Highest Posterior Density

The highest-posterior-density (HPD) interval with probability content (1 — a) for any estimated parameter i is... 

The posterior density function p(0 V) found by Box and Draper (1965) is a power of the determinant i (0) whose elements appear in Eq. (7.1-3). These authors used contour plots of p[6 Y) to estimate highest-posterior-density (HPD) regions for a two-parameter model. This technique is useful for models with just a few parameters — say. not more than three. 

The value of q is multiplied by the tabulated values of the highest density regions for the degrees of freedom, and an appropriate confidence level (Table 26-4). The distribution curve (the posterior density of a) is skewed so that the interval is larger on one side of the standard deviation than on the other, with low values of the standard deviation less probable than high values. 

Computing highest probabiiity density (HPD) regions from marginal posterior distributions... 

Such a set is called a Highest Probability Density (HPD) region. We wish to find the HPD region that contains some fraction (1 — a) of the total maiginal posterior density i.e., we identify the contour value such that... 

Hi-receptors in the adrenal medulla stimulates the release of the two catecholamines noradrenaline and adrenaline as well as enkephalins. In the heart, histamine produces negative inotropic effects via Hr receptor stimulation, but these are normally masked by the positive effects of H2-receptor stimulation on heart rate and force of contraction. Histamine Hi-receptors are widely distributed in human brain and highest densities are found in neocortex, hippocampus, nucleus accumbens, thalamus and posterior hypothalamus where they predominantly excite neuronal activity. Histamine Hrreceptor stimulation can also activate peripheral sensory nerve endings leading to itching and a surrounding vasodilatation ( flare ) due to an axonal reflex and the consequent release of peptide neurotransmitters from collateral nerve endings. 

Autoradiographic studies have revealed that H3 receptors are located in most areas of the brain with the highest content in the basal ganglia and the cortex [9]. H3 receptors are distributed with moderate density in the hypothalamus, where the highest density of histaminergic nerve fibers are found [4-5], and were demonstrated in the anterior, medial and posterior hypothalamus [8]. However, their localization in hypothalamic areas primarily involved in the regulation of pituitary hormone secretion were not studied in detail. In the pituitary gland the H3 receptors were scarce. 

But for safety it is usually acceptable to demonstrate pessimistic predictions. We can then look for a worst case prior distribution that one could assume for the inference. We can show that such a worst case does exist, as follows. Consider a probability density function for the unknown pfd, Q, made of two probability masses a mass Pp in Q = 0 and a mass (1 — Pp) in Q = q. Now, if we assume gjv to be close to either end of the interval [0,1], reliability predictions after observing failure-free demands will be very high. Indeed, in the two limiting cases, predicted reliability will be 1 if gjv = 1, one test is enough to show that P Q = <1n) = 0 and thus Q = 0 with certainty and if = 0, Q = 0 with certainty to start with.In between these extreme values, successM tests will reduce P Q = qw) and increase P Q = 0), but still leave a non-zero probabihty of future failure. Thus, posterior reliability as a function of q is highest at the two ends of the interval, and must have a minimum somewhere in between. ... 

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