Inference for point processes (estimation of any type) Flashcards
(18 cards)
How is the Importance Weight Distribution given?
w_{θ,θ_0,n}(Y_m) = (h_θ(Y_m) / h_θ_0(Y_m)) / ∑_{i=0}^{n-1} h_θ(Y_i) / h_θ_0(Y_i), m= 0, . . . ,n−1,
What is the Importance sampling Formula?
𝔼_θ[k(X)] = 𝔼_θ_0[k(X)h_θ(X) / h_θ_0(X)] / (c_θ / c_θ_0)
How does a Toroidal Approximation work?
S ⊃ W is chosen to be a rectangle wrapped around a Torus, such that points at opposite edges are considered to be neighbours.
Conditional Approach
A smaller window is chosen, and the log likelihood is then found by conditioning on the points outside the new window.
The missing data problem is then avoided.
Newton-Raphson
For a starting value θ_0 = ˆθ^{(0)}, the Newton-Raphson iterations for maximising the log likelihood are given byˆθ^{(m+1)} = ˆθ^{(m)} + u( ˆθ^{(m)} ) j(ˆθ^{(m)} )^{−1}, m = 0, 1,
...
Similar iterations for maximising the approximate log likelihood are given byˆθ^{(m+1)} = ˆθ^{(m)} + u_{θ_0 , n} (ˆθ^{(m)} ) j_{θ_0 ,n} (ˆθ^{(m)} )^{−1}, m = 0, 1, ...
where an MCMC sample with importance sampling parameter θ_0 is used.
What is an MLE
It is the unique maximum, θ_(hat), of the log likelihood, l(θ)
It is a solution to u(θ)=0
Exponential family density
For x ∈ N_f and θ ∈ Θ, θ(x) = b(x) exp( θ · t(x) ) / c_θ,
where b : N_f → [0, ∞) and t : N_f → R^p are functions and · is the usual inner product.
Define the Stauss process in terms of the exponential family density
A Strauss process with fixed interaction range R > 0 is an exponential family model with b = 1, θ = (θ1, θ2) = (log β, log γ), t(x) = (n(x), s_R(x)), and Θ = R × (−∞, 0].
Normalising Constant
It is hard to calculate explicitly, therefore we try to use inference to solve for it.
(minimal) canonical sufficient statistic
The canonical sufficient statistic is defined as
V_θ (X) = log h_θ (x) / dθ = t(X)
If the density is identifiable, it is the minimal canonical sufficient statistic.
Importance Sampling Approximation of the Log Likelihood Ratio
For fixed θ_0 ∈ Θ,l_{θ_0, n(θ)} = log(h_θ (x) / h_{θ_0} (x)) − log 1 / n ∑_{m=0}^{n-1} h_θ (Y_m) / h_{θ_0} (Y_m)
Free Boundary Log Likelihood
For Y ∼ Poisson( ˜S, 1) and c_{θ,˜S} (∅) is the normalising constant for h_{θ,˜S} (·|∅), then
log E_θ [f_{θ, W} (x|X_{∂W} )|X_{∂˜S} = ∅] = log [ E[h_{θ,˜S} (x ∪ Y_{˜S\W} |∅)] ] − log [ c_{θ,˜S} (∅)]
Missing Data Log Likelihood
If W ⊂ S, the log likelihood function is given by the more complicated expressionl_{mis} (θ) = log E f_θ(x ∪ Y_{S\W} ) = log E h_θ (x ∪ Y_{S\W}) − log c_θ
where Y ∼ Poisson(S, 1).
However, as E h_θ (x ∪ Y_{S\W}) can not generally be calculated explicitly, we have a missing data problem.
Missing Data Approaches
It is a method for handling edge effects for Markov point processes.
Identifiability
f_θ ≠ f_θ˜ for different θ, θ˜ ∈ Θ
It is often required in the context of Markov point process densities (exponential families).
Which parameter is called the Importance Sampling Parameter?
θ_0
What is the Importance Sampling Estimator of 𝔼_θ[k(X)]
𝔼_{θ,θ_0,n}[k] = ∑_{m=0}^{n-1} k(Y_m) w_{θ,θ_0,n}(Y_m)How is the Importance Sampling Approximation given?
E_θ[k(X)] ~= ∑_{m=1}^{n} k(X_m) w_{θ,θ_0,n}(X_m),
where X_m is generated from f_{θ_0}
