Filtering, Smoothing, and Forecasting in dynamic linear models38

previous state. The filtering distribution is π(𝜃_𝑡|𝑦_1:𝑡). And π(𝜃_𝑡−𝑘|𝐷_𝑡), 𝑓𝑜𝑟 𝑘 ≥ 1 is called the k-step time t filtered distribution for the state vector. Smoothing is a

39

retrospective reconstruction of the time series system using the filtered distribution.

Forecasting is to compute the value or the state of the future based on the filtering density of time t state, θ_𝒕. π(𝑦_𝑡+1|𝑦_1:𝑡)is the one-step-ahead predictive density.

2.4.10 Generalized autoregressive moving-average models

The problems of non-Gaussian time series is encountered in many research fields such as biology, epidemiology, and medical studies, in that the observations are likely to be positive and sometimes are integers or counts data. Time series models for counts have been used by several authors ⁶⁸. The extension of the normal distributed time series to the suitable distribution is reasonable. Poisson autoregressive model is therefore considered.

Hsu et al. applied generalized Poisson autoregressive model on the Japanese encephalitis to evaluate the secular time trend, seasonal variation pattern, autoregressive order of Japanese encephalitis incident cases and temporal relationship of lagged temperature and precipitation¹³. They also adjusted vaccination rate, pig

density, and geographic area variation. The model form is written as follows.

log (𝐸(𝑦_𝑡|𝑦_𝑡−1, … 𝑦_𝑡−𝑙, 𝑥₁, 𝑥₂, 𝑥₃, … , 𝑥_𝑝, 𝑡, 𝑚₁, … , 𝑚₁₁)) = ln(𝑃𝑌) + 𝛼 + ∑^𝑛_𝑘=1𝑣_𝑘𝑇𝑀_𝑡−𝑘+ ∑^𝑚_𝑖=1𝜋_𝑖𝑅𝑎_𝑡−𝑖+ 𝛿₁𝑡 + 𝛼₃+ 𝛽₁𝑥₁+ ⋯ + 𝛽_𝑝𝑥_𝑝+

(2.41)

40 𝜔₁𝑚₁+ ⋯ + 𝜔₁₁𝑚₁₁+ ∑^𝑙_𝑗=1𝛾_𝑗𝑦_𝑡−𝑗

Where 𝑦_𝑡 is the number of Japanese encephalitis cases by month at time 𝑡.

𝑥₁, … , 𝑥_𝑝 are covariates such as pig density and geographic area. 𝑚₁, … , 𝑚₁₁ are

dummy seasonal variables. 𝑦_𝑡−𝑗 are autoregressive orders and ln(𝑃𝑌) = ln(Person − Years) is the offset. ∑^𝑛_𝑘=1𝑣_𝑘𝑇𝑀_𝑡−𝑘+ ∑^𝑚_𝑖=1𝜋_𝑖𝑅𝑎_𝑡−𝑖 is the

combinations of lagged temperature and precipitation. 𝑣_𝑘 , 𝜋_𝑖 , 𝛿₁ , 𝛽₁, … , 𝛽_𝑝, 𝜔₁, … , 𝜔₁₁are the corresponding regression coefficients ¹³.

The proposed model, albeit it has been extended form Gaussian data to non-Gaussian data, is the failure of taking into account the uncertainty of parameters, and correlated property and heterogeneity property due to hierarchical data.

Due to the limitation of the application of previous models for the infection data, we developed a novel Bayesian generalized linear mixed ARIMA model and applied it to HAIs.

41 Chapter 3 Materials and Methods

3.1 Setting

A cohort of healthcare-associated infections was followed during the period of January 1, 1994 and December 31, 2011 in an urban tertiary medical center in northern Taipei with 921-bed and approximately 27,000 inpatient admission annually.

3.2 Patient enrollment and Definition

Patients who fulfilled the criteria of healthcare-associated infections (HAIs) were eligible for this HAIs cohort since 1994. All confirmed cases and related factors were prospectively collected in the database.

Hospital information system

The nationwide health insurance has been launched since 1-March 1995 in Taiwan ⁶⁹. Before this significant change in medical system, the hospital information system, including inpatient, outpatient, emergency, prescription, bio-laboratory, etc., was established for automatic insurance reimbursement and payment. Besides those systems, the hospital-associated infection registry system also was separately set up to systematically collect the HAIs for quality control of hospital care which was

independently from insurance payment. But, those could link with admission of

42

hospitalized status and laboratory examination using chart number linkage. Therefore, the all hospitalized admission and infection status could be completed accessed since 1994. The database contains information about every admitted patient, includes, date of admission, date of discharge, and date of transfer, ward of admission, physician attending service, operation information if available, discharge condition, and the International Classification of Diseases (Ninth Revision) Clinical Modification (ICD-9-CM) codes of diagnoses, admission types (from emergency or outpatient

department),

Hospital-associated infection registry system

The infection control team established a nosocomial infection registry system since 1993. Ever since then, a central infection committee has coordinated the infection control and confirmed enrolled cases every week. HAIs were defined according to the U.S. Centers for Disease Control and Prevention standards. The U.S.

CDC defined an HAI as a localized or systemic condition resulting from an adverse reaction to the presence of an infectious agents or its toxins⁷⁰. HAIs were classified as urinary tract infection (UTI), surgical site infection (SSI), pneumonia (PNEU), bloodstream (Bacteremia), skin and soft tissue (SST), gastrointestinal system (GI), eye, ear, nose, throat, or moth (EENT), and other (central nervous, reproductive tract,

43

bone and joint, and cardiovascular system infections) ^70,71. The database contains comprehensive information about age, gender, infection sites, culture sampled date, indwelling bladder catheter, central venous catheter, admitted medical departments, and numbers of HAIs episodes during each admission. This registry system are used for clinical infectious diseases studies ^72,73.

Incident cases: Incident cases were those free of HAIs hospitalized patients attacked

by a new episode of HAI at least 48 hours after their admission in each month.

HAI episode: A HAI episode was defined as a new infection acquired in the hospital

documented after at least 48 hours admission in the hospital.

HAI-related 30-day death: For the incident cases who died within 30 days and

caused by the HAI were defined.

Patient-days: It was calculated since admission entrance and ended by the first

infection occur. The patient-days of non-infectious patients was reckoned the duration of hospitalization.

Incidence rate: The incidence rate was expressed as number of infection episodes per

1,000 patient-days.

Death rate: Death rate was the number of deaths due to HAIs per 100 HAIs patients.

44

Mortality rate: Mortality rate was calculated as the death cases per 1,000

patient-days which was also the product of the incidence rate and the death rate.

Culture: Microbiological specimens were collected as recommended by the CDC.

Thresholds of positive culture was defined according to the CDC for different infection sites^70,71.

3.3 Study design

This study was designed as an incidence-death follow-up cohort. The current study focused on the incidence part of the cohort (Figure 1). Those incident cases were followed up for 30 days, and those who died within 30 days and caused by the HAI episode were defined as HAIs related 30-day death. Since the seasonal variations exist in hospital admissions ¹⁷, this study use disease incidence as a measurement unit instead of counts.

During the study period of twenty-years, the hospital policy-maker conducted infection control interventions. The intervention programs of the hospital were Plan-Do-Check-Act (PDCA) program, Hygiene Programs, Centers for Disease Control, R.O.C. (Taiwan) (CDC) Hand-hygiene project and the urinary tract infection Quality Improvement Program of Taiwan Joint Commission on Hospital Accreditation

45

(TJCHA), and bundle care program were conducted by the hospital infection control team. The details of each intervention were described as follows.

PDCA program: The hospital infection control decision-makers initiated a program

using Edward Deming’s Plan-Do-Check-Act (PDCA) cycle-combined monitoring for

each warning outbreak of HAI incidence since 2005. It was a warning-feedback system that the infection control team monitored the infection rate of wards. If the infection rate was above the 95% confidence interval, the ward-based unit should launch the act to improve the infection rate.

Hygiene programs: They were the combined facility improvement and continuing

education programs. The infection control team both introduced an alcohol-based hand rub disinfection site for each ICU bed and set hand hygiene stands at ward entrances since 2007. They also set a red warning line to separate the ICU patients care from the healthcare member paperwork station.

CDC/TJCHA project: Centers for Disease Control, R.O.C. (Taiwan) (Taiwan CDC)

and Taiwan Joint Commission on Hospital Accreditation (TJCHA) together setup the standards for the hospital healthcare-associated infection control audit since 2008.

The Minister of Health Dr. Yeh approved Taiwan CDC to initiate National Hand Hygiene Campaign in May, 2009. The goals of the National Hand Hygiene Campaign

46

were to form the culture of patient safety and to encourage hospitals to improve continually by evaluating performance indicators regularly. Shin Kong Wu Ho-Su Memorial Hospital participated the auditing program since 2010. In order to meet the standards of the auditing, the hospital increased the numbers of hand-hygiene stands in the ward, rewarded the healthcare staffs if they reached the points of infection control, and launched serial education classes. We also joined the Catheter-related Urinary Tract Infection Quality Improvement Project conducted by TJCHA. The catheter use rate and the indwelling time were monitored in the intensive care units.

Bundle care program: It was a multidisciplinary approach to decrease the

catheter-related infection in intensive care unit since July, 2011.

3.4 Covariates

All confirmed cases and related factors were prospectively collected in the database.

Patient characteristics: age, gender, date of birth, admission, diagnosis of HAI,

discharge, transfer to ICU, and death

Infection-related covariates: infection sites, cultures sampled date, indwelling

47

bladder catheter and central venous catheter, microorganism cultures

Disease conditions: the International Classification of Diseases (Ninth Revision)

Clinical Modification (ICD-9-CM) codes of diagnoses, admission types (from emergency or outpatient department), admitted medical departments, numbers of HAIs episodes during each admission, and cause of death.

Intervention programs indicators: Intervention of PDCA, Hygiene programs,

Taiwan Centers for Disease Control (CDC) National Hand Hygiene Campaign and the urinary tract infection quality improvement program of Taiwan Joint Commission on Hospital Accreditation (TJCHA) called CDC/TJCHA, and Bundle care program.

48 Chapter 4 Model specification

Generalized linear time series models were used for analysis of HAIs incidence. Moreover, Bayesian state-space model was used for evaluating the time-dependent incidences and for forecasting future incidences in HAIs. Both were delineated as follows.

4.1 Generalized linear time series model

To accommodate various distributions of the outcome of interest covering from continuous to discrete types, a generalized linear time-series model following Zeger et al⁷⁴ study and applied to Japanese encephalitis by Hsu et al¹³ was proposed in my thesis (Figure 4.1.1, Figure 4.1.2).

Under the context of GLM, three components are often defined, including (1) Random component

It is the distribution of the outcome Y and expressed by E(Y) = μ (2) Link function

It is a function defined by h(μ), which can take varies kinds of distributions, mainly including identity function for normal distribution, logistic function for binomial distribution, logarithm function for Poisson distribution and so on.

(3) Systemic components

This part includes the major components covered in time-series analysis such as

49

seasonal variation, time trend, the cyclic change, and autoregressive orders. The

generalized form is written as follows.

h(μ) = η (4-1)

η represents systematic components. If the outcome of Y is specified by a

Poisson distribution, the equation (4-1) is expressed by

log 𝜇 = ln(𝑃𝑌) + 𝛽₀+ 𝛽₁𝑋₁+𝛽₂𝑋₂+ ⋯ + 𝛽_𝑝𝑋_𝑝+ 𝑔(𝑡) + 𝑠(𝑡) +

∑^𝑙_𝑗=1𝑟_𝑗𝑦_𝑡−𝑗 (4-2)

𝑋₁, . . , 𝑋_𝑝 represent covariates such as age and gender.

𝑔(𝑡) is a polynomial function of time trend.

𝑠(𝑡) is a function of seasonality. Trigonometric function is one of choices. Here,

we used dummy variables to denote spring, summer, autumn, and winter denoted by 𝑠₁− 𝑠₃. Trigonometric function is used in Bayesian approaches for

comparison.

50

4.2 Decomposition method with generalized linear time-series model

We extended the time series to the Poisson seasonal model in order to have a better fit with the HAIs incidence data. The algorithm consists of four steps. Firstly, the incidence was checked for its seasonality and de-seasonalized if any. Secondly, the residual of the first step was used for trend detection. De-trend procedure was applied if trend existed. Third, cycling was checked if any. Finally, the left residual of the above three steps was checked for discernible patterns or autocorrelations.

The algorithm is written as follows:

Step1.

log (𝐸(𝑦_𝑡|𝑥₁, 𝑥₂, 𝑥₃, … , 𝑥_𝑝, 𝑠₁, 𝑠₂, 𝑠₃, 𝑠₄, 𝑡))

= ln(𝑃𝐷) + 𝛼₂+ 𝜔₁𝑠₁+ 𝜔₂𝑠₂+ 𝜔₃𝑠₃+ 𝜔₄𝑠₄ + 𝜀₁

(4-3)

Step2.

𝐸(𝜀₁|𝑡) = 𝛼₂+ 𝛿₁(𝑡 − 𝑡̅) + 𝛿₂(𝑡 − 𝑡̅)² + 𝛿₃(𝑡 − 𝑡̅)³+ 𝜀₂ (4-4)

Step3.

𝐸(𝜀₂|𝑐) = 𝛼₃+ 𝛽₁𝑥₁+ ⋯ + 𝛽_𝑝𝑥_𝑝+ 𝜀₃ (4-5)

51 Step4.

Residual plots for 𝜀₃ to check

(a) Autocorrelations: autoregressive order, sample autocorrelation function plots (SAC) using ARIMA method, Ljung-Box statistics.

(b) Normality: Normal plot

4.3 Bayesian Dynamic linear models (DLM)

We are not only interested in the population microbiology on understanding temporal fluctuations in abundance of specific microorganisms, but also in the impact on its relation to the diseases. For instance, as the antibiotics-resistant microorganism grows rapidly, it may increase the possibility to invade the hosts, causes diseases, and the incidence of diseases will increase. However, this is the simplified model for microorganism-host interactions. There are still several interactions between microorganisms, environment, and hosts. Factors involving in the interactions make the interpretation of incidence-microorganism abundance status a complex phenomenon. Crucially, our understanding of disease incidence is also affected by errors in the observation process. Separating true microorganism abundance (biological signals) from observation error in data is of great importance.

52

In this study, we aimed at evaluating the population dynamics of HAIs

microorganisms and its relationships with disease incidence. The disease incidence survey of HAIs time series sometimes include errors, and yet most previous studies have not explicitly been able to account for errors. Therefore, analysis that can manage the source of errors are needed in disease control.

Following the dynamic linear model mentioned in the literature in Chapter 2, we have used state-space model based on the equation (2-21) and (2-22).

To simplify the notation, we use state-space model with the form expressed by

𝑦_𝑡 = 𝑋_𝑡^𝑇𝛽 + 𝑒₁^𝑇𝐶_𝑡 (4-6)

, which is so-called observation equation, and

𝐶_𝑡 = 𝐺𝐶_𝑡−1+ 𝑒₁𝑣_𝑡 (4-7)

, which is so-called state equation. 𝑒₁is a px1 vector with a one in the first row and zeros elsewhere. Note that 𝑌_𝑡is the observed number of disease, 𝐶_𝑡 is the true pathogen state. 𝑌_𝑝is expressed as

𝑌_𝑝 = 𝑋_𝑝𝛽 + 𝐶_𝑝 (4-8)

53

Σ_𝑝can be written in terms of ϕ (autocorrelation) with vector form like the

following

Vec(Σ_𝑝) = Vec(GΣ_𝑝𝐺^𝑇) + Vec(𝑒₁𝑒₁^𝑇)

= (G ⊗ G)Vec(Σ_𝑝) + Vec(𝑒₁𝑒₁^𝑇)

= [I − (G ⊗ G)]⁻¹Vec(𝑒₁𝑒₁^𝑇) (4-10)

The joint posterior distribution of the parameters is

P(β, ϕ, 𝜎_𝑣²) ∝ (1

54 The derivation of P(𝜎_𝑣²|y, β, ϕ) is straightforward by assuming that the prior distribution for _𝜎¹

The conditional posterior distribution of β and 𝜎_𝑣² can be simulated by Gibbs sampling scheme.

To reflect the dependence of Σ_𝑝 on ϕ with Σ_𝑝(ϕ), let 𝑦_𝑡^∗ = 𝑦_𝑡− 𝑋_𝑡^𝑇𝛽

𝑦_𝑡^∗ = 𝜙₁𝑦_𝑡−1^∗ + ⋯ + 𝜙_𝑝𝑦_𝑡−𝑝^∗ + 𝑣_𝑡

55

Where 𝑆_𝜙 is the region in which the process is stationary. Note that this distribution is a non-standard distribution but it may be sampled with an Metropolis-Hasting algorithm after specifying the prior distribution of 𝜙like 𝜙~N(𝜙₀, Φ₀). A proposal

candidate density may be obtained by multiplying 𝑃(𝜙)by the terms involving 𝑦_𝑡^∗, 𝑡 > 𝑝 with the distribution specified by 𝜙~N(𝜙̂, Φ̂ ).

56

stationary region, which is subject to the usual Metropolis-Hasting acceptance region.

The state-space form consists of the observation equation when regression

coefficients evolve randomly through time following the equations (4-6) and (4-7).

𝑦_𝑡 = 𝑋_𝑡^𝑇𝛽_𝑡+ 𝑣_𝑡 (4-17)

and the transition equation

𝛽_𝑡 = 𝛽_𝑡−1+ 𝑒_𝑡 (4-18)

where t = 1, … , T, 𝑦_𝑡 and 𝑣_𝑡 are scalars, 𝑋_𝑡, 𝛽_𝑡, and 𝑒_𝑡 are k x 1 vectors,

𝑣_𝑡~𝑁(0, 𝜎²), and 𝑒_𝑡~𝑁_𝑘(0, Σ). 𝑣_𝑡 and 𝑒_𝑡 are assumed to be independent with each

other through time.

To generalize this model as a vector autoregression, this model is defined as Y = (𝑦_𝑇, 𝑦_𝑇−1, … , 𝑦₁)^𝑇

This likelihood function is

𝑓(𝑦|𝛽, 𝜎²) ∝ 1 𝜎²

𝑇2

exp [− 1

2𝜎²(𝑦 − X𝛽)^𝑇(𝑦 − X𝛽)] (4-19)

57

The prior distribution based on the transition equation for β can be defined by the kT x kT matrix F

The priors for the remaining parameters are adopted with standard forms such as

1

𝜎²~Gamma (^𝛼₂⁰,^𝛿₂⁰) and Σ⁻¹~𝑊_𝑘(𝑢₀, 𝑠₀).

The joint posterior distribution is P(β, 𝜎², Σ|y) ∝_𝜎¹₂

The following conditional posterior distributions can be sampled with a Gibbs

algorithm.

58

4.4 Bayesian Generalized Time-series Model

We can also combine observation equation with state-space equation to develop a unified framework for the extension of DLM model to generalized linear time-series model with Bayesian underpinning.

59

4.4.1 Bayesian Generalized Autoregressive Poisson Regression Model

Under the context of generalized linear model together with autoregressive model proposed in Chapter 2, a p-order autoregressive Poisson regression model is proposed to include autoregressive order, two classical components of time-series model (time trend and seasonal variation),

log μ_t = β₀+ β₁S₁+ β₂S₂+ β₃S₃+ β₄f(t) + β₅X + ϕ₁y_t−1+ ϕ₂y_t−2 + ⋯ + ϕ_βy_t−p+ u_t

(4-23)

μ_t = E(y_t): The expected count of HAIs

f(t) represents the p polynomial time trend function

X includes age, gender, intervention, department and site of infection

The Doodle of Bayesian Directed Acyclic Graphic Model (DAG)

The proposed model is framework in the doodle of Bayesian directed acyclic graphic model (DAG).

60 Doodle

f(t) represents the p polynomial time trend function

X includes age, gender, intervention, department and site of infection

γ(t) is the observed count of HAI at time t. It is determined by μ(t), expected numbers of counts, with Poisson distribution (μ_t). The parent node of γ_t is the μ_t. μ_t is linked through a logical expression governed by the equation(4-23). β_s and ϕ_s are the corresponding regression coefficients.

The posterior distribution of P(θ|y) is formed in proportion to P(θ), prior distribution, and the likelihood function ℓ(y|θ), where θ = (β₁− β_k, ϕ₁− ϕ_p) .

61

4.4.2 Bayesian Generalized Moving Average Model

In a similar vein, a q-order moving average Poisson regression model is proposed with the model form like the following,

log( μ_t) = β₀+ β₁S₁+ β₂S₂+ β₃S₃+ β₄f(t) + β₅X + u_t− θ₁u_t−1

− θ₂u_t−2− ⋯ − θ_pu_t−p

(4-24)

The doodle of Bayesian DAG model and also the posterior distribution are derived in a similar manner as done for Bayesian AR process.

4.4.3 Bayesian Generalized ARMA Poisson Regression Model Fixed –effect Model correlated property, partly because of hierarchical (multilevel) data and partly because of the spread of transmission infection disease. To cope with this correlated property, we resort to the adoption of random-effect approach expressed as a linear mixed model and used in a longitudinal follow-up study.

The model form is modified from the equation (4-24) and expressed as follows.

62

logit μ_t = β_0k+ β₁S₁+ β₂S₂+ β₃S₃+ β₄f(t) + β₅X + ϕ₁y_t−1+ ϕ₂y_t−2 + ⋯ + ϕ_βy_t−pu_t− θ₁u_t−1− θ₂u_t−2− ⋯ − θ_pu_t−p

(4-26)

β_0k~Ν(0, σ_k²)

k represent department (hierarchy) or site of infection.

This model is often so-called random-intercept model in a classical mixed model.

Doodle

63 4.5 Estimation of parameters

There are two main methods to estimate the parameters of generalized time series model and dynamic linear model, including maximum likelihood method and Bayesian approach.

4.5.1 Maximum Likelihood Estimate (MLEs)

We can specify the probability distribution of the data given the states and parameters for the observation equation and specify the probability distribution of the state conditional on the previous state in the previous time point in the process

equation. The maximum likelihood estimates (MLE) of the parameters can be obtained provided data and initial state available. However, in some situations, in nonlinear and non-Gaussian distributions, MLE is not easily feasible.

4.5.2 Bayesian Markov chain Monte Carlo methods

In Bayesian inference, the posterior distributions of the parameters are often analytically intractable in that it is difficult to derive in closed form summaries of the posterior. To deal with this problem, one of the methods is to resort to simulation methods. Complicated computational problems also arise in the condition of

non-64

linear distributions. Bayesian simulation-based methods such as Markov chain Monte Carlo (MCMC) approach with Gibbs sampling and Metropolis-Hasting algorithm for Bayesian dynamic models.

Markov Chain Monte Carlo (MCMC) Method⁷⁵

The MCMC method is to generate stationary distribution underpinning the Markov chain model for which the parameter θ in the time j +1 follows the transition kernel K(θ | θ ^(j)), indicating that the distribution of parameters in the time j +1 is only dependent on the distribution of parameter θ in time j. According to the theory of Markov chain, the Markov chain under regular situations may reach the equilibrium and independent of initial distribution after a long run of transitions. This implies that

if a stationary distribution, S(θ), can be identified one can infer if θj comes from S(θ).

θ j + 1 also comes from S(θ).

Gibbs Sampler

The posterior distribution for the random effect model consists of all parameters involved, 1-5, 1-p, 1-p, b is denoted by

65 p (θ | y) = p(1-5, 1-p, 1-p, b| y) --- (1)

The Gibbs sampler is a method of estimating these marginal posterior distributions.

The procedure is as follows.

Suppose θ is k-dimension denoted by θ = {1-5, 1-p, 1-p, }

1) Assign starting value

θ ⁰ = {1 (0),..., p (0), τ⁽⁰⁾ }

2) Step 1: Update 1 by sampling

1(1) ∼ P{1 | 2(0),..., p(0), τ⁽⁰⁾}

3) Step 2: Update 2 by sampling

66

2(1) ∼ P{2 | 1(1), 3(0),..., p(0), τ⁽⁰⁾ }

4) Step 3: Update 3 by sampling

3(1) ∼ P{3|1(1), 2(1), 4(1)... , p(0), τ⁽⁰⁾ }

………..

……….

4) Step k: Update τ by sampling

τ⁽¹⁾ ∼ P{τ | 1(1), 2(1), 4(1)... , p(1) }

This completes one iteration of the Gibbs sampler, which yields a new realization for θ which is given by 1(1), 2(1), 3(1)... , p(1), τ⁽¹⁾. The above procedure is then repeated to get a second realization 1(2), 2(2), 3(2)... , p(2), τ⁽²⁾. This can be obtained by

sampling 1(2) from the full conditional probability of 1

67 P{1 | 2(1), 3(1),..., p(1), τ⁽¹⁾ }

Next, 2(2) is sampled from the full conditional probability:

P{2 | 1(2), 3(1),..., p(1), τ⁽¹⁾ }

This process continues until τ⁽²⁾ is sampled from the full conditional probability:

P{τ | 1(2), 2(2),... , p(2) }

This completes the second iteration.

This whole process is repeated for r iterations. The final iteration yields the

realization 1(r), 2(r),..., p(r), τ^(r). Geman and Geman show that, for very large values of r, the distribution of the sample 1(r), 2(r),..., p(r), τ^(r) becomes close to the marginal posterior distribution of 1, 2,..., p, τ.

In addition to marginal posterior density, disease prediction using the Bayesian

68 approach needs predictive distribution.

The fundamental idea of getting predictive distribution is that given the observed

data r one can integrate out relevant regression parameters to get predictive distribution, P(^new |), which is given by

p(𝜇^𝑛𝑒𝑤|𝜇) = ∫ p(𝜇^𝑛𝑒𝑤|𝜃, 𝜇)p(𝜃|𝜇)𝑑𝜃

However, like marginal posterior density, to integrate out relevant parameters requires

However, like marginal posterior density, to integrate out relevant parameters requires

在文檔中 醫療照護相關感染長期趨勢統計模式分析 (頁 76-0)