What Values Can Be Probabilities

Probability Value

Measures, Performance Assessment and Enhancement

In Time Frequency Analysis, 2003

Remark:

In the probability theory all results are derived for the probability values p_i , assuming that Σ _i p _i = 1 and p_i ≥ 0. The aforementioned assumptions are made in classical indicate assay for the point power. Since a full general TFD commonly does not satisfy both ${\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\rho }_{\mathrm{10}}\left(t,f\right)=1}}$ , the obtained measures of TFD concentration may just formally look similar the original entropies or classical signal analysis forms, while they can take different behavior and backdrop. ³

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Designing of Latent Dirichlet Allocation Based Prediction Model to Find Midlife Crisis of Losing Jobs due to Prolonged Lockdown for COVID-19

Basabdatta Das , ... Abhijit Das , in Cyber-Physical Systems, 2022

thirteen.3.3.1 Formulation of Dirichlet distribution

In order to railroad train our model, nosotros demand to choose our posterior probability and prior probability values. As the dataset is the bag of words, which form a multinomial distribution, nosotros need a conjugate prior to this distribution. Therefore we formulate the Dirichlet distribution, which is conjugate to the multinomial distribution. To speak specifically, Dirichlet distribution is a distribution of beta distribution, and also tin exist derived from gamma distribution. The parameters α and β (Blei, Ng and, Hashemite kingdom of jordan) are tuned so that we can filter our dataset to have more precise results. In club to do this, nosotros follow the equation Eq. (13.1).

(thirteen.i) $\mathrm{Dir}\left(p\mathrm{;}\alpha \right)=\frac{\mathrm{i}}{B\left(\alpha \right)}\prod _{\mathrm{t}=1}^{\mathrm{|}\alpha \mathrm{|}}{p}_t^{{\alpha }_t-1}$

where the normalizing abiding is the multinomial beta function, which can be expressed in terms of gamma function (Blei, Ng and, Hashemite kingdom of jordan) in Eq. (13.2)

(xiii.2) $B\left(\mathrm{\alpha }\right)=\frac{{\prod }_{\mathrm{i}=\mathrm{i}}^{\mathrm{|}\alpha \mathrm{|}}\mathrm{\Gamma }\left({\alpha }_{\mathrm{i}}\right)}{\mathrm{\Gamma }(\sum _{\mathrm{i}=\mathrm{one}}^{\mathrm{|}\alpha \mathrm{|}}{\alpha }_{\mathrm{i}})}$

In application, we need to find a elementary generative model, which may determine that the word w_i (i∈ 1,ii, …., n) from the searched tweet is a depressive tagged give-and-take and reflects task loss. Our model must assign nonzero probability to w_i. As well, it should also satisfy exchangeability. Each word is assigned a unique integer xϵ [0, ∞), and C_ten is the word count in the Dirichlet procedure.

The probability that the word w_i+1 is depressive is

$\frac{C_x}{\left(C+\alpha \right)}$

and the probability that the discussion west_i+1 is an optimistic i is

$\frac{\alpha }{C+\alpha }$

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Elementary Probability and Statistics

Prasanta S. Bandyopadhyay , Steve Red , in Philosophy of Statistics, 2011

4.1 Understanding information in terms of objects, variables, and scales

In deductive logic, we attribute truth-values to propositions. In probability theory, nosotros attribute probability values both to events and propositions. In statistics, information which stand up for our observations about the globe lie at its core. In order for data to exist converted into some linguistic communication which is free from any ambiguity, so that what the data could replenish us with reliable data nigh the world, nosotros take recourse to language of mathematical statistics.

The discussion of data in most introductory statistics textbooks typically starts with the definition of a population as a collection of objects of involvement to an investigator. The investigator wishes to learn something about selected properties of the population. Such properties are determined by the characteristics of the individuals who make upwards the population and these characteristics are referred to as variables because their values vary over the individuals in the population. These characteristics tin be measured on selected members of the population. If an investigator has access to all members of a population so he has conducted a census. A census is rarely possible and an investigator will select instead a subset of the population called a sample. Obviously, the sample must be representative of the population if it is to be used to draw inferences to the population from which information technology was fatigued.

An important concept in statistics is the idea of a information distribution which is a list of the values and the number of times (frequency) or proportion of the fourth dimension (relative frequency) those values occur.

Variables can be classified into four bones types — nominal, ordinal, interval, and ratio. Nominal and ordinal variables are described as qualitative while interval and ratio scale variables are quantitative.

Nominal variables differ in kind but. For case, political political party identification is a nominal variable whose "values" are labels; e.g. Democrat, Republican, Green Party. These values do not differ in any quantitative sense. This remains truthful even if we represent Democrats by 1, Republicans by 2 and so on. The numbers remain simply labels identifying grouping membership without implying that 1 is superior to two. Because this scaling is not liable to quantification does not hateful that information technology has no value. In fact, it helps usa to summarize a large corporeality of information into a relatively small set up of not-overlapping groups of individuals who share a common characteristic.

Sometimes the values of a qualitative variable can be placed in a rank order. The latter might stand for the quality of toys received in different overseas cargos. Each toy in a batch receives a quality rating (Depression, Medium, and High). They could also exist given numerical codes (e.1000. ane for high quality, two for medium quality, and 3 for low quality). This ordinal ranking implies a bureaucracy of quality in a batch of toys received from overseas. This ranking must satisfy the law of transitivity implying that if 1 is better than 2 and 2 is better than three then 1 must exist ameliorate than 3. Since both nominal and ordinal scales are designated equally qualitative variables, they are regarded as not-metric scales.

Interval scale variables are quantitative variables with an arbitrarily divers zippo value. Put some other style, a value of 0 does non mean the absence of whatever is being measured. Temperature measured in degrees Celsius is an interval scale variable. This is a metric scale in which for example the difference between 2 and 5 is the aforementioned as the difference between 48 and 51.

In dissimilarity to interval scale data, in "ratio" calibration data, aught is actually a pointer of "zero" scored on the scale but every bit nosotros see zero on a speedometer which signifies no movement of a car. Temperature measured in degrees Kelvin is a ratio scale variable because a value of 0 implies the absence of all motion at the diminutive level.

Mathematical operations make sense with quantitative information whereas this is non true in general of qualitative data. This should non be taken to mean that qualitative information cannot exist analyzed using quantitative methods, however. For example, gender is a qualitative variable and information technology makes no sense to talk near the "average" gender in a population just it makes a lot of sense to talk about the proportions of men and women in a population of interest.

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26th European Symposium on Computer Aided Process Technology

Brigitta Nagy , ... Dimitrios I. Gerogiorgis , in Estimator Aided Chemic Applied science, 2016

iii.2 Statistical hypothesis testing

The null hypothesis H ₀ (similarity of trends between drug substances and formulations) is accepted when the p-value (probability of observed or more extreme results under H ₀) is smaller than the significance level (5% in this written report), and rejected when it is larger. MWW exam results computed are mostly consistent for 2008-2013 and shown in Fig. two. Definite H ₀ rejection is observed for import and export prices as well every bit export value, indicating a potent incentive to explore the economical bear on of CPM implementation.

Effigy ii. Mann-Whitney-Wilcoxon (MWW) exam results for all nine parameters and all years.

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Probabilistic Temporal Reasoning

Steve Hanks , David Madigan , in Foundations of Artificial Intelligence, 2005

10.5.iii Incremental model construction

The techniques discussed higher up were based on the implicit supposition that a (graphical) model was constructed in total prior to solution. Furthermore, the algorithms computed a probability value for every node in the graph, thus providing data about the state of every arrangement variable at every bespeak in fourth dimension. For many applications this information is non necessary: all that is needed is the value of a few query variables that are relevant to some prediction or controlling situation. Work on incremental model construction starts with a compositional representation of the organization in the form of rules, model fragments, or other noesis base, and computes the value of a query expression trying to instantiate simply those parts of the network necessary to compute the query probability accurately. In [Ngo et al., 1995], the underlying system representation takes the form of sentences in a temporal probabilistic logic, and constructs a Bayesian network for a particular query. The resulting network, which should include merely those parts of the network relevant to the query, can exist solved by standard methods or whatsoever of the special-purpose algorithms discussed above.

In [Hanks and McDermott, 1994] the underlying arrangement representation consists of STRLPS-like rules with a probabilistic component (Section 10.iii.2). The system takes equally input a query formula along with a probability threshold. The algorithm does not compute the exact probability of the query formula; rather it answers whether or not that probability is less than, greater than, or equal to, the threshold. The justification for this approach is that in controlling or planning situations, the exact value of the query variables is usually unimportant—all that matters is what side of the threshold the probability lies. For example, a decision rule for planning an outing might be to schedule the trip merely if the probability of rain is beneath xx%.

The algorithm in [Hanks and McDermott, 1994] works as follows: suppose the query formula is a unmarried state variable P@t, and the input threshold is τ. The algorithm computes an estimate of [electronic mail protected] based on its current set of evidence. (Initially the prove fix is empty, and gauge is the prior for P@t). The estimate is compared to the threshold, and the algorithm computes an answer to the question "what evidence would cause the current judge of P@t to change with respect to τ?"

Prove and rules tin be irrelevant for a number of reasons. Kickoff, they can exist of the wrong sort (positive prove nearly P and rules that make P true are both irrelevant if the electric current approximate is already greater than τ). A dominion or slice of bear witness can also be as well tenuous to exist interesting, either because it is temporally too remote from the query time signal, or because its "noise" factor is as well large. In either case, the evidence or rule can be ignored if its issue on the current gauge is weak enough that fifty-fifty if it were considered, information technology would non change the current estimate from greater than τ to less than τ, or vice versa.

Once the relevant evidence has been characterized, a search through the temporal database is initiated. If the search yields no evidence, and the current qualitative estimate is returned. If new evidence is institute, the guess is updated and the process is repeated.

There is an attribute of dynamic model construction in [Nicholson and Brady, 1994] every bit well, though this work differs from the first two in that it constructs the network in response to incoming ascertainment data rather than in response to queries.

For work on learning dynamic probabilistic model structure from training data, come across, for instance, [Friedman et al., 1998], and the references therein.

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The basics of natural language processing

Chenguang Zhu , in Machine Reading Comprehension, 2021

ii.4.2 Evaluation of language models

The language model establishes the probability model for text, that is, $P\left(w_1{w}_2\dots {\mathrm{due\; west}}_g\right)$ . So a linguistic communication model is evaluated by the probability value it assigns to test text unseen during grooming.

In the evaluation, all sentences in the test prepare are concatenated together to make a single discussion sequence: ${\mathrm{due\; west}}_1,{\mathrm{west}}_2,\dots ,w_{\mathrm{Northward}}$ , which includes the special symbols <s> and </s>. A language model should maximize the probability $P\left({\mathrm{westward}}_{\mathrm{i}}{w}_2\dots {\mathrm{westward}}_{\mathrm{Due\; north}}\right)$ . Nonetheless, equally this probability favors shorter sentences, we use the perplexity metric to normalize it by the number of words:

$P\mathrm{eastward}rplexity\left(w_1{w}_{\mathrm{ii}}\dots w_N\right)=P{\left(w_1{w}_2\dots w_{\mathrm{Due\; north}}\right)}^{-\frac{1}{N}}=\sqrt[\mathrm{North}]{\frac{1}{P\left(w_{\mathrm{i}}{w}_2\dots {\mathrm{west}}_N\right)}}$

For case, in the bigram language model, $Perp\mathrm{fifty}\mathrm{east}xity\left({\mathrm{westward}}_1{w}_{\mathrm{ii}}\dots {\mathrm{due\; west}}_N\right)=\sqrt[\mathrm{Northward}]{{\prod }_{i=\mathrm{ane}}^N\frac{1}{P\left({\mathrm{westward}}_i|{\mathrm{due\; west}}_{i-\mathrm{one}}\right)}}$ .

Since perplexity is a negative ability of probability, it should be minimized to maximize the original probability. On the public benchmark dataset Penn Tree Bank, the currently best language model can achieve a perplexity score around 35.8 [4].

It's worth noting that factors similar the dataset size and inclusion of punctuations can have a significant impact on the perplexity score. Therefore beyond perplexity, a language model can be evaluated by checking whether it helps with other downstream NLP tasks.

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Incorporating Dubiousness into Data Integration

AnHai Doan , ... Zachary Ives , in Principles of Information Integration, 2012

13.1.2 From Uncertainty to Probabilities

A probabilistic model for representing incertitude has many positives. Withal, a natural question is how i goes from confidence levels in data, mappings, queries, or schemas to actual probability values. Subsequently all, for example, converting a string edit altitude score to a probability requires a model of how typographical errors or string modifications are introduced. Such a model is probable highly dependent on the particular data and application — and thus unavailable to us.

The respond to the question of where probabilities come from is typically application specific, and often non formally justified. In the best cases, we do have probabilistic information about distributions, error rates, etc. to build from. In a few of these cases, we may even have models of how information values correlate.

However, in many other cases, we simply have a subjective confidence level that gets converted to a [0,1] interval and gets interpreted as a probability. Much as in Spider web search, the ultimate question is whether the system assigns a higher score to answers equanimous from good (high-confidence) values than to poor ones — non whether we accept a mathematically solid foundation to the generation of the underlying scores.

Within this section, our focus has been on representing uncertainty associated with information. We next describe how we tin can ascribe uncertainty to another cardinal ingredient in data integration, namely, schema mappings.

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Type I and Type II Error

Alesha E. Doan , in Encyclopedia of Social Measurement, 2005

Alternatives to α: P Value and Confidence Intervals

Instead of setting the α level, which is ofttimes capricious or washed out of convention, a researcher can use a exam statistic (e.g., the t statistic) to find the p value. The p value is the probability value; it provides the verbal probability of committing a blazon I fault (the p value is also referred to every bit the observed or verbal level of significance). More specifically, the p value is defined as the everyman significance level at which the null hypothesis can exist rejected. Using the test statistic, a researcher can locate the exact probability of obtaining that test statistic past looking on the appropriate statistical tabular array. As the value of the test statistic increases, the p value decreases, allowing a researcher to pass up the zero hypothesis with greater assurance.

Another option in lieu of relying on α is to use a confidence interval approach to hypothesis testing. Conviction intervals tin can be synthetic around betoken estimates using the standard error of the estimate. Confidence intervals indicate the probability that the true population coefficient is contained in the range of estimated values from the empirical assay. The width of a confidence interval is proportional to the standard mistake of the estimator. For case, the larger the standard error of the estimate, the larger the conviction interval, and therefore the less sure the researcher tin exist that the true value of the unknown parameter has been accurately estimated.

The null hypothesis is frequently prepare every bit an empirical straw human considering the objective of empirical research is to find back up for the alternative hypothesis (hence the conventional wisdom that null findings are non newsworthy findings). The null hypothesis may reflect a fairly cool scenario that is actually used to dramatize the significance of empirical findings. Consequently, some econometricians argue for the use of confidence intervals, which focus attending on the magnitude of the coefficients (findings) rather than on the rejection of the zero hypothesis. According to De Long and Lang (1992) "if all or almost all null hypotheses are false, at that place is piffling point in concentrating on whether or not an estimate is indistinguishable from its predicted value under the null" (p. 1257).

Both of these options present alternatives to simply choosing a level of significance. The p value yields an exact probability of committing a type I fault, which provides the researcher with enough data to decide whether or not to reject the naught hypothesis based on the given p value. Using confidence intervals differs in approach by concentrating on the magnitude of the findings rather than the probability of committing a blazon I error. Every approach to hypothesis testing—using α, p values, or confidence intervals—contains some amount of trade-offs. Ultimately, a researcher must decide which approach, or combination thereof, suits his or her research fashion.

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A logical reasoning framework for modelling and merging uncertain semi-structured information

Anthony Hunter , Weiru Liu , in Modern Information Processing, 2006

Abstract

Semi-structured information in xml can be merged in a logic-based framework [7,9]. This framework has been extended to deal with uncertainty, in the form of probability values, degrees of beliefs, or necessity measures, in the xml documents [8]. In this newspaper, we discuss how this logical framework tin can be used to model and reason with structured scientific cognition on the Web in medical and bioscience domains. We volition demonstrate how multiple summaritive and evaluative knowledge under doubtfulness tin can be merged to obtain less conflicting and better confirmed results in response to users queries. We will also show how reliability of a source can exist integrated into this structure. A number of examples are deployed to illustrate potential applications of the framework.

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Statistical and Syntactic Design Recognition

Anke Meyer-Baese , Volker Schmid , in Pattern Recognition and Point Assay in Medical Imaging (Second Edition), 2014

vi.iii.1 Bayes Determination Theory

Bayes decision theory represents a fundamental statistical approach to the trouble of pattern classification. This technique is based on the assumption that the conclusion problem is formulated in probabilistic terms, and that all relevant probability values are given. In this section, nosotros develop the fundamentals of this theory.

A uncomplicated introduction to this approach can be given by an example which focuses on the two-class case ${\omega }_1\text{,}{\omega }_2$ . The a priori probabilities $P({\omega }_{\mathrm{ane}})$ and $P({\omega }_2)$ are causeless to exist known since they tin can be easily determined from the available data set. Too known are the pdfs $p({\mathbf{x}}_i\mid {\omega }_i)$ , $i=1\text{,}\mathrm{two}$ . $p({\mathbf{x}}_i\mid {\omega }_i)$ is also known under the name of the likelihood function of ${\omega }_i$ with respect to $\mathbf{x}$ .

Recalling the Bayes rule, nosotros have

(6.1) $P({\omega }_i\mid \mathbf{x})=\frac{p(\mathbf{ten}\mid {\omega }_i)P({\omega }_i)}{p(\mathbf{x})}$

where $p(\mathbf{x})$ is the pdf of $\mathbf{ten}$ , and for which it holds

(6.2) $p(\mathbf{x})={\displaystyle \sum _{i=1}^2}p(\mathbf{10}\mid {\omega }_i)P({\omega }_i)$

The Bayes nomenclature rule tin now be stated for the 2-class case ${\omega }_{\mathrm{ane}}\text{,}{\omega }_2$

(6.3) $\begin{array}{cc}\hfill & \text{If}P({\omega }_1\mid \mathbf{x})>P({\omega }_{\mathrm{ii}}\mid \mathbf{x})\text{,}\mathbf{x}\text{is assigned to}{\omega }_1\hfill \\ \hfill & \text{If}P({\omega }_{\mathrm{one}}\mid \mathbf{ten})<P({\omega }_{\mathrm{ii}}\mid \mathbf{x})\text{,}\mathbf{x}\text{is assigned to}{\omega }_2\hfill \end{array}$

Nosotros immediately can conclude from above that a feature vector can be either assigned to 1 class or the other. Equivalently, nosotros now tin can write

(half dozen.4) $p(\mathbf{x}\mid {\omega }_{\mathrm{i}})P({\omega }_{\mathrm{ane}})\gtrless p(\mathbf{x}\mid {\omega }_2)P({\omega }_2)$

This corresponds to determining the maximum of the conditional pdfs evaluated at $\mathbf{x}$ . Figure vi.1 visualizes two equiprobable classes and the conditional pdfs $p(x\mid {\omega }_i)\text{,}i=1\text{,}2$ as function of $x$ . The dotted line at ${\mathrm{10}}_0$ corresponds to a threshold splitting the one-dimensional feature infinite into two regions $R_{\mathrm{one}}$ and $R_2$ . Based on the Bayes nomenclature rule, all values of $\mathrm{10}\in R_1$ are assigned to class ${\omega }_1$ , while all values $\mathrm{10}\in R_2$ are assigned to class ${\omega }_2$ .

Figure 6.1. Two equiprobable classes and the corresponding regions $R_{\mathrm{i}}$ and $R_{\mathrm{ii}}$ .

The probability of the conclusion error is given past

(6.5) $P_{\mathrm{eastward}}={\int }_{-\infty }^{x_0}p(\mathrm{10}\mid {\omega }_2)\mathit{dx}+{\int }_{x_0}^{+\infty }p(x\mid {\omega }_{\mathrm{one}})\mathit{dx}$

The Bayes classification rule achieves a minimal error probability. In [84] it was shown that the classification mistake is minimal, if the partition of the feature set into the two regions $R_1$ and $R_{\mathrm{two}}$ is called such that

(half-dozen.half dozen) $\begin{array}{c}\hfill R_1:P({\omega }_1\mid \mathbf{x})>P({\omega }_{\mathrm{two}}\mid \mathbf{x})\\ \hfill R_2:P({\omega }_2\mid \mathbf{10})>P({\omega }_{\mathrm{ane}}\mid \mathbf{x})\end{array}$

The generalization for $M$ classes ${\omega }_{\mathrm{one}}\text{,}{\omega }_{\mathrm{two}}\text{,}\dots \text{,}{\omega }_M$ is very simple. A feature vector $\mathbf{ten}$ is assigned to form ${\omega }_i$ if

(six.7) $P({\omega }_i\mid \mathbf{10})>P({\omega }_j\mid \mathbf{10})\forall j\ne i$

Every time nosotros assign an object to a class, we risk making an error. In multiclass problems, some misclassifications can have more serious repercussions than others. A quantitative way to measure this is given by a so-called cost function. Let $\mathrm{Fifty}(i\text{,}j)$ be the cost (or "loss") of assigning an object to class $i$ when information technology really belongs to class $j$ .

From the above, we see that a different classification possibility is achieved by defining a so-chosen cost term $L(i\text{,}j)$ with $i\text{,}j=1\text{,}2\text{,}\dots \text{,}\mathrm{Yard}$ . The penalization term is equal to zippo, $L(i\text{,}j)=0$ , if the characteristic vector $\mathbf{10}$ is correctly assigned to its class, and larger than goose egg, $L(i\text{,}j)>0$ , if assigned to grade ${\omega }_j$ instead of the correct grade ${\omega }_i$ . In other words, there is only loss if misclassification occurs.

The conditional loss term $R_i(\mathbf{x})$ with respect to the form assignment of $\mathbf{ten}$ is

(six.8) $R_i(\mathbf{x})={\displaystyle \sum _{j=\mathrm{i}}^M}L(i\text{,}j)P({\omega }_j\mid \mathbf{x})$

or equivalently,

(6.9) $R_i(\mathbf{x})={\displaystyle \sum _{j=\mathrm{i}}^M}L(i\text{,}j)p(\mathbf{ten}\mid {\omega }_j)P({\omega }_j)$

For practical applications nosotros choose $L(i\text{,}j)=0$ for $i=j$ , and $L(i\text{,}j)=1$ for $i\ne j$ .

Thus, given the characteristic vector, there is a certain risk involved in assigning the object to any grouping.

Based on the to a higher place definitions, we obtain a slightly inverse Bayes classification dominion: a feature vector $\mathbf{x}$ is assigned to a course ${\omega }_i$ for which $R_i(\mathbf{x})$ is minimal.

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What Values Can Be Probabilities,

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