unconditional probability of E is not bounded by those values. The paradox of non-conglomerability is the counterintuitive—and controversial—claim that a rational agent’s subjective probability function can be non-conglomerable. In this paper, I present a qualitative analogue of the paradox. I show that, under antecedently plausible assumptions, an analogue of the paradox arises for rational comparative confidence. As I show, the qualitative paradox raises its own distinctive set of philosophical issues.

The Effect of Industry Restructuring on Peer Firms

I study the bond price reaction of a merged firms peers, in order to better understand how the market responds to a restructuring. I argue that a merger announcement may signal the possibility of a merger wave to the industry, and in doing so, increase the conditional probability that peer firms might themselves be acquired in the future. Consistent with this hypothesis, I show that price reactions are stronger for firms that have a higher unconditional probability of being acquired ex-ante. Furthermore I show that following an initial merger, when the signal about a potentially upcoming wave should be the strongest, the peer bond returns are the highest. In addition, I document that, cross-sectionally, the abnormal returns I observe from peer bondholders are concentrated among firms that have the highest expected coinsurance benefit from a potential acquisition. I show that this is the case for bondholders between firms and within firms.

Probability of an Error in Estimation of States of a Modulated Synchronous Flow of Physical Events

A flow of physical events (photons, electrons, etc.) is considered. One of the mathematical models of such flows is a modulated synchronous doubly stochastic flow of events. Analytical results for conditional and unconditional probabilities of erroneous decision in optimal estimation of flow states upon the criterion of the a posteriori probability maximum are presented.

Probability of Error in Estimating States of a Flow of Physical Events

A flow of physical events (photons, electrons, etc.) is considered. One of the mathematical models of such flows is the MAP flow of events. Analytical results for conditional and unconditional probabilities of erroneous decision in optimal estimation of states of the MAP flow of events are presented.

Effects of early intervention on dietary intake and its mediating role on cognitive functioning: a randomised controlled trial.

To investigate the impact of an early intervention programme, Preparing for Life, on dietary intake between 12 and 36 months of age, and the mediating role played by diet on cognitive functioning. A randomised controlled trial evaluation of a community-based home visiting programme. The intervention involved biweekly visits from mentors from pregnancy until age 5 years and parent training at age 2 years. Dietary intake was assessed at 12, 18, 24 and 36 months using an FFQ to calculate the proportion meeting dietary recommendations. Cognitive functioning was measured at 24 and 36 months. Treatment effects were estimated using conventional χ 2 tests, permutation testing, inverse probability weighting and the stepdown procedure. Mediation analysis examined the indirect effect of the intervention on cognitive functioning via its effect on dietary intake. Socio-economically disadvantaged communities in Dublin, Republic of Ireland. Pregnant women (n 233) were assigned to the intervention (n 115) or control (n 118) group using an unconditional probability randomisation strategy. Positive treatment effects were observed for meeting dietary recommendations for protein foods at 24 (OR=2·52) and 36 (OR=2·42) months, and all food groups at 24 (OR=3·92) months. There were no effects on grain, dairy, fruit and vegetable, or fatty/sugary food recommendations in most models. The conventional and more novel methods yielded similar results. Mediation analysis indicated that 13 % of the intervention's effect on cognitive functioning was mediated by 36-month protein food consumption. The study demonstrates some potential to alter early childhood dietary patterns through community-based intervention programmes.

Contagion in Financial Systems: A Bayesian Network Approach

We develop a structural default model for interconnected financial institutions in a probabilistic framework. For all possible network structures we characterize the joint default distribution of the system using Bayesian network methodologies. Particular emphasis is given to the treatment and consequences of cyclic financial linkages. We further demonstrate how Bayesian network theory can be applied to detect contagion channels within the financial network, to measure the systemic importance of selected entities on others, and to compute conditional or unconditional probabilities of default for single or multiple institutions.

Bubble Investing: Learning from History

History is important to the study of financial bubbles precisely because they are extremely rare events, but history can be misleading. The rarity of bubbles in the historical record makes the sample size for inference small. Restricting attention to crashes that followed a large increase in market level makes negative historical outcomes salient. In this paper I examine the frequency of large, sudden increases in market value in a broad panel data of world equity markets extending from the beginning of the 20th century. I find the probability of a crash conditional on a boom is only slightly higher than the unconditional probability. The chances that a market gave back it gains following a doubling in value are about 10%. In simple terms, bubbles are booms that went bad. Not all booms are bad.

Effects of Bayesian Model Selection on Frequentist Performances: An Alternative Approach

It is quite common in statistical modeling to select a model and make inference as if the model had been known in advance; i.e. ignoring model selection uncertainty. The resulted estimator is called post-model selection estimator (PMSE) whose properties are hard to derive. Conditioning on data at hand (as it is usually the case), Bayesian model selection is free of this phenomenon. This paper is concerned with the properties of Bayesian estimator obtained after model selection when the frequentist (long run) performances of the resulted Bayesian estimator are of interest. The proposed method, using Bayesian decision theory, is based on the well known Bayesian model averaging (BMA)’s machinery; and outperforms PMSE and BMA. It is shown that if the unconditional model selection probability is equal to model prior, then the proposed approach reduces BMA. The method is illustrated using Bernoulli trials.

The $f$-Sensitivity Index

This paper presents a general multivariate $f$-sensitivity index, rooted in the $f$-divergence between the unconditional and conditional probability measures of a stochastic response, for global sensitivity analysis. Unlike the variance-based Sobol index, the $f$-sensitivity index is applicable to random input following dependent as well as independent probability distributions. Since the class of $f$-divergences supports a wide variety of divergence or distance measures, a plethora of $f$-sensitivity indices are possible, affording diverse choices to sensitivity analysis. Commonly used sensitivity indices or measures, such as mutual information, squared-loss mutual information, and Borgonovo's importance measure, are shown to be special cases of the proposed sensitivity index. New theoretical results, revealing fundamental properties of the $f$-sensitivity index and establishing important inequalities, are presented. Three new approximate methods, depending on how the probability densities of a stochastic response are determined, are proposed to estimate the sensitivity index. Four numerical examples, including a computationally intensive stochastic boundary-value problem, illustrate these methods and explain when one method is more relevant than the others.

Dynamics of morphological evolution in experimental Escherichia coli populations.

Here, we applied a two-stage clonal expansion model of morphological (cell-size) evolution to a long-term evolution experiment with Escherichia coli. Using this model, we derived the incidence function of the appearance of cell-size stability, the waiting time until this morphological stability, and the conditional and unconditional probabilities of morphological stability. After assessing the parameter values, we verified that the calculated waiting time was consistent with the experimental results, demonstrating the effectiveness of the two-stage model. According to the relative contributions of parameters to the incidence function and the waiting time, cell-size evolution is largely determined by the promotion rate, i.e., the clonal expansion rate of selectively advantageous organisms. This rate plays a prominent role in the evolution of cell size in experimental populations, whereas all other evolutionary forces were found to be less influential.

Finding Success in Tragedy: Forced Entrepreneurs after Corporate Bankruptcy

Using linked employer-employee data that covers most of U.S. firms and contains full resumes of workers we document how worker displacement of high-skilled labor leads to both self-employment and new employer firm creation. To causally identify the effect of forced unemployment on entrepreneurship, we compare workers in firms that experienced different bankruptcy outcomes as measured by the severity of layoffs around the filing date. The identification assumption is that two employees in the same occupation, with the same years of experience, education, and firm tenure, do not anticipate the bankruptcy outcome when joining the firm at least 4 years prior to the first bankruptcy filing. We find that forced unemployment increases entrepreneurship rates by seven-fold relative to the unconditional probability of entrepreneurship in normal times. Remarkably, 36% of these firms hire multiple employees. We further confirm that our results are causal by using the historical toughness of bankruptcy judges, who are randomly assigned to each case within a bankruptcy court. We also show that the effect of forced unemployment on self-employment and employer-firm creation is twice as large for workers with high-skilled occupations. Given the success of the firms in our sample in both survival and employment growth, our results highlight the potential productivity gains from reallocating skilled labor away from distressed firms and towards entrepreneurship.

The geographical spillover of armed conflict in Sub-Saharan Africa

Anecdotal accounts of the geographical spread of war inevitably involve Sub-Saharan African (SSA) countries. But is conflict spillover effectively stronger in SSA than elsewhere? To answer this question, we estimate models of civil war onset comparing SSA against the rest of the world (RoW). We find that in SSA a neighbour at war increases the probability of civil war onset by at least 1%. This is not negligible, considering that the unconditional probability of civil war onset is 1.1% in the global sample and 1.5% in the SSA sample. The spillover effect in the RoW is three times smaller than in SSA and, in general, statistically not different from zero. The results are robust to changes in the definition of neighbourhood and the inclusion of regional variables in the estimating equations to account for clustering effects. Finally, we provide evidence that refugee inflows and the artificial separation of ethnic groups explain part, but not all, of the spillover effect in SSA.

Forecasting seeing and parameters of long-exposure images by means of ARIMA

Atmospheric turbulence is the one of the major limiting factors for ground-based astronomical observations. In this paper, the problem of short-term forecasting seeing is discussed. The real data that were obtained by atmospheric optical turbulence (OT) measurements above Mount Shatdzhatmaz in 2007--2013 have been analysed. Linear auto-regressive integrated moving average (ARIMA) models are used for the forecasting. A new procedure for forecasting the image characteristics of direct astronomical observations (central image intensity, full width at half maximum, radius encircling 80% of the energy) has been proposed. Probability density functions of the forecast of these quantities are 1.5--2 times thinner than the respective unconditional probability density functions. Overall, this study found that the described technique could adequately describe temporal stochastic variations of the OT power.

Seismic Performance Evaluation Framework Considering Maximum and Residual Inter-Story Drift Ratios: Application to Non-Code Conforming Reinforced Concrete Buildings in Victoria, BC, Canada

This paper presents a seismic performance evaluation framework using two engineering demand parameters, i.e. maximum and residual inter-story drift ratios, and with consideration of mainshock-aftershock (MSAS) earthquake sequences. The evaluation is undertaken within a performance-based earthquake engineering framework in which seismic demand limits are defined with respect to the earthquake return period. A set of 2-, 4-, 8-, and 12-story non-ductile reinforced concrete buildings, located in Victoria, British Colombia, Canada, is considered as a case study. Using 50 mainshock and MSAS earthquake records (two horizontal components per record), incremental dynamic analysis is performed, and the joint probability distribution of maximum and residual inter-story drift ratios is modeled using a novel copula technique. The results are assessed both for collapse and non-collapse limit states. From the results, it can be shown that the collapse assessment of 4- to 12-story buildings is not sensitive to the consideration of MSAS seismic input, whereas for the 2-story building, a 13% difference in the median collapse capacity is caused by the MSAS. For unconditional probability of unsatisfactory seismic performance, which accounts for both collapse and non-collapse limit states, the life safety performance objective is achieved, but it fails to satisfy the collapse prevention performance objective. The results highlight the need for the consideration of seismic retrofitting for the non-ductile reinforced concrete structures.

Sample size planning for phase II trials based on success probabilities for phase III.

In recent years, high failure rates in phase III trials were observed. One of the main reasons is overoptimistic assumptions for the planning of phase III resulting from limited phase II information and/or unawareness of realistic success probabilities. We present an approach for planning a phase II trial in a time-to-event setting that considers the whole phase II/III clinical development programme. We derive stopping boundaries after phase II that minimise the number of events under side conditions for the conditional probabilities of correct go/no-go decision after phase II as well as the conditional success probabilities for phase III. In addition, we give general recommendations for the choice of phase II sample size. Our simulations show that unconditional probabilities of go/no-go decision as well as the unconditional success probabilities for phase III are influenced by the number of events observed in phase II. However, choosing more than 150 events in phase II seems not necessary as the impact on these probabilities then becomes quite small. We recommend considering aspects like the number of compounds in phase II and the resources available when determining the sample size. The lower the number of compounds and the lower the resources are for phase III, the higher the investment for phase II should be.

Part sojourn time distribution in a two-machine line

The time that a part may spend in a buffer between successive operations is limited in some manufacturing processes. Parts that wait too long must be reworked or discarded due to the risk of quality degradation. In this paper, we present an analytic formulation for the steady-state probability distribution of the time a part spends in a two-machine, one-buffer transfer line (the part sojourn time). To do so, we develop a set of recurrence equations for the conditional probability of a part’s sojourn time, given the number of parts already in the buffer when it arrives and the state of the downstream machine. Then we compute the unconditional probabilities of the part sojourn time using the total probability theorem. Numerical results are provided to demonstrate how the shape of the distribution depends on machine reliability and the buffer size. The analytic formulation is also applied to approximately compute the part sojourn time distribution in a given buffer of a long line. Comparison with simulation shows good agreement.

Reflecting Random Flights

We consider random flights in $\mathbb{R}^d$ reflecting on the surface of a sphere $\mathbb{S}^{d-1}_R,$ with center at the origin and with radius $R,$ where reflection is performed by means of circular inversion. Random flights studied in this paper are motions where the orientation of the deviations are uniformly distributed on the unit-radius sphere $\mathbb{S}^{d-1}_1$. We obtain the explicit probability distributions of the position of the moving particle when the number of changes of direction is fixed and equal to $n\geq 1$. We show that these distributions involve functions which are solutions of the Euler-Darboux-Poisson equation. The unconditional probability distributions of the reflecting random flights are obtained by suitably randomizing $n$ by means of a fractional-type Poisson process. Random flights reflecting on hyperplanes according to the optical reflection form are considered and the related distributional properties derived.

Forecasting Coherent Volatility Breakouts

The paper develops an algorithm for making long-term (up to three months ahead) predictions of volatility reversals based on long memory properties of financial time series. The approach for computing fractal dimension using sequence of the minimal covers with decreasing scale is used to decompose volatility into two dynamic components: specific and structural. We introduce two separate models for both, based on different principles and capable of catching long uptrends in volatility. To test statistical significance of its abilities we introduce several estimators of conditional and unconditional probabilities of reversals in observed and predicted dynamic components of volatility. Our results could be used for forecasting points of market transition to an unstable state.