Caution
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Bayesian Statistics: From Concept to Data Analysis
Oren Bochman
Exponential Distribution, Homework
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---
title: "Homework Alternative Priors - M4L11HW1"
subtitle: "Bayesian Statistics: From Concept to Data Analysis"
categories:
- Bayesian Statistics
keywords:
- Exponential Distribution
- Homework
---
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::: {#exr-alt-priors-1}
Suppose we flip a coin five times to estimate $\theta$, the probability of obtaining heads. We use a Bernoulli likelihood for the data and a non-informative (and improper) $\mathrm{Beta}(0,0)$ prior for $\theta$. We observe the following sequence: $(H, H, H, T, H)$.
Because we observed at least one H and at least one T, the posterior is proper. What is the posterior distribution for $\theta$
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#### Solution:
Beta(4,1)
We observed four "successes" and one "failure," and these counts are the parameters of the posterior beta distribution.
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::: {#exr-alt-priors-2}
Continuing the previous question, what is the posterior mean for $\theta$ ? Round your answer to one decimal place.
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#### Solution:
$\bar{y} = 0.8 $
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::: {#exr-alt-priors-3}
Consider again the thermometer calibration problem from Lesson 10.
Assume a normal likelihood with unknown mean $\theta$ and known variance $\sigma^2=0.25$. Now use the non-informative (and improper) flat prior for $\theta$ across all real numbers. This is equivalent to a conjugate normal prior with variance equal to $\infty$.
You collect the following $n=5$ measurements: (94.6, 95.4, 96.2, 94.9, 95.9). What is the posterior distribution for $\theta$?
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#### Solution:
Recall from the lesson that with a flat prior on $\theta$ , the posterior distribution is
$$
\mathcal{N}(\bar{y},\frac{\sigma^2}{n})=\mathcal{N}(95.4,0.05)
$$
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::: {#exr-alt-priors-4}
Which of the following graphs shows the Jeffreys prior for a Bernoulli/binomial success probability p?
Hint: The Jeffreys prior in this case is $\mathrm{Beta}(1/2, 1/2)$.
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#### Solution:
Beta(1/2, 1/2).
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::: {#exr-alt-priors-5}
Scientist A studies the probability of a certain outcome of an experiment and calls it $\theta$ . To be non-informative, he assumes a $Uniform(0,1)$ prior for $\theta$ .
Scientist B studies the same outcome of the same experiment using the same data, but wishes to model the odds $\phi= \frac{\theta}{1−\theta}$. Scientist B places a uniform distribution on $\phi$. If she reports her inferences in terms of the probability $\theta$, will they be equivalent to the inferences made by Scientist A?
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#### Solution:
No, they did not use the Jeffreys prior.
The uniform prior on $\theta$ implies the following prior PDF for
$$
f(\phi)= \frac{1}{(1+\phi)^2} I_{\{\phi≥0\}}
$$
which is not the uniform prior used by Scientist B.
They would obtain equivalent inferences if they both use the Jeffrey's prior.
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