The Maple Syrup Problem: The Normal-Normal Conjugate

2019 ◽  
pp. 172-190
Author(s):  
Therese M. Donovan ◽  
Ruth M. Mickey
Keyword(s):  
Normal Distribution ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Unknown Parameter ◽  
Distributed Data ◽  
Maple Syrup ◽  
New Information ◽  
The Mean ◽  
Two Parameters ◽  
Parameter Values

In this chapter, Bayesian methods are used to estimate the two parameters that identify a normal distribution, μ‎ and σ‎. Many Bayesian analyses consider alternative parameter values as hypotheses. The prior distribution for an unknown parameter can be represented by a continuous probability density function when the number of hypotheses is infinite. In the “Maple Syrup Problem,” a normal distribution is used as the prior distribution of μ‎, the mean number of millions of gallons of maple syrup produced in Vermont in a year. The amount of syrup produced in multiple years is determined, and assumed to follow a normal distribution with known σ‎. The prior distribution is updated to the posterior distribution in light of this new information. In short, a normal prior distribution + normally distributed data → normal posterior distribution.

The Shark Attack Problem: The Gamma-Poisson Conjugate

2019 ◽  
pp. 150-171
Author(s):  
Therese M. Donovan ◽  
Ruth M. Mickey
Keyword(s):  
Gamma Distribution ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Unknown Parameter ◽  
New Information ◽  
Special Cases ◽  
Poisson Data ◽  
Shark Attacks ◽  
Parameter Values ◽  
Bayesian Prior

This chapter introduces the gamma-Poisson conjugate. Many Bayesian analyses consider alternative parameter values as hypotheses. The prior distribution for an unknown parameter can be represented by a continuous probability density function when the number of hypotheses is infinite. There are special cases where a Bayesian prior probability distribution for an unknown parameter of interest can be quickly updated to a posterior distribution of the same form as the prior. In the “Shark Attack Problem,” a gamma distribution is used as the prior distribution of λ‎, the mean number of shark attacks in a given year. Poisson data are then collected to determine the number of attacks in a given year. The prior distribution is updated to the posterior distribution in light of this new information. In short, a gamma prior distribution + Poisson data → gamma posterior distribution. The gamma distribution is said to be “conjugate to” the Poisson distribution.

The Maple Syrup Problem Revisited: MCMC with Gibbs Sampling

2019 ◽  
pp. 247-266
Author(s):  
Therese M. Donovan ◽  
Ruth M. Mickey
Keyword(s):  
Monte Carlo ◽  
Gibbs Sampling ◽  
Posterior Distribution ◽  
Unknown Parameter ◽  
The Other ◽  
Sampling Step ◽  
Maple Syrup ◽  
Two Parameters ◽  
Special Case ◽  
Joint Posterior Distribution

This chapter introduces Markov Chain Monte Carlo (MCMC) with Gibbs sampling, revisiting the “Maple Syrup Problem” of Chapter 12, where the goal was to estimate the two parameters of a normal distribution, μ‎ and σ‎. Chapter 12 used the normal-normal conjugate to derive the posterior distribution for the unknown parameter μ‎; the parameter σ‎ was assumed to be known. This chapter uses MCMC with Gibbs sampling to estimate the joint posterior distribution of both μ‎ and σ‎. Gibbs sampling is a special case of the Metropolis–Hastings algorithm. The chapter describes MCMC with Gibbs sampling step by step, which requires (1) computing the posterior distribution of a given parameter, conditional on the value of the other parameter, and (2) drawing a sample from the posterior distribution. In this chapter, Gibbs sampling makes use of the conjugate solutions to decompose the joint posterior distribution into full conditional distributions for each parameter.

The White House Problem: The Beta-Binomial Conjugate

2019 ◽  
pp. 133-149
Author(s):  
Therese M. Donovan ◽  
Ruth M. Mickey
Keyword(s):  
Posterior Distribution ◽  
Beta Distribution ◽  
Prior Distribution ◽  
Unknown Parameter ◽  
Fixed Number ◽  
Binomial Data ◽  
White House ◽  
New Information ◽  
Famous Person ◽  
Special Cases

This chapter introduces the beta-binomial conjugate. There are special cases where a Bayesian prior probability distribution for an unknown parameter of interest can be quickly updated to a posterior distribution of the same form as the prior. In the “White House Problem,” a beta distribution is used to set the priors for all hypotheses of p, the probability that a famous person can get into the White House without an invitation. Binomial data are then collected, and provide the number of times a famous person gained entry out of a fixed number of attempts. The prior distribution is updated to a posterior distribution (also a beta distribution) in light of this new information. In short, a beta prior distribution for the unknown parameter + binomial data → beta posterior distribution for the unknown parameter, p. The beta distribution is said to be “conjugate to” the binomial distribution.

scholarly journals ANALISIS METODE BAYESIAN PADA SISTEM ANTREAN RAWAT JALAN DI RSUP Dr. KARIADI DENGAN DISTRIBUSI SAMPEL POISSON DAN GEOMETRIK

2021 ◽  
Vol 10 (3) ◽  
pp. 413-422
Author(s):  
Nur Azizah ◽  
Sugito Sugito ◽  
Hasbi Yasin
Keyword(s):  
Internal Medicine ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Bayesian Method ◽  
Likelihood Function ◽  
Queuing System ◽  
Hospital Service ◽  
Queuing Model ◽  
Sample Distribution ◽  
New Information

Hospital service facilities cannot be separated from queuing events. Queues are an unavoidable part of life, but they can be minimized with a good system. The purpose of this study was to find out how the queuing system at Dr. Kariadi. Bayesian method is used to combine previous research and this research in order to obtain new information. The sample distribution and prior distribution obtained from previous studies are combined with the sample likelihood function to obtain a posterior distribution. After calculating the posterior distribution, it was found that the queuing model in the outpatient installation at Dr. Kariadi Semarang is (G/G/c): (GD/∞/∞) where each polyclinic has met steady state conditions and the level of busyness is greater than the unemployment rate so that the queuing system at Dr. Kariadi is categorized as good, except in internal medicine poly. 

scholarly journals Statistical Analysis of Basketball Teams Through Normal Distribution and R Programming

2020 ◽  
Vol 4 (9) ◽  
Author(s):  
Megan Wang
Keyword(s):  
Statistical Analysis ◽  
Standard Deviation ◽  
Probability Density Function ◽  
Normal Distribution ◽  
Sports Analytics ◽  
R Programming Language ◽  
General Improvement ◽  
The Mean ◽  
R Programming ◽  
Two Parameters

Basketball has existed for almost 130 years, becoming one of the most famous sports worldwide by affecting millions of lives and having national and global tournaments. With the general improvement of people's concern and love for sports competition, sports analytics’ role will become more prominent. Hence, this paper combines the relevant knowledge of statistics and typical basketball competition cases from NBA, expounding the application of statistics in sports competition. The paper first examines the importance of normal distribution (also called Gaussian distribution) in statistics through its probability density function and the function's graph. The function has two parameters: the mean for the maximum and standard deviation for the distance away from the mean[1]. By compiling datasets of past teams and individuals for their basketball performances and making simple calculations of their standard deviation and mean, the paper constructs normal distribution graphs using the R programming language. Finally, the paper examines the Real Plus-Minus value and its importance in basketball.

A simple model for showing effects of geometry on the ocean tides

1979 ◽  
Vol 367 (1731) ◽  
pp. 549-571 ◽  
Keyword(s):  
Simple Model ◽  
Southern Ocean ◽  
Parameter Space ◽  
Resonance Line ◽  
Analytic Solutions ◽  
Dimensional Parameter ◽  
The Mean ◽  
Two Parameters ◽  
Parameter Values ◽  
Made In

There is a need for a simple model to show effects of ocean shape on the tides and, in particular, to show how the Atlantic tides interact with those of the Southern Ocean. In response to this need, the Atlantic and Southern Oceans are here represented by narrow canals which meet in a T-junction. Analytic solutions for this geometry are easily obtained. Rotation effects can be included by calculating the second terms in an expansion in a small parameter proportional to the widths of the canals, and this can produce a realistic configuration of cotidal lines. The solution is studied in a two dimensional parameter space, the two parameters corresponding to the ocean depth and the mean latitude of the Southern Ocean. The solution is very sensitive to parameter values near the resonance line, but also depends very much on position in parameter space relative to a special point on the resonance line where the equilibrium tide is orthogonal to the resonant free oscillation. With a small friction, solutions on one side of this point generally give southward propagation of tides in the Atlantic, while northward propagation is generally obtained for parameter values on the other side. The effect depends on the direction in which the phase of the free tide is shifted relative to that of the direct tide. Useful conclusions about some old controversies can be made in the light of these results.

Photo-electric Hβ and uvby photometry

1966 ◽  
Vol 24 ◽  
pp. 170-180
Author(s):  
D. L. Crawford
Keyword(s):  
Narrow Band ◽  
Line Strength ◽  
Interference Filter ◽  
B Stars ◽  
Relative Line ◽  
The Mean ◽  
F Stars ◽  
Two Parameters ◽  
Filter Photometry ◽  
The Continuum

Early in the 1950's Strömgren (1, 2, 3, 4, 5) introduced medium to narrow-band interference filter photometry at the McDonald Observatory. He used six interference filters to obtain two parameters of astrophysical interest. These parameters he calledlandc, for line and continuum hydrogen absorption. The first measured empirically the absorption line strength of Hβby means of a filter of half width 35Å centered on Hβand compared to the mean of two filters situated in the continuum near Hβ. The second index measured empirically the Balmer discontinuity by means of a filter situated below the Balmer discontinuity and two above it. He showed that these two indices could accurately predict the spectral type and luminosity of both B stars and A and F stars. He later derived (6) an indexmfrom the same filters. This index was a measure of the relative line blanketing near 4100Å compared to two filters above 4500Å. These three indices confirmed earlier work by many people, including Lindblad and Becker. References to this earlier work and to the systems discussed today can be found in Strömgren's article inBasic Astronomical Data(7).

Prediction of Haemoglobin Level by Some Probability Distribution

2020 ◽  
Vol 9 (1) ◽  
pp. 84-88
Author(s):  
Govinda Prasad Dhungana ◽  
Laxmi Prasad Sapkota
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Goodness Of Fit ◽  
Continuous Variable ◽  
Hemoglobin Level ◽  
Chi Square ◽  
Chi Square Test ◽  
Log Normal ◽  
Two Parameters ◽  
Best Fit

 Hemoglobin level is a continuous variable. So, it follows some theoretical probability distribution Normal, Log-normal, Gamma and Weibull distribution having two parameters. There is low variation in observed and expected frequency of Normal distribution in bar diagram. Similarly, calculated value of chi-square test (goodness of fit) is observed which is lower in Normal distribution. Furthermore, plot of PDFof Normal distribution covers larger area of histogram than all of other distribution. Hence Normal distribution is the best fit to predict the hemoglobin level in future.

Essai d'application de modèles météorologiques au calcul du moment d'apparition de deux phases phénologiques | Application of Meteorological Models for the Calculation of the Beginning of Two Phenological Phases

2000 ◽  
Vol 151 (10) ◽  
pp. 385-397
Author(s):  
Bernard Primault
Keyword(s):  
Sunshine Duration ◽  
Fruit Trees ◽  
Swiss Alps ◽  
Winter Solstice ◽  
Phenological Data ◽  
Thermal Thresholds ◽  
The Mean ◽  
Two Parameters ◽  
Meteorological Models ◽  
Phenological Phases

Many years ago, a model was elaborated to calculate the«beginning of the vegetation's period», based on temperatures only (7 days with +5 °C temperature or more). The results were correlated with phenological data: the beginning of shoots with regard to spruce and larch. The results were not satisfying, therefore, the value of the two parameters of the first model were modified without changing the second one. The result, however, was again not satisfying. Research then focused on the influence of cumulated temperatures over thermal thresholds. Nevertheless, the results were still not satisfying. The blossoming of fruit trees is influenced by the mean temperature of a given period before the winter solstice. Based on this knowledge, the study evaluated whether forest trees could also be influenced by temperature or sunshine duration of a given period in the rear autumn. The investigation was carried through from the first of January on as well as from the date of snow melt of the following year. In agricultural meteorology, the temperature sums are often interrelated with the sunshine duration, precipitation or both. However,the results were disappointing. All these calculations were made for three stations situated between 570 and 1560 m above sea-level. This allowed to draw curves of variation of the two first parameters (number of days and temperature) separately for each species observed. It was finally possible to specify the thus determined curves with data of three other stations situated between the first ones. This allows to calculate the flushing of the two tree species, if direct phenological observation is lacking. This method, however, is only applicable for the northern part of the Swiss Alps.

Limit tolerances for sums of measurement errors

2018 ◽  
Vol 934 (4) ◽  
pp. 59-62
Author(s):  
V.I. Salnikov
Keyword(s):  
Normal Distribution ◽  
Mean Square Error ◽  
Gaussian Distribution ◽  
Measurement Errors ◽  
Theory And Practice ◽  
Mean Square ◽  
The Mean ◽  
Limiting Values ◽  
Limiting Value

The question of calculating the limiting values of residuals in geodesic constructions is considered in the case when the limiting value for measurement errors is assumed equal to 3m, ie ∆рred = 3m, where m is the mean square error of the measurement. Larger errors are rejected. At present, the limiting value for the residual is calculated by the formula 3m√n, where n is the number of measurements. The article draws attention to two contradictions between theory and practice arising from the use of this formula. First, the formula is derived from the classical law of the normal Gaussian distribution, and it is applied to the truncated law of the normal distribution. And, secondly, as shown in [1], when ∆рred = 2m, the sums of errors naturally take the value equal to ?pred, after which the number of errors in the sum starts anew. This article establishes its validity for ∆рred = 3m. A table of comparative values of the tolerances valid and recommended for more stringent ones is given. The article gives a graph of applied and recommended tolerances for ∆рred = 3m.

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