Profile likelihood function

The profile likelihood function of 9i is shown in three-dimensional space in Figure 1.9. The two-dimensional profile likelihood function is found by projecting it back to the f X 9i plane and is shown in Figure 1.10. (It is like the "shadow" the curve L 9i, 02 9i, data) would project on the f x6i plane from a light source infinitely far away in the 02 direction.) The profile likelihood function may lose some information about 9 compared to the joint likelihood function. Note that the maximum profile likelihood value of 9 will be the same as its maximum likelihood value. However, confidence intervals based on profile likelihood may not be the same as those based on the joint likelihood. 

The Bayesian posterior estimator for 0i found from the marginal posterior will be the same as that found from the joint posterior when we arc using the posterior mean as our estimator. For this example, the Bayesian posterior density of 0i found by marginalizing 02 out of the joint posterior density, and the profile likelihood function of 01 turn out to have the same shape. This will not always be the case. For instance, suppose we wanted to do inference on 02, and regarded 0i as the nuisance parameter. 

Figure 1.15 shows both the profile likelihood function and the marginal posterior density in 2D for 02 for this case. Clearly they have different shapes despite coming from the same two-dimensional function. 

ML is the approach most commonly used to fit a distribution of a given type (Madgett 1998 Vose 2000). An advantage of ML estimation is that it is part of a broad statistical framework of likelihood-based statistical methodology, which provides statistical hypothesis tests (likelihood-ratio tests) and confidence intervals (Wald and profile likelihood intervals) as well as point estimates (Meeker and Escobar 1995). MLEs are invariant under parameter transformations (the MLE for some 1-to-l function of a parameter is obtained by applying the function to the untransformed parameter). In most situations of interest to risk assessors, MLEs are consistent and sufficient (a distribution for which sufficient statistics fewer than n do not exist, MLEs or otherwise, is the Weibull distribution, which is not an exponential family). When MLEs are biased, the bias ordinarily disappears asymptotically (as data accumulate). ML may or may not require numerical optimization skills (for optimization of the likelihood function), depending on the distributional model. 

Furthermore, when alternative approaches are applied in computing parameter estimates, the question to be addressed here is Do these other approaches yield similar parameter and random effects estimates and conclusions An example of addressing this second point would be estimating the parameters of a population pharmacokinetic (PPK) model by the standard maximum likelihood approach and then confirming the estimates by either constructing the profile likelihood plot (i.e., mapping the objective function), using the bootstrap (4, 9) to estimate 95% confidence intervals, or the jackknife method (7, 26, 27) and bootstrap to estimate standard errors of the estimate (4, 9). When the relative standard errors are small and alternative approaches produce similar results, then we conclude the model is reliable. 

Where Cf is a structural parameter that counts available reactive carbon sites and c, is a coefficient that account for distribution of reactive carbon sites types and catalytic effects and thus a variation in c, may change kinetic parameters. The gas composition vector (F) in J X) may beside the CO2 partial pressure also include such gas partial pressures as KOH since the likelihood that a catalytic site is activated is a function of the partial pressure of the catalyst, the site-catalyst attraction forces and the temperature. Since from Eq. (2) the structural profile invariance SPI assumption is by no means obvious we suggest that a temperature and partial pressure range are always given for the validity of the structural profile invariance assumption. Only if the invariant structural profile assumption is approximately valid a reference profile (/ /) can be used to eliminate the structural profile to form a normalised reactivity (R ) and to determine kinetics up to a constant... 

As well as these general motif databases, there are some more specialized sequence analysis tools for identifying particular structural or functional features. One well-studied example is that of coiled coils. The program COILS " allows one to compare a sequence against profiles derived from known parallel two-stranded coiled coils, to assess the likelihood that the query sequence is indeed a coiled coil. More involved analyses based on pairwise interactions have been developed, such as PAIRCOIL. Several families of coiled coils are actually well recognized by their PROSITE motifs. However, the leucine zipper PROSITE motif, L-X(6)-L-X(6)-L-X(6)-L, is poorly discriminating, and for this specific family of coiled coils, the program TRESPASSER provides a more refined identification... 

HMMER [96] is a freely distributable collection of software for protein-sequence analysis using profile HMMs. A profile HMM [97] is a statistical model of a multiple alignment of sequences drawn from a putative protein family. It captures position-specific information about the relative degree of conservation of different columns in an alignment and the relative likelihood of particular residues occurring in specific positions. Profile HMMs can thus capture the essential features of a structural or functional domain. 

Figure 7.13 Likelihood profile for clearance using the data in Table 7.4. Ninety percent CI is indicated by the arrows. The dotted—dashed line indicates the cut-off at which the change in objective function becomes statistically significant 2.70 in this case.

Fig. 2 Three-dimensional simulation of an object observed through a microscope, in the presence of optical blur and noise. The object consists of five spheres with different diameters but the same fluorescence density A initial object B point spread function (PSF) of the microscope C, D lateral and axial cross sections of the object after convolution with the microscope s PSF E lateral cross section of the object after blurring and the addition of noise F intensity profiles and percentage of the object s maximum intensity of original (dashed),hhxned (black) and blurred-i-noisy (red) data. Following blurring, the smaller the object, the weaker its maximum intensity is likely to be. Noise reduces the likelihood of detecting small and highly attenuated objects...

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