In Python generate 4096 flips of the coin using the binomial function of the random module...

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In Python generate 4096 "flips" of the coin using the binomial function of the random module in numpy: import numpy.random flips = numpy.random.binomial(1, bias, 4096) Where bias is the value of the bias weighting you chose. In python write a function for the likelihood function for the flips. You'll want to be able to compute the function for a set of bias weightings, a number of flips, and a number of heads in that number of flips. You must: 1. Compute the various probabilities as (natural) logarithms. 2. Normalize the binomial function 3. Use the approximation: In(N!) ~ N In N - N + ln(2+N)/2. 4. Sample the bias weighting parameter H rather finely: AH = 0.001 works well. (In Python use numpy.arange(0.001, 1.0, 0.001) Choose an adequate prior and write a function for it, the prior on H is given by: H (0 -1/2H+3/8 (0.25 1/2H - 1/8 (0.5 -H+1 <H≤ 0.25 ) ≤H≤ 0.5 ) ≤H≤ 0.75 ) (0.75 ≤H≤ 1 ) use this prior and compare it to a prior from using the principal of indifference 10 P(H) = Construct the (normalized) posterior for the flips and write a function for it. Plot the posterior PDF of H2 and determine the most probable value of the bias weighting from the posterior Determine the best estimate and uncertainty of the bias weighting under the Gaussian assumption for the shape of the posterior (with uniform prior) Do this after 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, and 4096 flips. (Make a table of the most probable value of Q2, the best estimate, and the uncertainty as a function of the number of flips of Q3.) Does the Gaussian best estimate match the most probable value within the Gaussian uncertainty at each number of flips In Python generate 4096 "flips" of the coin using the binomial function of the random module in numpy: import numpy.random flips = numpy.random.binomial(1, bias, 4096) Where bias is the value of the bias weighting you chose. In python write a function for the likelihood function for the flips. You'll want to be able to compute the function for a set of bias weightings, a number of flips, and a number of heads in that number of flips. You must: 1. Compute the various probabilities as (natural) logarithms. 2. Normalize the binomial function 3. Use the approximation: In(N!) ~ N In N - N + ln(2+N)/2. 4. Sample the bias weighting parameter H rather finely: AH = 0.001 works well. (In Python use numpy.arange(0.001, 1.0, 0.001) Choose an adequate prior and write a function for it, the prior on H is given by: H (0 -1/2H+3/8 (0.25 1/2H - 1/8 (0.5 -H+1 <H≤ 0.25 ) ≤H≤ 0.5 ) ≤H≤ 0.75 ) (0.75 ≤H≤ 1 ) use this prior and compare it to a prior from using the principal of indifference 10 P(H) = Construct the (normalized) posterior for the flips and write a function for it. Plot the posterior PDF of H2 and determine the most probable value of the bias weighting from the posterior Determine the best estimate and uncertainty of the bias weighting under the Gaussian assumption for the shape of the posterior (with uniform prior) Do this after 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, and 4096 flips. (Make a table of the most probable value of Q2, the best estimate, and the uncertainty as a function of the number of flips of Q3.) Does the Gaussian best estimate match the most probable value within the Gaussian uncertainty at each number of flips

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