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refs #92 doc updats
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AnthonyLim23 committed Aug 8, 2024
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12 changes: 6 additions & 6 deletions docs/source/cf_methods.rst
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Expand Up @@ -16,7 +16,7 @@ Bayesian inference is used to calculate the whole posterior probability distribu
The equation for the posterior probability can be written as

.. math::
:name: eq:post
:label: post
P(\underline{\theta}| D, M) = P(D | \underline{\theta}, M)\frac{P(D | \underline{\theta}, M)}{P(D | M),
Expand All @@ -35,7 +35,7 @@ Essentially these methods will sample the PDF directly, allowing them to generat
Bayesian model selection use Bayes theorm to calculate the probability, :math:`P` of the data :math:`D` given the model :math:`M`

.. math::
:name: eq:int
:label: int
P(D|M) = \int_\Omega P(D| \underline{\theta}, M)P( \underline{\theta}|M)\mathrm{d\underline{\theta}}.
Expand All @@ -44,7 +44,7 @@ where the :math:`\underline{\theta}` are the parameters and the integral is over
quickBayes
----------

The quickBayes method makes a series of assumptions to reduce :ref:`the full PDF evaluation <eq:int>` to a single analytic equation.
The quickBayes method makes a series of assumptions to reduce :ref:`the full PDF evaluation <int>` to a single analytic equation.
The full theory is discussed here.
The key assumptions are:

Expand All @@ -71,13 +71,13 @@ where :math:`P(D | M)` is the probability of the data given the model, :math:`P(
The probability of the data will be the same for all models, so by taking a ratio the term can be removed

.. math::
:name: eq:odds
:label: eq: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)}
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)`.
Then equation :numref:`eq:odds` can be simplified to
Then equation :numref:`odds` can be simplified to

.. math::
O_{21} = \frac{P(D | M_2)}{P(D | M_1)},
Expand All @@ -93,7 +93,7 @@ To evaluate the odds factor, the probability of the data given the model needs t
This is written as

.. math::
:name: eq:P(D|M)
:label: P(D|M)
P(D | M) = \int_\omega d\undeline{\theta} \quad P(D| \underline{\theta}, M)P(\underline{\theta} | M)
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