on 27-Nov-2019 (Wed)

The FluSight challenge focuses on forecasts of the weighted percentage of doctor’s office visits where the patient showed symptoms of an ILI in a particular US Health and Human Services (HHS) region.

Equation 5 decomposes logit(π j,t ) into an SIR model component logit(I j,t ), a discrep- ancy component common to all flu seasons μ t , and a discrepancy component specific to each flu season δ j,t .

We assign an informative prior to I j,0 , β j and ρ j via empirical Bayes by fitting a multivari- ate Gaussian distribution to the parameter estimates of the fitted SIR models shown in Figure 4 and truncating to respect known and/or assumed boundary constraints.

We specify the μ_t process as a reverse-random walk.

The specific hyperparameters, however, were chosen somewhat by trial and error based on the visual plausibility of realizations drawn from the prior predictive distribution. Though our choices yielded good forecasts (see Section 5.3), hyperparameter selection might be better optimized via cross-validation over the training data.

We impose temporal structure on δ j,t and encourage it to be close to zero for all t = T by modeling δ j,t with the following autoregressive, reverse random-walk

The mean in (12) reflects our prior belief that the constant determining how much temporal structure is kept (where 1 is a full random walk and 0 is no temporal structure) is near 0.9.

“[Statistical] algorithmic invention is a free-wheeling and adventurous enterprise, with inference playing catch-up as it strives to assess the accuracy, good or bad, of some hot new algorithmic methodology.” [Efron and Hastie, 2016, pxvi]

There are no statistical models that don’t originate inside our minds. So there is no arbiter to determine the “true” statistical model for (X, Y ): we may expect to disagree about the statistical model for (X, Y ), between ourselves, and even within ourselves from one time- point to another.

Parametric model: model a random variable X as the triple E = {X, Θ, f_X(x |θ)} where only the finite dimensional parameter θ ∈ Θ is unknown.

Definition 2 (Statistic; estimator) Any function of a random variable X is termed a statistic. If T is a statistic then T = t(X) is a random variable and t = t(x) the corresponding value of the random variable when X = x. In general, T is a vector. A statistic designed to estimate θ is termed an estimator.

Typically, estimators can be divided into two types: point estimators and set estimators.

• A point estimator which maps from the sample space X to a point in the parameter space Θ.
• A set estimator which maps from X to a set in Θ.

Likelihood: The likelihood for θ given observations x is L_X(θ; x) = f_X(x |θ), θ ∈ Θ regarded as a function of θ for fixed x.

If we are interested in inferences about θ = (α, β) following the observation of the data, then it seems reasonable that these should be invariant to the choice of measurement scale: it should not matter whether x or y was recorded.

Cox and Hinkley (1974; p12) observe, if we are interested in comparing two possible values of θ using the likelihood, then we should consider the ratio of the likelihoods rather than, for example, the difference

The weak likelihood principle: If X = x and X = y are two observations for the experiment E_X = {X, Θ, f_X(x |θ)} such that L_X(θ; y) = c(x, y)L_X(θ; x) for all θ ∈ Θ then the inference about θ should be the same irrespective of whether X = x or X = y was observed.

The strong likelihood principle: Let E_X and E_Y be two experiments which have the same parameter θ. If X = x and Y = y are two observations such that L_Y (θ; y) = c(x, y)L_X(θ; x) for all θ ∈ Θ then the inference about θ should be the same irrespective of whether X = x or Y = y was observed.

Sufficient statistic: A statistic S = s(X) is sufficient for θ if the conditional distribution of X, given the value of s(X) (and θ), f_X|S(x | s, θ), does not depend upon θ.

Note that, in general, a sufficient statistic S is a vector and that if S is sufficient then so is any one-to-one function of S.

A companhia ou sociedade anônima terá o capital dividido em ações, e a responsabilidade dos sócios ou acionistas será limitada ao preço de emissão das ações subscritas ou adquiridas.

Art. 3º A sociedade será designada por denominação acompanhada das expressões "companhia" ou "sociedade anônima", expressas por extenso ou abreviadamente mas vedada a utilização da primeira ao final.