BATS - The Models

Bayesian Analysis for Time Series

Microarray Experiments

Though these sub-windows an user can  choose one of the three prior models implemented in BATS and can choose how to estimate the hyper-parameters.

We distinguish 3 Models

Model 1)

 

 

Model 2)

 

 

Model 3)

 

the marginal distribution of the noise Student T

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A user friendly software for Bayesian Analysis of Time Series Microarray Experiments.

Number of time-points

Number of replicates

Number of genes

 

Observed data

Gene    “true” functional profile

 

Noise. i.i.d.

We assume that genes are conditionally independent

 

And we place a prior on unknown parameters

 

                      Poisson truncated at         

 

Gene “non affected” by the treatment

Gene “affected” by the treatment

 

Gene’s specific variance (to be estimated from the data)

Prior probability of not being affected by the treatment (to be estimated from the data)

the marginal distribution of the noise is Gaussian

the marginal distribution of the noise Double-exponential

 

Noise model in BATS

Data model in BATS

Model + observed data

 

Prior Information

 

Posterior Distribution

 

For 1)-3) it is analitically known.

Moreover hyper-parameters can be estimated from the data

 

C. Angelini, D. De Canditiis, M. Mutarelli, M. Pensky. A Bayesian Approach to Estimation and Testing in Time-course Microarray Experiments, Statistical Applications in Genetics and Molecular Biology: vol 6 : Iss. 1, Article 24, (2007).

For a detailed description of the statistical method implemented in BATS the users are referred  to the following paper

Global parameters

 

 

 

Can be either estimated from the data or chosen by the users

Gene specific parameters                        can be estimated from the data as in the empirical Bayes approach

 
Fumetto 4: The number of terms in the functional expansion (to be estimated from the data) Fumetto 4: Coefficients in the functional expansion (mixture for modeling differentially expressed and not differentially expressed) Fumetto 4: Model on observed data (the likelihood) Fumetto 4: Modeling the variance as a random variable allows to deal with different types of noise Fumetto 4: BAYESIAN INFERENCE Fumetto 4: Estimation of parameters Fumetto 4: Second, consider that we use a (Bayesian) functional approach, so we recommend 5-6 different time points to be available. Fumetto 4: We assume the gene expression time profile being a smooth curve Fumetto 4: The expansion is in an orthogonal system on [0,T]. Legendre and Fourier bases are currently available. Suggested value is n/2 where n is the number of different time points Fumetto 4: We assume the noise to be i.i.d zero mean and finite variance Fumetto 4: Bayesian Decision in terms of BF

For models 1)-3) the posterior distribition can be  analitically evaluated and it is possible to test the statistical hypothesis

                                                  

 

 

                                                  

 

 

VS

Statistical decision (both selection and ranking) can be taken by looking at the Bayes Factors of each gene
Fumetto 4: First observe that this approach handles the so called “One sample” experimental design in the time course ….Be sure to be under this condition