I cant get the step$\_$e to work properly as seen in the photo

def dice$\_$posterior$($sample$\_$draw: List$[$int$],$

die$\_$type$\_$counts: Tuple$[$int$],$

dice: Tuple$[$Die$])->$ float:

$""$ "Calculates the posterior probability of a type $1$ vs a type $2$ die,

based on the number of times each face appears in the draw, and the

relative numbers of type $1$ and type $2$ dice in the bag, as well as the

face probabilities for type $1$ and type $2$ dice. The single number returned

is the posterior probability of the Type $1$ die. Note: we expect a BagOfDice

object with only two dice.

$*"=$

Requiring only two dice with the sane number of faces simplifies the

problen a bit.

if len$($dice$)!=2$:

raise ValueError$(\text{'}$This code requires exactly $2$ dice'$)$

if dice$[0].$nun$\_$faces $!=$ dice$[1].$num$\_$faces:

raise ValueError$(\text{'}$This code requires two dice with the same number of faces'$)$

if len$($sample$\_$draw$)!=$ dice$[0].$num$\_$faces:

raise ValueError$(\text{'}$The sample draw is a list of observed counts for the $\backslash$

faces. Its length nust be equal to the number of faces $\backslash$

on the dice.$\text{'})$

yOUR CODE hERE. You may want to use your safe$\_$exponen$$ate$.$

prior$\_$die$\_1=$ die$\_$type$\_$counts$[0]/$ sum$($die$\_$type$\_$counts$)$

prior$\_$die$\_2=$ die$\_$type$\_$counts$[1]/$ sum$($die$\_$type$\_$counts$)$

$1$ikelihood$\_$die$\_1=$ np$.$prod$([$safe$\_$exponentiate$($dice$[0].$face$\_$probs$[$i$],$ sample$\_$draw$[$i$])$

for $1$ in range$($dice$[0].$num$\_$faces$)])$

likelihood$\_$die$\_2=$ np$.$prod$([$safe$\_$exponentiate$($dice$[1].$face$\_$probs$[1],$ sample$\_$draw$[[1])$

for $1$ in range$($dice$[1],$num$\_$faces$)])$

return $($likelihood$\_$die$\_1*$ prior$\_$die$\_1)/(($likelihood$\_$die$\_1*$ prior$\_$die$\_1)$

$(1$ikelihood$\_$die$\_2*$ prior$\_$die$\_2))$

def e$\_$step$($experiment$\_$data: List$[$NDArray$[$np$.$int$\_]],$

bag$\_$of$\_$dice: BagofDice$)->$ NDArray:

$-""$Performs the Expectation Step of the EM algorithm for the dice problem.

Given a set of sample rolls and a current estimate of the bag of

dice parameters, this function computes the expected counts of how

many times each face of each die was rolled, based on the current

dice parameters.

:paran experiment$\_$data: A list of numpy arrays. Each array has length equal

to the number of faces and records the number of times each face

was rolled for a given draw. The number of arrays is equal to the

number of draws in the data.$|$

:type experiment$\_$data: list of numpy arrays with integer entries

:paran bag$\_$of$\_$dice: A BagofDice object. This object stores the

parameters of the dice in the bag.

:type bag$\_$of$\_$dice: BagOfDice

:return: An array that is the same length as the number of dice in the bag.

Each entry in the array is an array of floats which represent the

expected number of times each face was rolled for the corresponding

die.

:rtype: np$.$array of np$.$arrays of floats

$-""$

$\backslash$ Initialize the expected counts object for each die

max$\_$number$\_$of$\_$faces $=$ max$([$len$($die$)$ for die in bag$\_$of$\_$dice.dice$])$

$\backslash$ Initialize expected$\_$counts to zero. It is a list of lists. The number

w$\}\backslash$mathrm$\{$ of inner lists is equal to the number of dice and the length of each

$*\}\backslash$mathrm$\{$ inner list is the number of faces of the die with the most faces.

expected$\_$counts $=$ np$.$zeros$(($len$($bag$\_$of$\_$dice$),$ max$\_$number$\_$of$\_$faces$))$

$\backslash$ Iterate over draws. For each draw, calculate the the posterior probability

that each die type was rolled on that draw by calling dice$\_$posterior.

Then combine the posterior for each die type with the observed counts for

the current draw to get the expected counts for each die type on this dram.

To get the total expected counts for each type, you sum the expected

$*\}\backslash$mathrm$\{$ counts for each type over all the draws.

PUT YOUR CODE HERE, FOLLOWING THE DIRECTIONS ABOVE

die$\_$type$\_$counts $=$ np$.$array$($bag$\_$of$\_$dice.die$\_$priors$)*$ len$($experiment$\_$data$)$

for dray in experiment$\_$data:

posteriors $=[$

dice$\_$posterior$($draw$,$ die$\_$type$\_$counts$)$

for die$\_$idx in range$($len$($bag$\_$of$\_$dice$))$

$]$

for die$\_$idx in range$($len$($bag$\_$of$\_$dice$))$:

expected$\_$counts$[$die$\_$idx$]+=$ posteriors$[$die$\_$idx$]*$ draw

return expected$\_$counts