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{
 "worksheets": [
  {
   "cells": [
    {
     "metadata": {},
     "cell_type": "markdown",
     "source": "# Setting MNIW Priors #\n\nFor an autoregressive model with $k$ lags and parameters $(A,\\Sigma)$, for each time $t$ if we write the vector of lagged observations as $\\tilde{y} \\triangleq (y_{t-1}, y_{t-2}, \\ldots, y_{t-k})$ then we can write the generative model as\n$y | \\tilde{y},A,\\Sigma \\sim \\mathcal{N}(A \\tilde{y}, \\Sigma)$\nso that the likelihood is\n\n$$ p(y|\\tilde{y},A,\\Sigma) \\propto \\exp \\left\\{ -\\frac{1}{2}(y-A\\tilde{y})^T \\Sigma^{-1} (y-A\\tilde{y}) \\right\\}. $$\n\nThe natural conjugate prior over $(A,\\Sigma)$ is the Matrix-Normal-Inverse-Wishart (MNIW) with hyperparameters $(\\nu_0,S_0,M_0,K_0)$, where $(\\Sigma,A) \\sim \\text{MNIW}(\\nu_0,S_0,M_0,K_0)$ if\n\n$$ \\Sigma \\sim \\text{InvWishart}(\\nu_0,S_0) $$\n$$ A | \\Sigma \\sim \\text{MN}(M_0,\\Sigma,K_0). $$\n\nTherefore to set a prior over $(\\Sigma,A)$ we must choose the hyperparameters $(\\nu_0,S_0,M_0,K_0)$. 

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#!/usr/bin/env python
from __future__ import division
import sys
from dateutil.parser import parse
from dateutil.relativedelta import relativedelta
from datetime import datetime

from collections import OrderedDict

# from https://en.wikipedia.org/wiki/Year

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from __future__ import division
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib import rc, rcParams

np.random.seed(0)
rc('font',**{'family':'sans-serif','sans-serif':['Arial'],'size':16})

def myboxplot(boxdata,pointdata):

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from __future__ import division
import numpy as np
from matplotlib import pyplot as plt

import pyhsmm
from pyhsmm.util.text import progprint_xrange

obs_hypparams = dict(mu_0=np.zeros(2),sigma_0=np.eye(2),kappa_0=0.05,nu_0=4)

### generate data

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#include "bar.h"

void blah(int sz, double *x) {
    for (int i=0; i<sz; i++) {
        x[i] = (double) 2*i;
    }
}

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from __future__ import division
from sympy import MatrixSymbol, Matrix, latex, simplify
import baker

def SymmetricMatrix(name,n):
    J = Matrix(MatrixSymbol(name,n,n))
    for i in range(n):
        for j in range(i):
            J[i,j] = J[j,i]
    return J

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from __future__ import division
import baker
from pylab import *

def rand_psd(n):
    a = randn(n,n)
    return a.dot(a.T)

def ssor_split(J):
    # NOTE: J = L + D + U

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function A = grid(n,m)
    if ~exist('m','var'), m=n; end
    A = kron(eye(n),path(m)) + kron(path(n),eye(m));
end

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