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refs #92 doc updats
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AnthonyLim23 committed Aug 8, 2024
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Showing 1 changed file with 3 additions and 5 deletions.
8 changes: 3 additions & 5 deletions docs/source/cf_methods.rst
Original file line number Diff line number Diff line change
Expand Up @@ -16,9 +16,7 @@ Bayesian inference is used to calculate the whole posterior probability distribu
The equation for the posterior probability can be written as

.. math::
:label: post
P(\underline{\theta}| D, M) = P(D | \underline{\theta}, M)\frac{P(D | \underline{\theta}, M)}{P(D | M),
P(\underline{\theta} | D, M) = P(D | \underline{\theta}, M)\frac{P(D | \underline{\theta}, M)}{P(D | M)},
where :math:`\underline{\theta}` is a vector of model parameters, :math:`M` is the model and :math:`D` is the data.
:math:`P(\underline{\theta} | M)` is the prior distribution and represents current knowledge of the system.
Expand Down Expand Up @@ -73,7 +71,7 @@ The probability of the data will be the same for all models, so by taking a rati
.. math::
:label: odds
O_{21} = \frac{P(M_2 | D)}{P(M_1 | D) = \frac{P(D | M_2)P(M_2)}{P(D | M_1)P(M_1)}
O_{21} = \frac{P(M_2 | D)}{P(M_1 | D)} = \frac{P(D | M_2)P(M_2)}{P(D | M_1)P(M_1)}
where :math:`0_{21}` is the odds factor for models two (:math:`M_2`) and one (:math:`M_1`).
Assuming that there is no prior knowledge then :math:`P(M_1) \approx P(M_2)`.
Expand All @@ -93,7 +91,7 @@ To evaluate the odds factor, the probability of the data given the model needs t
This is written as

.. math::
P(D | M) = \int_\omega d\undeline{\theta} \quad P(D| \underline{\theta}, M)P(\underline{\theta} | M)
P(D | M) = \int_\omega d\underline{\theta} \quad P(D| \underline{\theta}, M)P(\underline{\theta} | M)
where the integral over :math:`\omega` is over the available parameter space for :math:`\underline{\theta}`.

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