ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O B-Model I-Model I-Model I-Model O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Tag O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O
tokens: 2009 ). 3 Test model A simple 1-D test model is now defined. This allows numerous simulated data sets to be created so that the p -values computed according to Sects. 2.1 and 2.2 can be compared and any differeces established with high statistical confidence. 3.1 Hubble expansion The null hypothesis is that the local Universe is undergoing an isotropic, cold Hubble expansion with Hubble parameter h 0 = 70 km s −1 Mpc −1 . Moreover throughout the local Universe there exists a population of perfect standard candles of absolute bolometric magnitude M = −19.0. Finally, we suppose that the redshifts z of these standard candles are measured exactly, but that their apparent bolometric magnitudes m have normally-distributed measurement errors with σ m = 0.3 mag. 3.2 Ideal samples In order to test p -values when the null hypothesis is correct, we need data sets consistent with the above description of the Hubble flow. We define the local Universe as extending to z max = 0.2, so that to a good approximation the geometry is Euclidean. If we suppose the standard candles to be uniformly distributed in space, a sample { z i } is created with the formula where the x i are independent random numbers ∈ (0, 1). The distances of these n standard candles are and their measured apparent bolometric magnitudes are where the z G are independent gaussian variates from . The n pairs comprise a data set D from which the posterior probability density Λ( h | D ) of the Hubble parameter h can be inferred. Moreover, the global GOF between Λ( h | D ) and D can be tested with the p -values of Sects. 2.1 and 2.2 . 3.3 Imperfect samples In order to test p -values when the null hypothesis is false, we need to modify either the model or the data. If we continue to make inferences based on the Hubble flow defined in Sect. 3.1 , we can introduce a violation of the null hypothesis by contaminating the data sample with an unrecognized second class of standard candles with a different absolute bolometric magnitude. Thus the data set D now comprises n 1 pairs with given by Eq. ( 15 ) and n 2 pairs with given by and where n = n 1 + n 2 . Thus, the sample is now contaminated with a second population of standard candles that are brighter by Δ M magnitudes. The investigator is unaware of this, and so carries out a Bayesian analysis assuming an ideal sample. 4 Comparison of <italic>p</italic>-values The Bayesian p -values defined in Sect. 2 are now computed for simulated data samples derived as described in Sect. 3 . Fig. 1 Comparison of p -values. For J = 200 ideal samples D j , is plotted against log p B . 4.1 Null hypothesis correct In order to compare p -values in this case, J independent ideal samples D j are created as described in Sect. 3.2 , and then p -values for each D j computed as described in Sects. 2.1 and 2.2 . With a constant prior π ( h ) in the interval ( h 1 , h 2 ) with h 1 = 64 and h 2 = 76 km s −1 Mpc −1 , the posterior density Λ( h | D j ) is obtained from Eq. ( 1 )