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posterior dis look like what using pi and x

p(pi|x) = f(x|pi)p(pi)

posterior odds

prior odds

whats the corresponding bayes factor

= P(H1|x)/P(H0|x)

= P(H1)/P(H0)

= BF10 = Posterior odds/prior odds

if we have

zi, a, b, pi and y then what is the joint posterior dis of all parameters

P(a,b,pi,z | y) proportional to

P(y, z | a, b, pi) P(z| pi) p(a) p(b) p(pi)

p(zi = 1| pi, a, b, yi) how can we find this using bayes theorm

= p(zi = 1|pi, a, b)p(yi| pi, a, b, zi = 1) /

SUM(1, j=0) p(zi = j|pi, a, b)p(yi|pi, a ,b , zi = j)

say this equals qi then how can we sample zi

from bernoulli dis with probability qi

yi |µ, σ2 ∼ N(µ, σ² ),

p(µ) ∝ 1 and σ² ∼ Gamma(α, β).

then full conditional posterior distribution

p(µ|σ 2 , y1, . . . , y11)====

p(µ) (∏n, i=1)f(yi|µ, σ² )

as we only care about mu

what is the steps for using metropolis hasting algorith for say σ²

Let σ^2t be the t-th iterate of σ² in the Metropolis-Hastings algorithm. For sampling σ² at step (t + 1) of the algorithm, we do the following:

• Propose a new sample σ^2* from a proposal density q(σ^2*| σ^2t)

• Accept σ^2* as the new sample with probability r

what would r be in the case where

r = p(σ^2* | µ, y) q(σ^2t |σ^2*) / p(σ^2t | µ, y) q(σ^2* |σ^2t)

Why would a prior for the regression coefficients in a model chosen be N(0, 100)

just a lucky guess that seems reasonable lol

In The matrix of the natural log Bayes Factors

how would you calculate the value in row m1 and collumn m2.

Thus what would be the value in row m2 and collumn m1

= log marginal likelihood of m1 - log marginal likelihood of m2 = alpha

thus the second is

= - log(BF12) = -(alpha) easyyyyy

In BF_ab how do you choose a and b

a is the row b is the collumn

how do you write log marginal likelihood of m2 in probabiloty

logP(y|m2)

how do you chose what models fits best to the data

highest marginal likelihood.

say model B, then compare this to second best model (model C) by looking as

exp(BF_BC)

if

exp(logBF₃₂) = BF₃₂ = 780 what does this mean

data are about 780 times more likely under Model 3 than under Model 2.

prefer model 3!

What should I conclude if effective sample sizes from an MCMC output are slightly below the total number of iterations? Should I worry about convergence?

If effective sample sizes are slightly below the total number of MCMC iterations (e.g. 7,000–9,000 out of 10,000), this typically does not indicate a problem. It may just reflect some autocorrelation in the chain.
However, if the effective size is very low (e.g. a few hundred), that might indicate poor mixing or lack of convergence, and you'd want to run longer chains, thin them, or re-diagnose.

✅ If sizes are moderately high (e.g. 7,500+), you're probably fine — but you could run longer for reassurance, especially for key parameters.

what is the general form of joint posterior dis

p(parameters∣data)∝p(data∣parameters)⋅p(parameters)

if we have joint posterior distribution

p(α, β, γ | u, v, w, x, y, z) ∝ exp {− 1/2 [ uα² + vβ² + wγ² − xαβ − yβγ − zαγ]}

Write down the posterior full conditional densities for α, β, γ needed, in order to sample from their joint posterior distribution using a Gibbs sampler. also how would you find them

p( α| u, v, w, x, y, z, β, γ),

p(β|u, v, w, x, y, z, α, γ) and

p( γ|u, v, w, x, y, z, α, β)

you would find them by taking the terms from the joint posterior density with only that term in it

we have two models

M1 and M2

what is BF12

in terms of maarginal likelihood

P(y|M1)/P(y|M2)

what is P(y|M)

marginal likellihood

p(y∣M)=∫p(y∣θ,M) p(θ∣M) dθ=  int( {product [f(yi|θ)]} p(θ) )

if we are intergrating the beta pdf so

∫ Beta(a,b) =

B(a,b)

B(a, b) = gamma(a)gamma(b)/gamma(a+ b), and gamma(a + 1) = a x gamma(a)

when can the marginal likelihood p(y∣M3) not be computed analytically

if the priors are not conjugate to the likelihood

how do we know when to do inverse gamma

if it looks like it follows a gamma

but our coef, say x is 1/x and not x, then its inverse gamma.

meaning if we have

(x)^{-(n+p)/2 - a - 1} …. as our pdf

then this looks like it follows

inv-gamma((n+p)/2 + a, …)

in the r package/code

MCMCregress(…..)

what are the default priors for

σ²

and

β

σ²∼Inverse-Gamma(ν₀/2, (v₀ x σ²₀)/2​​)

β ∼N(0​,σ²B0^-1)

if log marginal bayes factor is = 1 which model do we prefer

the simpler one

using the code bvs1$modelsprob

what does the last collumn this gives tell us

(prob)

and how do we choose a model/variables using this

gives a list of the posterior prob of each of the models given the observed data.

we choose the model with the highest posterior prob, and choose the variables in that model.

exp(θxi)^zi =

exp(θxizi)

how do you do you write down the complete data likelihood of a model

we do the sum of the f(y|…)

but plug in ANYTHING we know

so say we now have a zi function follows bernoulli(pi) of being 1 and 0 if not then multiply any pi^zi and (1-pi)^1-zi

same for f1 and f2

if logBF_ij

gives us a positive then

gives us a negative then

A positive log BF means model M_i​ is better than M_j
A negative log BF means model M_j​ is better than M_i

what does the BVS code do in r

does baysian variable selection, which checks all possible combinations of predictors and gives the posterior probability of each model.

say we had 5 predictors this would mean if we were to complete this mannually we would need to do 2⁵ BFs which would be longgggg init

Inclusion probability in R means???

is the chance that a variable is included in the best models, based on the data

BF =

posterior odds/ prior odds

cade pgamma(2, 0.1, 0.5) tells us what

λ≤ 2 given λ ~ Gamma(0.1,0.5)

Posterior Predictive Distribution:

p(y~​∣y)=

∫p(y~​∣θ)p(θ∣y)dθ

by definition of the gamma function:

∫^inf ​λ^a−1e^−bλdλ=

Γ(a)/b^a

importance sampling estimator equation

ɸ^ = 1/M ( sum(m, i=1)wi ɸ(xi))

what is wi =

f(xi) / g(xi) or h(xi)

how do you then find the maximum value of the impartance sampling weight wi

log(wi)

take deriv with respect to x

solve to find max x

plug this x back into our wi and thats it baby

in a MCMCregress code in R,

if we state that V = 1/10

and then that B0 = V

what exactly is this telling us

and how can we use this to find the priors on B

that the prior precision is 1/10

and that the priors on B are N(0, 1/B0)

= N(0, 10I) where I is the identity matrix

because the precision is the inverse of the variance, and it tells us how stongly we believe to prior for the regression coef B.

if there is no specification on the prior precision what do we set the priors of B to be

~ N(0, I)

in MCMCregression

what prior do we assume for σ²

~ inv_gamma(with default hyper parameter settings)

what happens when we set B0 = 0

means we have infinite variance, and know nothing about B and the prior is improper so we cant calculate marginal likelihood.

how do we choose which model is best using there log marginal likelihood

choose the model with the highest log marginal likelihood

how do we compare two models using the log bayes factor

say we have log(BF34) = 3.668

then BF34 = ex(3.668) = 39.17 which shows that model 3 is a significant improvment from model 4

if we have 3 parameters how many possible models do we have/ covariate combinations

2³ = 8

okay so say we have 3 covariates and we have done BF for 4 possible models, if this enough to choose the best model. if not what should we do to find the best one

no because we havent tested every possible combination of models.

instead we should carry out a bayesian variable selection procedure e.g BVS

what does BVS calculate

posterior probability of a model

we choose the model with the highest one.

if μ∼N(3,1)

then what are the prior odds of

odds:

P(μ≤3)/P(μ>3)

= 0.5 / 0.5 = 1

because the normal dis is symetric around it mean then it have the same probability of being greater or being less then 3.

how do you perform a BF using hyposthesis

BF10 = Posterior odds / prior odds =

P(H1|data)/P(H0|data) / P(H1)/P(H0)

if very very large we favour H1

if we know µ|Y = 6 ~ Normal(5, 1/3).

then how do we calculate

P(µ ≤ 3|Y = 6)

Standardize so

𝐳 = 3-5/sqrt(1/3)

then find P(Z ≤ 𝐳 )

potential scale reduction factor

PSRF > 1 means what 2 things

between chain variance is larger than within chain variance

lack of convergence, i.e., chains are still over-dispersed and haven’t mixed well.

At the end of a run of the Accept-Reject algorithm, we get a set of

independent draws from the target density.

what are the constaints that must hold for candidate density g(x)

if we aim to use it to sample from more complex dis f(x)

f and g have compatible supports (e.g if f(x) >0 then g(x)>0)

there is a constant M so that f(x)/g(x) <= M for all X