Give a detail calculation step of the amount of information that N gives about p as: H(p)-H(p|N)

Flipping a coin once, where P(Heads) = p. We calculate Fisher information for this experiment is FI(X) = px(1p) where X = (S, P) is the prob- ability space. S = {Heads, Tails}, P(Heads) =p, P(Tails) = 1 p. The result using entropy instead of Fisher information: Supposedly, FI(X) is the amount of information about p revealed by measuring N= the numbers of Heads in one experiment of flipping a coin. By using entropy, p should be a random variable. Suppose the possible values of p are only and 2. Then we will think of our exper- iment as having two parts: first, choose p; then flip the coin to get Heads or Tails. New probability space: possible outcomes are a = (, Heads), b = (}, Tails), c = (1 , Heads), d (2, Tails). Let say P(p }) = { and P(p = 3) = 1. Then P( Heads (p = j) = j,P( Tails (p = {) = {, P( Heads (p = 1) = 1,P( Tails p 2) = 1. The event Tails in our new sample space: Tails = {b,d}. And "Heads P({a,c} intersect {a,b} P({a}) = {a,c}. P( Heads (p= }) = P(Heads intersect p= }) P=(p=3) P(p=}) P(p=3) Plas. So P(a) = { x = 5. The amount of information that N gives about p is = = = Flipping a coin once, where P(Heads) = p. We calculate Fisher information for this experiment is FI(X) = px(1p) where X = (S, P) is the prob- ability space. S = {Heads, Tails}, P(Heads) =p, P(Tails) = 1 p. The result using entropy instead of Fisher information: Supposedly, FI(X) is the amount of information about p revealed by measuring N= the numbers of Heads in one experiment of flipping a coin. By using entropy, p should be a random variable. Suppose the possible values of p are only and 2. Then we will think of our exper- iment as having two parts: first, choose p; then flip the coin to get Heads or Tails. New probability space: possible outcomes are a = (, Heads), b = (}, Tails), c = (1 , Heads), d (2, Tails). Let say P(p }) = { and P(p = 3) = 1. Then P( Heads (p = j) = j,P( Tails (p = {) = {, P( Heads (p = 1) = 1,P( Tails p 2) = 1. The event Tails in our new sample space: Tails = {b,d}. And "Heads P({a,c} intersect {a,b} P({a}) = {a,c}. P( Heads (p= }) = P(Heads intersect p= }) P=(p=3) P(p=}) P(p=3) Plas. So P(a) = { x = 5. The amount of information that N gives about p is = = =