Priors in Fabber¶

Each parameter in a Fabber model has a prior which describes our existing knowledge of the parameter’s value before we see any data.

A model must provide a set of priors for all of it’s parameters and in general it is not good practice to modify them - especially in light of knowledge derived from the data as this undermines the Bayesian principles.

Nevertheless there are a number of options that can be set for priors which can be used reasonably.

Spatial priors¶

A spatial prior applies spatial regularization to the parameter so that the spatial variation in it’s value is limited by the information present in the data. This has the effect of smoothing the parameter map in areas where there is not enough information in the data to justify more detail.

This can be beneficial since it produces smoother parameter maps with clearer structure but done in a principled way which treats each parameter independently and applies a degree of smoothing related to the information in the data.

A spatial prior would be defined as follows:

--PSP_byname1=myparam
--PSP_byname1_type=M

The first options specifies which named parameter any additional --PSP_byname1_* options refer to. The second option sets the prior type as M which is the most common type of spatial prior. Other supported types are m, P and p.

Spatial priors are normally only applied to a single parameter which is representative of the overall scale of the data. Since all the parameters are linked in the model, the result will generally be that all parameters are smoothed appropriately.

The following descriptions of the spatial prior types are based on Penny et al 2004.

Markov random field spatial prior (type M)¶

In this case the spatial matrix \(S^TS\) is defined as 1 for nearest neighbour voxels and 0 otherwise. The actual number of nearest neighbours is used so there is no bias at boundaries (e.g. at the surface of the volume)

Markov random field spatial prior without boundary correction (type m)¶

In this case the spatial matrix \(S^TS\) is defined as 1 for nearest neighbour voxels and 0 otherwise. The number of nearest neighbours is defined by the number of spatial dimensions (i.e. 8 for 3D spatial inference). This can cause bias at the image/mask boundaries hence spatial prior type M is generally used instead.

ARD priors¶

Automatic Relevance Detection (ARD) is a type of prior in which a parameter’s value can ‘collapse’ to zero if there is not sufficient information in the data to justify it having a nonzero value. This is useful for parameters which may be relevant only in certain parts of the data, for example an arterial signal component which only exists in large arteries.

An ARD prior would be defined as follows:

--PSP_byname1=myparam
--PSP_byname1_type=A

Image priors¶

An image prior is a prior whose mean value may be different at each voxel. For example if the tissue’s local T1 value is a model parameter, it may be useful to use a T1 map calculated by some independent means (e.g. VFA or MOLLI sequences) to provide the prior value at each voxel, while still allowing for the possibility of variation.

Image priors can be specified as follows:

--PSP_byname1=myparam
--PSP_byname1_type=I
--PSP_byname1_prec=100

Note that the precision can be specified, this controls how free the model is to vary the parameter. Choosing a high precision (e.g. 1e6) effectively makes the image ‘ground truth’. In this case we have given a precision of 100 which translates into a standard deviation of 0.1, allowing some variation in the inferred value but ensuring it will remain close to the image value.

Customizing priors¶

Warning

Customizing priors, especially in response to information from the data is opposed to the Bayesian methodology and should not be done unless you have good reason!

It is possible to override the model’s built-in priors and specify their mean and precision directly. This is done as follows:

--PSP_byname1=myparam
--PSP_byname1_mean=1.5
--PSP_byname1_prec=0.1

This would set the prior for parameter ‘myparam’ to have a mean of 1.5 and a precision of 0.1 (variance=10).

While this is normally discouraged, there are cases where it may be appropriate, for example when studying a population whose physiological parameters are known to differ systematically from the average, or for similar reasons to allow a parameter to vary more from the ‘standard’ prior value than the model normally allows.

Parameter transformations¶

Parameter transformations can be used when the default Gaussian distribution does not seem appropriate for a parameter. An example would be a parameter which for physical reasons cannot be negative. In this case we might guess that a log-normal distribution would be more appropriate. This can be handled in Fabber by telling the core inference engine to work with the log of the parameter value (which is distributed as a Gaussian) and transform it to the actual value when evaluating the model.

Warning

Transformations are normally built into the model where they are appropriate. Inappropriate transformations can lead to numerical instability and poor fitting.

Since transformations are transparent to the model they can be modified as follows:

--PSP_byname1=myparam
--PSP_byname1_trans=L

This sets the parameter named myparam to have a log-transform.

Prior mean/precision and transformations¶

A natural question is how should the prior mean and variance be modified when using a transformation. For example suppose we have a parameter representing a transit time and it’s normal prior has a mean of 1.3s and a precision of 5. Unfortunately this defines a Gaussian which has a significant probability of being negative, which is probably not physically reasonable.

We might choose to apply a log-transform to this parameter to avoid this problem. But what should the mean and variance of the underlying Gaussian distribution (i.e. the distribution of the log of the value) be.

We might naively assume that the same transform applies fir the mean, however this is not the case. If we choose \(log(1.3)\) as our mean we are modelling the prior as a log-normal distribution with a geometric mean of 1.3, which is subtly different.