Bayesian Analysis of Oxygen Consumption Rates in Athletes
The sports scientist measures the rate of oxygen consumption in athletes after exercise, with a sample mean of 2.25 litres per minute and a standard deviation of 1.6. Using Bayesian analysis with vague prior knowledge, a posterior distribution is obtained. The 95% Bayesian confidence interval is calculated to be (1.997, 2.503), indicating a precise estimation compared to frequentist methods.
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MAS2317 Presentation Jake McGlen
14. A sports scientist measures the rate of oxygen consumption of 10 randomly chosen athletes immediately after exercise. The sample mean is ?= 2.25 litres per minute and she assumes a Normal distribution for these measurements with a standard deviation of 1 6 litres per minute. Assuming vague prior knowledge, obtain the posterior distribution for and hence construct a 95% Bayesian confidence interval for and comment on your results.
The Normal Distribution is given as ~?( , 2). in this case is known by using ?= 2.25 and is told to us to be 1 6. Therefore the normal distribution is given by ~?(2.25 ,1 the posterior distribution is |x~?(B ,1 6 ). If the prior is known to be ~?(b ,1 ? ), then ? ); Where ? =??+?? ? ? = 2.25. ?+?? ,? = ? + ?? ,? = 10,? = 6 and
If we now proceed to use vague prior knowledge about , we let the prior variance tend to (d 0). This causes B ?and D ?? as d 0, creating the posterior distribution of |x~?( ?, ?? ). Using the information from the previous slide, the distribution is |x~? 2.25, 60 . 1 1
The mean does not change for a normal distribution. The posterior distributions shape is much more compact than the original prior distribution as the variance is ten times smaller This makes the distribution more precise.
We now need to find the 95% Bayesian confidence interval for and comment on the results. For a 95% interval we use the formula 2.25 1.96 60. This gives the answer (1.997, 2.503). This confidence interval shows the probability of capturing the true value of to be 0.95. A range of 0.506 is a precise region for . This is a smaller range than would have been given if we used the frequentist prior distribution.
