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Though hypotheses and theories ..

Both statistical theory and counting/measurement theory are necessary to make inferences about reality.

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Distinguishing Between Hypotheses, Theories and Facts

In sum, classical procedures employ the data to narrow down a setof hypotheses. Put in such general terms, it becomes apparent thatclassical procedures provide a response to the problem ofinduction. The data are used to get from a weak general statementabout the target system to a stronger one, namely from a set ofcandidate hypotheses to a subset of them. The central concern in thephilosophy of statistics is how we are to understand these procedures,and how we might justify them. Notice that the pattern of classicalstatistics resembles that of eliminative induction: in viewof the data we discard some of the candidate hypotheses. Indeedclassical statistics is often seen in loose association with Popper'sfalsificationism, but this association is somewhat misleading. Inclassical procedures statistical hypotheses are discarded when theyrender the observed sample too improbable, which of course differsfrom discarding hypotheses that deem the observed sampleimpossible.

This is indeed a drawback of the classical theory of testing statistical hypotheses.

On the other hand, the null hypothesis is straightforward -- what is the probability that our treated and untreated samples are from the same population (that the treatment or predictor has no effect)? There is only one set of statistical probabilities -- calculation of chance effects. Instead of directly testing H, we test H. If we can reject H, (and factors are under control), we can accept H. To put it another way, the fate of the research hypothesis depends upon what happens to H.

Scientific Hypothesis, Theories and Laws

The notion of physical probability is connected to one of the majortheories of statistical method, which has come to be calledclassical statistics. It was developed roughly in the firsthalf of the 20th century, mostly by mathematicians and workingscientists like Fisher (1925, 1935, 1956), Wald (1939, 1950), Neymanand Pearson (1928, 1933, 1967), and refined by very many classicalstatisticians of the last few decades. The key characteristic of thistheory of statistics aligns naturally with viewing probabilities asphysical chances, hence pertaining to observable and repeatableevents. Physical probability cannot meaningfully be attributed tostatistical hypotheses, since hypotheses do not have tendencies orfrequencies with which they come about: they are categorically true orfalse, once and for all. Attributing probability to a hypothesis seems to entail thatthe probability is read epistemically.

Classical statistics is often called frequentist, owing tothe centrality of frequencies of events in classical procedures andthe prominence of the frequentist interpretation of probabilitydeveloped by von Mises. In this interpretation, chances arefrequencies, or proportions in a class of similar events oritems. They are best thought of as analogous to other physicalquantities, like mass and energy. It deserves emphasis thatfrequencies are thus conceptually prior to chances . In propensitytheory the probability of an individual event or item is viewed as atendency in nature, so that the frequencies, or the proportions in aclass of similar events or items, manifest as a consequence of the lawof large numbers. In the frequentist theory, by contrast, theproportions lay down, indeed define what the chances are. Thisleads to a central problem for frequentist probability, theso-called reference class problem: it is not clear whatclass to associate with an individual event or item (cf. Reichenbach1949, Hajek 2007). One may argue that the class needs to be as narrowas it can be, but in the extreme case of a singleton class of events,the chances of course trivialize to zero or one. Since classicalstatistics employs non-trivial probabilities that attach to the singlecase in its procedures, a fully frequentists understanding ofstatistics is arguably in need of a response to the reference classproblem.

A theory is a based upon a hypothesis and backed by evidence

This distinction should not be confused with that betweenobjective and subjective probability. Both physical and epistemicprobability can be given an objective and subjective character, in thesense that both can be taken as dependent or independent of aknowing subject and her conceptual apparatus. For more details on theinterpretation of probability, the reader is invited to consultGalavotti (2005), Gillies (2000), Mellor (2005), von Plato (1994), theanthology by Eagle (2010), the handbook of Hajek and Hitchcock(forthcoming), or indeed the entry on .In this context the key point is that the interpretations can all beconnected to foundational programmes for statisticalprocedures. Although the match is not exact, the two major typesspecified above can be associated with the two major theories ofstatistics, classical and Bayesian statistics, respectively.

Probabilities may be taken to represent doxastic attitudesin the sense that they specify opinions about data and hypotheses ofan idealized rational agent. The probability then expresses the strengthor degree of belief, for instance regarding the correctness of thenext guess of the tea tasting lady. They may also be taken asdecision-theoretic, i.e., as part of a more elaboraterepresentation of the agent, which determines her dispositions towardsdecisions and actions about the data and the hypotheses. Oftentimes adecision-theoretic representation involves doxastic attitudesalongside preferential and perhaps other ones. In that case, theprobability may for instance express a willingness to bet on the ladybeing correct. Finally, the probabilities may be taken aslogical. More precisely, a probabilistic model may betaken as a logic, i.e., a formal representation that fixes a normativeideal for uncertain reasoning. According to this latter option,probability values over data and hypotheses have a role that iscomparable to the role of truth values in deductive logic: they serveto secure a notion of valid inference, without carrying the suggestionthat the numerical values refer to anything psychologicallysalient.

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  • Statistical hypothesis testing - Wikipedia

    The next step is to compute the relevant statistic based on the null hypothesis and the random sample of size n.

  • A hypothesis (plural hypotheses ..

    Even though the words "hypothesis" and "theory" are ..

  • be explained with the available scientific theories

    01/02/2014 · Understand the definitions of scientific hypothesis, model, theory, ..

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Herzberg | Statistical Hypothesis Testing | Motivation

The epistemic view on probability came into development in the 19thand the first half of the 20th century, first by the hand of De Morgan(1847) and Boole (1854), later by Keynes (1921), Ramsey (1926) and deFinetti (1937), and by decision theorists, philosophers andinductive logicians such as Carnap (1950), Savage (1962), Levi (1980),and Jeffrey (1992). Important proponents of these views in statisticswere Jeffreys (1961), Edwards (1972), Lindley (1965), Good (1983),Jaynes (2003) as well as very many Bayesian philosophers andstatisticians of the last few decades (e.g., Goldstein 2006, Kadane2011, Berger 2006, Dawid 2004). All of these have a view that placesprobabilities somewhere in the realm of the epistemic rather than thephysical, i.e., not as part of a model of the world but rather as ameans to model a representing system like the human mind.

Is there a valid part to Herzberg's two-part theory

While there is large variation in how statistical procedures andinferences are organized, they all agree on the use of modernmeasure-theoretic probability theory (Kolmogorov ), or a nearkin, as the means to express hypotheses and relate them to data. Byitself, a probability function is simply a particular kind ofmathematical function, used to express the measure of a set(cf. Billingsley 1995).

It will be fairly obvious that theories I to V are ..

For present concerns the important point is that each of theseepistemic interpretations of the probability calculus comes with itsown set of foundational programs for statistics. On the whole,epistemic probability is most naturally associated with Bayesianstatistics, the second major theory of statistical methods (Press2002, Berger 2006, Gelman et al 2013). The keycharacteristic of Bayesian statistics flows directly from theepistemic interpretation: under this interpretation it becomespossible to assign probability to a statistical hypothesis and torelate this probability, understood as an expression of how stronglywe believe the hypothesis, to the probabilities of events. Bayesianstatistics allows us to express how our epistemic attitudes towards astatistical hypothesis, be it logical, decision-theoretic, ordoxastic, changes under the impact of data.

Hypothesis | Hypothesis | Statistical Hypothesis Testing

The take-home message is that the Bayesian method allows us toexpress our epistemic attitudes to statistical hypotheses in terms ofa probability assignment, and that the data impact on this epistemicattitude in a regulated fashion.

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