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Cell[TextData[{
  "Data analysis for the paper \"An Extensive Mitochondrial Bottleneck Occurs \
During Oogenesis in a Non-Mammalian Vertebrate\" by Jonci N Wolff, Daniel J\n\
White, Michael Woodhams, Helen E White and Neil J Gemmell.\n\nThe data \
analysis has been done in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ". As this is an expensive commercial package, many will not have access to \
it or be familiar with it. I have assumed the reader is familiar with several \
computer programming languages, and I give additional explainations where \
such experience does not suffice to understand ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s syntax. (",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " can be used like a procedural language, but it is more friendly to being \
used in functional and logical (e.g. Prolog) programming idioms.)\n\n\
Predefine some variables to avoid warning messages when variable names are \
similar:"
}], "Text"],

Cell[BoxData[{
    \(\(Off[General::spell, General::spell1];\)\), "\[IndentingNewLine]", 
    \(\({rawData, \ mData, meData, eData, fData, mfData, efData, normal, 
        norm, \[Sigma]PDFmData, \[Sigma]PDFeData, \[Sigma]PDFfData, \
\[Sigma]PDFmeData, \[Sigma]PDFmfData, \[Sigma]LogPDFmData, \
\[Sigma]LogPDFmeData, \[Sigma]LogPDFmfData, \[Sigma]LogPDFefData, \
\[Sigma]LogPDFeData, \[Sigma]LogPDFfData, \[Sigma]LogPDFtmData, \
\[Sigma]PDFtmData, \[Sigma]LogPDFtmeData, \[Sigma]PDFtmeData, \
\[Sigma]LogPDFtmfData, \[Sigma]PDFtmfData, tmData, mean, median, mode, var, 
        varM, varC, vard, nMin, nMax, grid, logInterp3Dtmef, logInterp3Dtme, 
        logInterp3Dtmf, interp1Dtmef, interp1Dtme, interp1Dtmf, 
        logInterp3Dme, interp1Dme, logInterp3Dmf, interp1Dmf, mean, meanM, 
        meanC, tmeData, tmfData, xMin, xMax, yMin, 
        yMax};\)\), "\[IndentingNewLine]", 
    \(\(On[General::spell, General::spell1];\)\)}], "Input"],

Cell["\<\
Turn off some numerical warnings which will otherwise be obtrusive:\
\>", "Text"],

Cell[BoxData[
    \(\(Off[NIntegrate::ncvb, NIntegrate::slwcon];\)\)], "Input"],

Cell["\<\
This is the raw data. Each individual has a list of its name followed by its \
heteroplasmy measurements in parts-per-thousand. Each family is a list of \
individual lists, with the mother coming first. The whole stricture is a list \
of families.\
\>", "Text"],

Cell[BoxData[
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              216}, {"\<e18\>", 224}, {"\<e19\>", 310}, {"\<e20\>", 
              214}, {"\<e21\>"}, {"\<e22\>"}, {"\<e23\>"}, {"\<e24\>"}, \
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{"\<f1 \>", 248, 279, 278}, {"\<f2 \>", 408}, {"\<f3 \>", 278}, {"\<f4 \>", 
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              294}, {"\<f16\>", 347}, {"\<f17\>", 359}, {"\<f18\>", 
              357}, {"\<f19\>", 278}, {"\<f20\>", 389}}};\)\)], "Input"],

Cell["\<\
Now we strip of the names to leave just numbers in 'data',  convert \
parts-per-thousand to a fraction, and make some subsets: mData is mothers \
only, eData eggs only, fData fry only, meData mothers and eggs, mfData \
mothers and fry.\
\>", "Text"],

Cell[BoxData[{
    \(\(data\  = \ 
        Map[Rest, rawData, {2}]/1000. ;\)\), "\[IndentingNewLine]", 
    \(\(mData\  = \ 
        data\[LeftDoubleBracket]
          All, {1}\[RightDoubleBracket];\)\), "\[IndentingNewLine]", 
    \(\(eData\  = \ 
        data\[LeftDoubleBracket]All, 
          Range[2, 31]\[RightDoubleBracket];\)\), "\[IndentingNewLine]", 
    \(\(fData\  = \ 
        data\[LeftDoubleBracket]All, 
          Range[32, 51]\[RightDoubleBracket];\)\), "\[IndentingNewLine]", 
    \(\(meData\  = \ 
        data\[LeftDoubleBracket]All, 
          Range[31]\[RightDoubleBracket];\)\), "\[IndentingNewLine]", 
    \(\(mfData\  = \ 
        data\[LeftDoubleBracket]All, 
          Join[{1}, Range[32, 51]]\[RightDoubleBracket];\)\)}], "Input"],

Cell["For example, here's the data for family 1, mother and eggs:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(meData\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input"],

Cell[BoxData[
    \({{0.634`, 0.654`, 0.637`, 0.627`, 0.632`, 0.616`, 0.63`, 0.638`, 
        0.645`, 0.639`, 0.616`, 0.636`, 
        0.63`}, {}, {}, {0.597`}, {0.63`}, {0.654`}, {0.597`}, {0.578`, 
        0.611`}, {0.6900000000000001`}, {0.5`}, {0.608`, 0.597`, 
        0.589`}, {0.5740000000000001`, 0.5710000000000001`, 0.578`}, {0.669`, 
        0.637`, 0.654`}, {0.667`, 0.653`, 0.636`}, {0.77`, 0.758`, 
        0.772`}, {0.638`, 0.628`, 0.605`}, {0.68`, 0.648`, 0.649`}, {0.627`, 
        0.641`, 0.625`}, {0.5660000000000001`, 0.5640000000000001`, 
        0.5630000000000001`}, {0.638`, 0.624`, 0.616`}, {0.666`, 0.664`, 
        0.674`}, {0.651`, 0.629`, 0.615`}, {0.686`, 0.66`, 0.685`}, {0.686`, 
        0.67`, 0.656`}, {0.71`, 0.679`, 0.7000000000000001`}, {0.656`, 
        0.656`, 0.64`}, {0.725`, 0.704`, 0.708`}, {0.621`, 0.633`, 
        0.631`}, {0.672`, 0.653`, 0.649`}, {0.647`, 0.649`, 0.633`}, {0.634`, 
        0.612`, 0.621`}}\)], "Output"]
}, Open  ]],

Cell["This is the Gaussian/Normal distribution function", "Text"],

Cell[BoxData[
    \(\(normal[x_, \[Mu]_, var_] := 
        1/\((\ Sqrt[var\ 2\ Pi])\)*
          Exp[\(-\((x - \[Mu])\)^2\)/\((2  var)\)];\)\)], "Input"],

Cell["\<\
Now calculate the likelihood of a given measurement error. It is possible to \
do the following analytically, but just throwing numerical integrations at \
the problem is simpler to understand.\
\>", "Text"],

Cell["\<\
'likelihood\[Mu]\[Sigma]' is the likelihood of the data (a list of numbers) \
for the given \[Mu] and \[Sigma]. (We apply the 'normal' function to each \
element of 'data', then multiply the results.\
\>", "Text"],

Cell[BoxData[
    \(\(likelihood\[Mu]\[Sigma][\[Mu]_, \[Sigma]_, data_List] := 
        Times\  @@ \ Map[normal[#, \[Mu], \[Sigma]^2] &, data];\)\)], "Input"],

Cell["\<\
This function finds the log likelihood for a value of \[Sigma] given a list \
of measurements from a single individual. We eliminate \[Mu] by integrating \
over it. (In principle we integrate from -\[Infinity] to +\[Infinity], but I \
omit the range where the function is known to be very small.)\
\>", "Text"],

Cell[BoxData[
    \(\(logLikelihood\[Sigma]1[\[Sigma]_, data_List] := 
        If[Length[data] < 2, 0, 
          Log[NIntegrate[
              likelihood\[Mu]\[Sigma][\[Mu], \[Sigma], data], {\[Mu], 
                Min[data] - 5*\[Sigma], 
                Max[data] + 5*\[Sigma]}]]];\)\)], "Input"],

Cell["\<\
Now 'array' is a list of lists (data from a single family), and each list \
(data from an individual) has the same measurement error \[Sigma] but they \
all have different unknown means. The log likelihood of \[Sigma] is just sum \
of the log likelihoods for each individual:\
\>", "Text"],

Cell[BoxData[
    \(\(logLikelihood\[Sigma]2[\[Sigma]_, array_List] := 
        Plus @@ Map[logLikelihood\[Sigma]1[\[Sigma], #] &, 
            array];\)\)], "Input"],

Cell["\<\
Finally logLikelihood\[Sigma]3 takes a list of data from all families and \
applies logLikelihood\[Sigma]2 to each family and adds the results.\
\>", "Text"],

Cell[BoxData[
    \(\(logLikelihood\[Sigma]3[\[Sigma]_, array_List] := 
        Plus @@ Map[logLikelihood\[Sigma]2[\[Sigma], #] &, 
            array];\)\)], "Input"],

Cell["\<\
As an illustration,  here is a plot of the log likelihood if we combine all \
the data (i.e. assume all measurements have the same error):\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(Plot[
        logLikelihood\[Sigma]3[\[Sigma], data], {\[Sigma], 0.01, 
          0.05}];\)\)], "Input"],

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Cell["\<\
This is a bit slow to calculate, as it requires integration, so we evaluate \
it on some grid points and create an interpolating function over those grid \
points to save time.

This procedure takes an input function and returns two approximating \
functions: one approximates the input function, the other is the exponential \
of the input function normalized to integate to one. (I.e. if the input \
function is a log likelihood, the second approximating function is the \
corresponding posterior probability, assuming a flat prior.)\
\>", "Text"],

Cell[BoxData[
    \(\(interpolationFromLogLikelihood[f_, min_, max_] := 
        Module[\[IndentingNewLine]{data, logInterp, norm, 
            dataMax}, \[IndentingNewLine]\ 
          data = \ Table[{x, f[x]}, {x, min, 
                max, \((max - min)\)/50}]; \[IndentingNewLine]logInterp\  = \ 
            Interpolation[data]; \[IndentingNewLine] (*\ 
            Scale\ data\ to\ avoid\ over/
                underflow\ *) \[IndentingNewLine]dataMax\  = \ 
            Max[data\[LeftDoubleBracket]All, 
                2\[RightDoubleBracket]]; \[IndentingNewLine]norm\  = \ 
            NIntegrate[
              Exp[logInterp[x] - dataMax], {x, min, 
                max}]; \[IndentingNewLine]linInterp\  = \ 
            Interpolation[
              Map[{#\[LeftDoubleBracket]1\[RightDoubleBracket], 
                    Exp[#\[LeftDoubleBracket]2\[RightDoubleBracket] - 
                          dataMax]/norm} &, 
                data]]; \[IndentingNewLine]Return[{logInterp, 
              linInterp}];\[IndentingNewLine]];\)\)], "Input"],

Cell["\<\
And we create interpolating functions for our various subsets of the data. \
(This takes a few seconds to calculate. The warning message is not \
significant.)\
\>", "Text"],

Cell[BoxData[{
    \(min\[Sigma] = 0.005; \ max\[Sigma] = 0.035;\), "\[IndentingNewLine]", 
    \(\({\[Sigma]LogPDFallData, \[Sigma]PDFallData\ } = \ 
        interpolationFromLogLikelihood[logLikelihood\[Sigma]3[#, data] &, 
          min\[Sigma], max\[Sigma]];\)\), "\[IndentingNewLine]", 
    \(\({\[Sigma]LogPDFmData, \[Sigma]PDFmData}\  = \ 
        interpolationFromLogLikelihood[logLikelihood\[Sigma]3[#, mData] &, 
          min\[Sigma], max\[Sigma]];\)\), "\[IndentingNewLine]", 
    \(\({\[Sigma]LogPDFeData, \[Sigma]PDFeData}\  = \ 
        interpolationFromLogLikelihood[logLikelihood\[Sigma]3[#, eData] &, 
          min\[Sigma], max\[Sigma]];\)\), "\[IndentingNewLine]", 
    \(\({\[Sigma]LogPDFfData, \[Sigma]PDFfData}\  = \ 
        interpolationFromLogLikelihood[logLikelihood\[Sigma]3[#, fData] &, 
          min\[Sigma], max\[Sigma]];\)\), "\[IndentingNewLine]", 
    \(\({\[Sigma]LogPDFmeData, \[Sigma]PDFmeData}\  = \ 
        interpolationFromLogLikelihood[logLikelihood\[Sigma]3[#, meData] &, 
          min\[Sigma], max\[Sigma]];\)\), "\[IndentingNewLine]", 
    \(\({\[Sigma]LogPDFmfData, \[Sigma]PDFmfData}\  = \ 
        interpolationFromLogLikelihood[logLikelihood\[Sigma]3[#, mfData] &, 
          min\[Sigma], max\[Sigma]];\)\)}], "Input"],

Cell["For example, for all the data lumped together:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Plot[\[Sigma]LogPDFallData[x], {x, 0.005, 
          0.035}];\)\), "\[IndentingNewLine]", 
    \(\(Plot[\[Sigma]PDFallData[x], {x, 0.005, 0.035}, 
        PlotRange\  \[Rule] \ All];\)\), "\[IndentingNewLine]", 
    \(\)}], "Input"],

Cell[CellGroupData[{

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Cell["\<\
and we see that they are not (Peaks are fry, eggs, mothers from left to \
right.)\
\>", "Text"],

Cell["\<\
This function finds mean, median, mode, 95% confidence intervals on a \
probability density function. (Confidence interval is from 2.5th percentile \
to 97.5th percentile.)\
\>", "Text"],

Cell[BoxData[{
    \(\(stats[f_, min_, max_] := 
        Module[{answer, x, \[Mu]}, \[IndentingNewLine]Off[NIntegrate::nlim, 
            NIntegrate::slwcon, NIntegrate::ncvb, 
            FindMaximum::lstol]; \[IndentingNewLine]\[Mu] = 
            NIntegrate[
              x\ f[x], {x, min, 
                max}]; \[IndentingNewLine]answer = {{mean \[Rule] \[Mu]}, \
\[IndentingNewLine]FindRoot[
                NIntegrate[f[x], {x, min, median}] \[Equal] 
                  0.5, {median, \[Mu], min, 
                  max}], \[IndentingNewLine]\(FindMaximum[
                  f[mode], {mode, \[Mu], min, 
                    max}]\)\[LeftDoubleBracket]2\[RightDoubleBracket], \
\[IndentingNewLine]FindRoot[
                NIntegrate[f[x], {x, min, lower}] \[Equal] 
                  0.025, {lower, \[Mu], min, \ 
                  max}], \[IndentingNewLine]FindRoot[
                NIntegrate[f[x], {x, min, upper}] \[Equal] 
                  1 - 0.025, {upper, \[Mu], min, \ 
                  max}]\[IndentingNewLine]}; \[IndentingNewLine]On[
            NIntegrate::nlim, NIntegrate::slwcon, NIntegrate::ncvb, 
            FindMaximum::lstol]; \
\[IndentingNewLine]answer\[IndentingNewLine]];\)\), "\[IndentingNewLine]", 
    \(\(stats[f_InterpolatingFunction] := 
        stats[f, f\[LeftDoubleBracket]1, 1, 1\[RightDoubleBracket], 
          f\[LeftDoubleBracket]1, 1, 2\[RightDoubleBracket]];\)\)}], "Input"],

Cell["\<\
And here are those distribution statistics for mothers, eggs and fry:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(stats[\[Sigma]PDFmData]\), "\[IndentingNewLine]", 
    \(stats[\[Sigma]PDFeData]\), "\[IndentingNewLine]", 
    \(stats[\[Sigma]PDFfData]\)}], "Input"],

Cell[BoxData[
    \({{mean \[Rule] 0.01915203137573301`}, {median \[Rule] 
          0.019006119683073714`}, {mode \[Rule] 
          0.018736198395496187`}, {lower \[Rule] 
          0.01592590071922597`}, {upper \[Rule] 
          0.02322276222513306`}}\)], "Output"],

Cell[BoxData[
    \({{mean \[Rule] 0.016686347023614177`}, {median \[Rule] 
          0.01655508256444523`}, {mode \[Rule] 
          0.016315213355911337`}, {lower \[Rule] 
          0.013827607568595865`}, {upper \[Rule] 
          0.020310614714731702`}}\)], "Output"],

Cell[BoxData[
    \({{mean \[Rule] 0.01180280185969781`}, {median \[Rule] 
          0.011678493868741657`}, {mode \[Rule] 
          0.011451500260893666`}, {lower \[Rule] 
          0.009482016516623826`}, {upper \[Rule] 
          0.014873227842482472`}}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Now we can move on to estimating N, the bottleneck genome number. (I use \
'n' instead of 'N' here as 'N' means something to ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ".\n\nNotation:\n",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  " = bottleneck genome number \n",
  Cell[BoxData[
      \(TraditionalForm\`m\)]],
  " = the mother's actual heteroplasmy ratio\n",
  Cell[BoxData[
      \(TraditionalForm\`c\)]],
  " = the child's actual heteroplasmy ratio\n",
  Cell[BoxData[
      \(TraditionalForm\`M\)]],
  " = mother's measured heteroplasmy ratio (including measurement error)\n",
  Cell[BoxData[
      \(TraditionalForm\`C\)]],
  " = child's measured heteroplasmy ratio\n",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_M\)]],
  " = measurement error in mother\n",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_C\)]],
  "= measurement error in child\n",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_d\^2\)]],
  " = 'drift variance' = ",
  Cell[BoxData[
      \(TraditionalForm\`E[\((m - c)\)^2]\)]],
  ", measuring the expected intergenerational change in heteroplasmy ratio\n\
",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\^2\)]],
  " = 'total variance' =  ",
  Cell[BoxData[
      \(TraditionalForm\`E[\((M - C)\)^2]\)]],
  ", measuring the expected intergenerational change in measured heteroplasmy \
ratio\n",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\&^\^2\)]],
  "= estimate of total variance (etc.)\n\nBy our model, the child's number of \
mutant genomes will be drawn from a binomial distribution of size ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  ". From the binomial distribution, the drift variance is ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\[Sigma]\_d\^2\)\(=\)\)\)]],
  Cell[BoxData[
      \(TraditionalForm\`\(m(1 - m)\)/n\)]],
  ", and except for small ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  ", we can take it to be a normal distribution. Similarly, we assume \
measurement errors are normally distributed, with zero mean. Given mother and \
child measurement variances ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_M\^2\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_C\^2\)]],
  ", the difference between mother and child measurement is distributed with \
mean of zero and variance ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\^2 = \[Sigma]\_M\^2 + \[Sigma]\_C\^2 + \
\[Sigma]\_d\^2\)]],
  "(assuming independence between the measurement errors and the drift in \
genome number) . Hence we can calculate a likelihood of a ",
  Cell[BoxData[
      \(TraditionalForm\`\((M, C)\)\)]],
  " measurement pair for a given ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  " if we know (or assume) values for ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_M\^2\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_C\^2\)]],
  "."
}], "Text"],

Cell["\<\
This comes from equation 3 in the paper. <Check that equation number has not \
changed.> It takes the measurements from one mother and the measurements from \
one child (egg or fry).\
\>", "Text"],

Cell[BoxData[
    \(\(logLikelihoodDiff[n_, errM_, errC_, Mlist_List, Clist_List] := 
        Module[{meanM, meanC, vard, varM, varC}, \[IndentingNewLine]If[
            Length[Clist] \[Equal] 0\  || \ 
              Length[Mlist] \[Equal] 0, \[IndentingNewLine]Return[
              0], \[IndentingNewLine]meanM\  = \ 
              Mean[Mlist]; \[IndentingNewLine]meanC\  = \ 
              Mean[Clist]; \[IndentingNewLine]vard\  = \ 
              meanM\ \((1 - meanM)\)/n; \[IndentingNewLine]varM\  = \ 
              errM^2/Length[Mlist]; \[IndentingNewLine]varC\  = \ 
              errC^2/Length[Clist]; \[IndentingNewLine]Return[
              Log[normal[meanM - meanC, 0, 
                  varM + varC + 
                    vard]]]\[IndentingNewLine]]\[IndentingNewLine]];\)\)], \
"Input"],

Cell["\<\
This function adds up the log likelihoods for all children in a single \
family:\
\>", "Text"],

Cell[BoxData[
    \(\(logLikelihoodFamily[n_, errM_, errC_, 
          familyData_] := \[IndentingNewLine]Plus @@ 
          Table[logLikelihoodDiff[n, errM, errC, 
              familyData\[LeftDoubleBracket]1\[RightDoubleBracket], 
              familyData\[LeftDoubleBracket]i\[RightDoubleBracket]], {i, 2, 
              Length[familyData]}];\)\)], "Input"],

Cell["And this one adds the log likelihoods for each family:", "Text"],

Cell[BoxData[
    \(\(logLikelihood[n_, \ errM_, \ errC_, \ data_]\  := \ 
        Plus @@ Map[logLikelihoodFamily[n, errM, errC, #] &, 
            data];\)\)], "Input"],

Cell["\<\
For example, here is a plot of the log likelihoods of n if we assume 2% \
measurement error for both mother and child, and we combine both eggs and fry \
in a single analysis:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(nMin = 30; \ nMax = 150;\), "\[IndentingNewLine]", 
    \(\(Plot[logLikelihood[n, 0.02, 0.02, data], {n, nMin, nMax}, 
        PlotRange \[Rule] {{0, nMax}, All}];\)\)}], "Input"],

Cell[GraphicsData["PostScript", "\<\
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For comparison, see what happens if we assume 1% measurement error instead:\
\>", "Text"],

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Cell[TextData[{
  "Because less of the difference between mother and child measurements is \
attributable to measurement error, the genetic drift must be greater hence n \
must be smaller - the peak shifts from about 85 to about 75.\n\nWe now want \
to eliminate the measurement uncertainties. We form a three dimensional grid \
over the plausible values of ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  ",  ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_M\)]],
  " and ",
  Cell[BoxData[
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  ". At each point we evaluate the overall likelihood (likelihood of ",
  Cell[BoxData[
      \(TraditionalForm\`\((n, \[Sigma]\_M, \[Sigma]\_C)\)\)]],
  " times likelihood of ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_M\)]],
  " times likelihood of ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_C\)]],
  ".)  The likelihoods of  ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_M\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma]\_C\)]],
  " come from the interpolated functions we generated earlier. \n\nThis \
function returns the grid of log likelihoods:"
}], "Text"],

Cell[BoxData[
    \(\(logLikelihoodGrid[logErrMInterp_, logErrCInterp_, data_] := 
        Table[{n, errM, errC, 
            logLikelihood[n, errM, errC, data] + logErrMInterp[errM] + 
              logErrCInterp[errC]}, {n, nMin, 
            nMax, \((nMax - nMin)\)/40}, {errM, min\[Sigma], 
            max\[Sigma], \((max\[Sigma] - min\[Sigma])\)/40}, {errC, 
            min\[Sigma], 
            max\[Sigma], \((max\[Sigma] - min\[Sigma])\)/40}];\)\)], "Input"],

Cell["\<\
And this function creates a (3D) interpolating function using the grid \
points:\
\>", "Text"],

Cell[BoxData[
    \(interpolateLogLikelihood[logErrMInterp_, logErrCInterp_, data_] := 
      Module[{grid, peak}, \[IndentingNewLine]grid\  = \ 
          Flatten[logLikelihoodGrid[logErrMInterp, logErrCInterp, data], 
            2]; \[IndentingNewLine]peak\  = \ 
          Max[grid\[LeftDoubleBracket]All, 
              4\[RightDoubleBracket]]; \[IndentingNewLine]grid\  = \ 
          Map[# - {0, 0, 0, peak} &, grid]; \[IndentingNewLine]Return[
          Interpolation[grid]];\[IndentingNewLine]]\)], "Input"],

Cell["\<\
It takes about 10 minutes to calculate these. (These timings are on an Intel \
E7300 2.66GHz CPU, a recent mid-range processor.)
This is for the mother-plus-eggs data:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["And this is the mother-plus-fry data:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell["\<\
And now we can integrate over the nuisance variables: (Warning messages are \
because the integrand is very small over most of the range.)\
\>", "Text"],

Cell[BoxData[
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          Exp[logInterp3D[n, errM, errC]], {errM, min\[Sigma], 
            max\[Sigma]}, {errC, min\[Sigma], max\[Sigma]}];\)\)], "Input"],

Cell[CellGroupData[{

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    RowBox[{\(NIntegrate::"slwcon"\), \(\(:\)\(\ \)\), "\<\"Numerical \
integration converging too slowly; suspect one of the following: singularity, \
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WorkingPrecision. If your integrand is oscillatory try using the option \
Method->Oscillatory in NIntegrate. \
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WorkingPrecision. If your integrand is oscillatory try using the option \
Method->Oscillatory in NIntegrate. \
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integration converging too slowly; suspect one of the following: singularity, \
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WorkingPrecision. If your integrand is oscillatory try using the option \
Method->Oscillatory in NIntegrate. \
\\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\
\\\", ButtonFrame->None, ButtonData:>\\\"NIntegrate::slwcon\\\"]\\)\"\>"}]], \
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Cell[BoxData[
    RowBox[{\(General::"stop"\), \(\(:\)\(\ \)\), "\<\"Further output of \
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ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
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Here are the two posterior distributions, plus statistics on the \
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\>", "Text"],

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*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)