Question

(1) The posterior density for β is given by [\pi(\ beta ∣y)\ propto \ prod__{i=1}∧{n}N\left(y_{i}\midx−​{i} \ beta, \ sigma ∧{2}\ right) \ cdot \ pi (\ beta )=0.51\ cdot 1\ left( \ beta =0\ right) +0.5\ cdot \ prod_\{i =1}∧{n} (3) The posterior density for β is proportional to: 0.51\ cdot 1(\beta=0)+0.5\cdot\prod_{i=1}∧{n}N\left(y_{i}\midx_{i}\beta, \sigma∧{2}\ right) N(\beta \mid0,\tau∧{2})] Since the marginal likelihood can be written as The posterior density for β can be written as: m\ left(y_\{1\}, \Idots, y_{n}\right) N\left(\beta\mid\\u,\psi∧{2}\right)] where v and ψ are given by the mode of the normal distribution: [\ nu =\ frac \( \left.\{\backslash \text { sum_\{i }=1\}^{\wedge}\{n\} x_{-}\{i\} y_{-}\{i\}\right\} \backslash \backslash \operatorname{sigma}^{\wedge}\{2\}+\mid \operatorname{tau}^{\wedge}\{2\} \backslash \) sum_\{i=1 }∧{n}x−​{i}}∧{2}}] [\pi∧{2}=\frac{\sigma∧{2}}\\sigma∧{2}+\tau∧{2}\ sum__ {i=1}∧{n}x−​{i}∧{2}}] (4) The posterior probability of β=0 given the data is given by: 1(\beta=0)}{m(y−​{1},\dots,y−​{n})}] (5) The term represents the Bayes factor which measures the relative strength of evidence for two models. Here, the two models being compared are the model with a non-zero mean for β (i.e., the posterior density for β ) and the model with a mean of zero for β (i.e., the prior density for β ). As the squared residuals \( [\backslash \text { sum_\{i }=1\}^{\wedge}\{n\} \backslash \) left(y_\{i }−0\ right )∧{2}] grow, the term increases, providing more evidence for the model with a non-zero mean for β. As , this term approaches infinity, providing increasingly strong evidence for the model with a non-zero mean for β.

Answer 1

The posterior density for β provides information about the probability of β being a particular value based on the data. The posterior probability of β=0 given the data can be calculated using the prior probability of β being 0 and the likelihood of the data given β. As the squared residuals increase, the evidence for a non-zero mean for β becomes stronger.

Step-by-step explanation:

The posterior density for β gives us information about the probability of β being a certain value after considering the data. It is represented by the equation:

[π(β|y) ∝ Π_{i=1}^{n} N(y_i|x_iβ, σ^2) ⋅ π(β) = 0.51 ⋅ 1(β=0) + 0.5 ⋅ Π_{i=1}^{n} N(y_i|x_iβ, σ^2) N(β|0,τ^2)]

To calculate the posterior probability of β=0 given the data, we use the equation:

[P(β=0|y) = P(β=0)/m(y_1, ..., y_n)]

The term 0.51 ⋅ 1(β=0) represents the prior probability of β being 0, and the term Π_{i=1}^{n} N(y_i|x_iβ, σ^2) represents the likelihood of the data given β. As the squared residuals (Σ_{i=1}^{n} (y_i−0)^2) increase, the posterior probability of β=0 decreases, providing more evidence for the model with a non-zero mean for β.