Chapter 8 – Theoretical Biophysics  363

Thus, the P(A) is the prior of A, P(B) is the evidence, P(B|A) is the likelihood for B given A,

and P(A|B) is the posterior for A given B (i.e., the probability for this occurring given prior

knowledge of the probability distributions of A and B).

Bayes’ theorem provides the basis for Bayesian inference as a method for the statistical

testing of different hypotheses, such that P(A|B) is then a specific model/​hypothesis. Thus,

we can say that

(8.96) P

P

P

P

P

model|data

data model

model

data

data model

(

) =

⋅(

)

(

)

=

⋅

(

|

)

(

|

) P

P

P

model

data model

model

allmodel

(

)

(

)

∑

(

|

)

P(data|model) is an experimentally observed dataset, P(model) is a prior for the given model,

and P(model|data) is the resultant prediction for posterior probability that that model

accounts for these data. Thus, the higher the values of P(model|data), the more likely it is that

the model is a good one, and so different models can thus be compared with each other and

ranked to decide which one is the most sensible to use.

For using Bayesian inference to discriminate between different models of diffusion for

single-​particle tracking data, we can define the following:

1 The likelihood, P(d|w,M), where d represents the spatial tracking data from a given

biomolecule and w is some general parameters of a specific diffusion model M. This is

the probability distribution of the data for a given parameter from this model.

2 The prior, P(w|M). This is the initial probability distribution function to any condi­

tioning by the data; priors represent any initial estimate of the system, such as distri­

bution of the parameters or the expected order of magnitude.

3 The posterior, P(w|d,M). This is the distribution of the parameter following the con­

ditioning by the data.

4 The evidence, P(d|M). This acts as a normalization factor and is a portable unitless

quantity.

One such analytical method that employs this strategy in inferring modes of diffusion

from molecular tracking data is called “Bayesian ranking of diffusion” (BARD). This uses

two stages in statistical inference: first, parameter inference, and second, model selec­

tion. The first stage infers the posterior distributions about each model parameter, which

is defined as

(8.97)

P w d M

P d w M P w M

P d M

( | ,

)

( |

,

) ( |

)

( |

)

=

The second stage is model selection:

(8.98)

P M d

P d M P M

P d

(

| )

( |

)

=

(

)

( )

P(M|d) is a number that is the model posterior, or probability. P(M) is a number that is the

model prior (model priors are usually flat, indicating that the initial assumption is that all

models are expected equally), P(d|M) is a number that is the model likelihood, and P(d) is a

number that is a normalizing factor that accounts for all possible models. This now generates

the posterior (i.e., probability) for a specific model. Linking the two stages is the term P(d|M),

the model likelihood, which is also the normalization term in the first stage. Comparing

calculated normalized P(d|M) values for each independent model then allows each model

to be ranked from the finite set of diffusion models tried in terms of the likelihood that it

accounts for the observed experimental tracking data (e.g., the mean square displacement vs

τ values for a given tracked molecule).