lab8_answered.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulating Language, Lab 8, Convergence to the Prior"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This simulation implements a simplified version of the language model from Kirby, Dowman & Griffiths (2007) using an explicit agent-based simulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "from IPython.display import set_matplotlib_formats\n",
    "set_matplotlib_formats('svg', 'pdf')\n",
    "\n",
    "from math import log, log1p, exp\n",
    "from scipy.special import logsumexp\n",
    "\n",
    "from numpy import mean # This is a handy function that calculate the average of a list"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Following Kirby, Dowman & Griffiths (2007), we assume a language is made up of a set of *variables*, each of which can exist in a number of different *variant* forms. This is a rather general characterisation that actually applies well to a number of linguistic phenomena. For example, we can think of the variables as different syntactic categories, and the variants as word orders. Alternatively, the variables could be verb-meanings and the variants different realisations of the past tense, and so on. Agents will produce (and learn from) data which simply exemplifies which variant they have for a particular variable (with the possibility of noise on transmission). We will group languages into two classes: regular languages (where the same variant is used for all variables) and irregular languages (where more than one variant is used)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "variables = 2           # The number of different variables in the language\n",
    "variants = 2            # The number of different variants each variable can take"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Functions for dealing with log probabilities\n",
    "\n",
    "Here are our standard functions for dealing with logs, as before."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def log_subtract(x,y):\n",
    "    return x + log1p(-exp(y - x))\n",
    "\n",
    "def normalize_logprobs(logprobs):\n",
    "    logtotal = logsumexp(logprobs) #calculates the summed log probabilities\n",
    "    normedlogs = []\n",
    "    for logp in logprobs:\n",
    "        normedlogs.append(logp - logtotal) #normalise - subtracting in the log domain\n",
    "                                        #equivalent to dividing in the normal domain\n",
    "    return normedlogs\n",
    " \n",
    "def log_roulette_wheel(normedlogs):\n",
    "    r = log(random.random()) #generate a random number in [0,1), then convert to log\n",
    "    accumulator = normedlogs[0]\n",
    "    for i in range(len(normedlogs)):\n",
    "        if r < accumulator:\n",
    "            return i\n",
    "        accumulator = logsumexp([accumulator, normedlogs[i + 1]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We're also going to have a function that works a little like the `log_roulette_wheel` but instead always picks the most probable index (instead of picking indices proportional to their probability). We sometimes call this a \"winner take all\" function. While `log_roulette_wheel` can be used to implement *sampling*, `wta` can be used to implement *MAP* hypothesis selection. If there is more than one winner, then we pick randomly among them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def wta(probs):\n",
    "    maxprob = max(probs) # Find the maximum probability (works if these are logs or not)\n",
    "    candidates = []\n",
    "    for i in range(len(probs)):\n",
    "        if probs[i] == maxprob:\n",
    "            candidates.append(i) # Make a list of all the indices with that maximum probability\n",
    "    return random.choice(candidates)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Production of data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def produce(language, log_error_probability):\n",
    "    variable = random.randrange(len(language)) # Pick a variant to produce\n",
    "    correct_variant = language[variable]\n",
    "    if log(random.random()) > log_error_probability:\n",
    "        return variable, correct_variant # Return the variable, variant pair\n",
    "    else:\n",
    "        possible_error_variants = list(range(variants))\n",
    "        possible_error_variants.remove(correct_variant)\n",
    "        error_variant = random.choice(possible_error_variants)\n",
    "        return variable, error_variant"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function produce takes a language, selects a random variable, and produces the relevant variant from the language, with a certain probability of error.\n",
    "\n",
    "- By looking at this code, can you tell how languages are represented in the simulation?\n",
    "- Can you see how errors on production work?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1.   * The way I do this is to find the `language` variable, which is one of the input arguments on the first line here, and then see what the function does with it. \n",
    "    * It gets used straight away - in the second line - to pick a `variable`. The function in that line is `random.randrange()`: this function picks a random **integer** between `0` and some number `n`, and the only *argument* it needs is an integer, which tells it how many integers to choose from (so if you want a number between `0` and `n`, you use `n+1` s the argument, because Python blah blah. \n",
    "    * This means that `len(language)` must be an integer (which we know must be true because `len()` returns an integer), and that we're asking `random.randrange()` to pick from a number which is equal to the length of the list of `language`.\n",
    "    * So, this must mean that the number of possible `variable`s is the same as the length of `language`\n",
    "    * So it's a pretty good bet that every `language` is a list of `variable`s, where the *index* of each item in the list represents a *different* `variable`.\n",
    "    * So the index is the variant, but what do the values of the list mean? \n",
    "    * The next line tells us what values mean: `correct_variant` is picked using the index of `language` which we picked in the previous line. \n",
    "    * So, to summarise - we know that `language` is a list, and we know that the *index* of every item in the list represents a different `variable`, and that the *value* of every item in the list represents the `variant`.\n",
    "    \n",
    "2.  * Stepping through the code, we first see an if statement which checks some log probabilities to make a decision. Let's ignore their loggyness as they're not important here.\n",
    "    * We check whether `random.random()` - a random number between 0 and 1 - is bigger than (log)`error_probability`, which is usually quite small, say 0.001 for example.\n",
    "    * If `error_probability` is quite small, then `random.random` will almost always be bigger, and in this case the function just returns the intended `variable`/`variant` pair and the function exits.\n",
    "    * In the rare case that it's actually smaller than `error_probability`, then... there's been an error! Then we go into the `else` condition, which will tell us what to do.\n",
    "    * We see that `possible_error_variants` creates a list with the same length as the number of possible `variant`s, and goes on to `remove` the `correct_variant` \n",
    "    * This means we will pick an `error_variant` which can be anything *except* the intended variant, which makes sense.\n",
    "    * `random.choice` now comes out to play, which simply picks a member of a list at random. In this case, it will pick an `error_variant`, whih can be any of the variants except the intended one.\n",
    "    * Finally, the function exits and returns a pair with the correct `variable` but the *wrong* `variant`, and the sneaky error has entered the system. That's it!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Classifying languages\n",
    "\n",
    "In this language model, prior probability is determined by language class: regular languages differ from irregular languages in their prior probability, and ultimately we are interested in the proportion of our simulated population who use regular languages. We therefore need a function to take a language and classify it as regular or not - the function `regular` does this.\n",
    "\n",
    "At this point, I should highlight a bit of a problematic bit of terminology that is used in the field. We've already seen the word \"regular\" in this course with our beta-binomial simulations, referring to whether or not something is given a variable label or a regular one. We're not using \"regular\" in that sense here. Instead we're using it in the same way linguists refer to \"regular verbs\" or \"irregular verbs\", for example. Regular verbs in English are ones that use a general strategy for forming the past tense (for example: *walk/walked*) whereas irregular verbs are ones that use an idiosyncratic strategy for forming the past tense (for example: *go/went*). Our language is \"regular\" in this sense if it uses the same strategy for each of its meanings. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def regular(language):\n",
    "    first_variant = language[0]\n",
    "    for variant in language:\n",
    "        if variant != first_variant:\n",
    "            return False # The language can only be regular if every variant is the same as the first\n",
    "    return True"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The purpose of this function is to check whether a particular language is regular or not. The function uses the fact that we know that, in a completely regular language, every `variable` must be the same type of `variant`. \n",
    "* So all we need to do here is check that every item in the list is the same type, i.e. just a list of the same value repeated every time. \n",
    "* There are a lot of ways that we *could* do this, but the one used here is sensible: \n",
    "    * `first_variant` records the value of the first item in the `language` list by checking index `0`\n",
    "    * The for loop sets up a check for every `variant` in the `language` list\n",
    "    * If at any point the `variant` being checked is not the same as `first_variant`, that means that the language has more than one value in it.\n",
    "    * In code language, if `variant != first_variant`, then we immediately know that the language is not regular, and can exit with an authoritative `False`, as there is no need to check any more items.\n",
    "    * On the other hand, if we manage to get all te way to the end of the list without finding any non-conformist variants, we finish the for loop knowing that the language must be regular, and can exit with `True`, hurrah"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Bayesian bits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def logprior(language, log_bias):\n",
    "    if regular(language):\n",
    "        number_of_regular_languages = variants\n",
    "        return log_bias - log(number_of_regular_languages) #subtracting logs = dividing\n",
    "    else:\n",
    "        number_of_irregular_languages = variants ** variables - variants # the double star here means raise to the power\n",
    "                                                                         # e.g. 4 ** 2 is four squared\n",
    "        return log_subtract(0, log_bias) - log(number_of_irregular_languages)\n",
    "        # log(1) is 0, so log_subtract(0, bias) is equivalent to (1 - bias) in the\n",
    "        # non-log domain"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* We know that a regular language is just a repeated list of the same value.\n",
    "* This means there is only one possible regular language for each variant: if languages are represented by a list of 5 variants, and the only variants are `0` and `1`, then the possible regular languages would be `[0,0,0,0,0]` and `[1,1,1,1,1]`.\n",
    "* So the number of possible regular languages is just the same as the number of variants, as we see in `number_of_regular_languages = variants`\n",
    "* How many *irregular* languages are there? Well, first we can calculate the total number of possible languages overall...\n",
    "* This is just like counting outcomes of a coin-flip or a dice-roll: the number of possibilities for the first variable (coin-flip/dice-roll), times the numer of possibilities for the second, times etc etc\n",
    "* So say languages are represented by 5 `variable`s, and there are 3 `variant`s, the total number of possible languages is `3 * 3 * 3 * 3 * 3`\n",
    "* A quicker way of writing that is $3^5$, or in Python `3**5`\n",
    "* So the total number of possible langages is `variants ** variables`\n",
    "* BUT: that includes all regular **and** irregular languages.\n",
    "* But since we know that the number of *regular* languages is just the same as the number of `variants`, we can just subtract that from the total number, hence `number_of_irregular_languages = variants ** variables - variants`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `logprior` returns the prior probability (as a log probability) of a particular language. The strength of preference for regular languages is given as the second argument - if bias is over 0.5 (when converted back from a log probability), regular languages have higher prior probability.\n",
    "\n",
    "- Why are we dividing the bias by the number of regular and irregular languages in this function? Check you understand how these numbers are calculated.\n",
    "- How does this function differ from the prior from the Kirby, Dowman & Griffiths (2007) paper? (Hint: consider the case of more than two variables.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. * First, it's important to know that this function calculates the prior probability of a *particular* language\n",
    "    * Second, `log_bias` is the *overall* bias of a system towards regular languages *in general*, i.e. not a particular regular language but *all* of them lumped together.\n",
    "    * This means that all of the prior probability which is allocated towards regular languages, `log_bias`, needs to be split between them.\n",
    "    * So, to calculate the bias towards any *particular* regular language, we need to spread the total regularity bias equally between them.\n",
    "    * So, when we divide `log_bias` by the numer of regular langugages, we're just sharing the total *general* bias for regular languages between the actual *particular* ones, that's all.\n",
    "    * Oh no its a log! Don't worry, remember that logs are our friends, and subtracting in logworld is the same as dividing normal numbers.\n",
    "    * But what about irregular languages? Well, if the bias towards regular languages *overall*, *in general*, is $bias$, then the bias towards *irregular* languages must be $1 -bias$.\n",
    "    * Oh dear, it's the dreaded logs again - what do we do with `log_bias`? Don't worry, all `log_subtract(0, log_bias)` is doing is $1 - bias$ but off in logland (if you're worried about why it's using `0` and not `1`, it's because $log(1) = 0$, but just don't worry anyway).\n",
    "    * Finally, the `- log(number_of_irregular_languages)` bit is just doing the same thing as above, i.e. sharing out the general bias - but this time towards irregular, not regular languages, evenly across all the possible irregular languages.\n",
    "    \n",
    "2. The prior in the KDS paper is a lot fancier, but put simply, it measures *how* regular/irregular languages are, and the regularity bias then applies differently to specific languages depending on how regular they are. So, for example, if there is a strong bias for regularity, it will result in a strong bias for not just the completely regular languages, but also quite a strong preference for the *almost* regular languages; equally, it would have a very strong preference against maximally irregular languages (i.e. with equal numbers of each variant), and a little less strong for the slightly less irregular ones."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "def loglikelihood(data, language, log_error_probability):\n",
    "    loglikelihoods = []\n",
    "    logp_correct = log_subtract(0, log_error_probability) #probability of producing correct form\n",
    "    logp_incorrect = log_error_probability - log(variants - 1) #logprob of each incorrect variant\n",
    "    for utterance in data:\n",
    "        variable = utterance[0]\n",
    "        variant = utterance[1]\n",
    "        if variant == language[variable]:\n",
    "            loglikelihoods.append(logp_correct)\n",
    "        else:\n",
    "            loglikelihoods.append(logp_incorrect)\n",
    "    return sum(loglikelihoods) #summing log likelihoods = multiplying likelihoods"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `loglikelihood` takes a language and a list of data and works out the (log) likelihood of the data given the language. We allows some small probability (given by the third argument) that a speaker will produce the ‘wrong’ variant, i.e. a variant other than that specified by their language."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Learning\n",
    "\n",
    "Bayesian learners calculate the posterior probability of each language based on some data, then select a language (‘learn’) based on those posterior probabilities. `learn` implements this. As discussed in the lecture, there are two ways you could select a language based on the posterior probability distribution:\n",
    "- You could pick the best language - i.e. the language with the highest posterior probability. This is called MAP (“maximum a posteriori”) learning.\n",
    "- Alternatively, you could pick a language probabilistically based on its posterior probability, without necessarily going for the best one every time (e.g. if language 0 has twice the posterior probability of language 1, you are twice as likely to pick it). This is called sampling (for “sampling from the posterior distribution”).\n",
    "\n",
    "The next bits of code implements both these ways of learning, using the `wta` function to do MAP learning and using `log_roulette_wheel` to do sampling (from previous labs, which assumed learners sample from the posterior). `all_languages` enumerates all possible languages for expressing `variables` variables and `variants` variants using a cute recursive method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[0, 0], [0, 1], [0, 2], [1, 0], [1, 1], [1, 2], [2, 0], [2, 1], [2, 2]]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def all_languages(variables, variants):\n",
    "    if variables == 0:\n",
    "        return [[]] # The list of all languages with zero variables is just one language, and that's empty\n",
    "    else:\n",
    "        result = [] # If we are looking for a list of languages with more than zero variables, \n",
    "                    # then we'll need to build a list\n",
    "        smaller_langs = all_languages(variables - 1, variants) # Let's first find all the languages with one \n",
    "                                                               # fewer variables\n",
    "        for language in smaller_langs: # For each of these smaller languages, we're going to have to create a more\n",
    "                                       # complex language by adding each of the possible variants\n",
    "            for variant in range(variants):\n",
    "                result.append(language + [variant])\n",
    "        return result\n",
    "    \n",
    "all_languages(2,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Don’t worry too much if you can’t figure out how it works, but you might get an idea if you figure out what steps it would take when called with different arguments, like `all_languages(0, 2)`, `all_languages(1, 2)`, `all_languages(2, 2)`, `all_languages(0, 3)`, `all_languages(1, 3)` and so on. Try that out in the cell below. You should also now be able to see if you were right with your answer to the question above as to how languages are represented in this code."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Honestly, don't worry about how the recursive function works - they're cool and interesting in their own right, and recursion is a big topic in language, but this recursive function is just a fancy way of building the list of all possible languages for the model. If your interest has been piqued, look up 'recursive functions Python simple explanation' or something like that on Google!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, `learn` implements hypothesis selection."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "def learn(data, log_bias, log_error_probability, learning_type):\n",
    "    list_of_all_languages = all_languages(variables, variants) # uses the parameters we set above\n",
    "    list_of_posteriors = []\n",
    "    for language in list_of_all_languages:\n",
    "        this_language_posterior = loglikelihood(data, language, log_error_probability) + logprior(language, log_bias)\n",
    "        list_of_posteriors.append(this_language_posterior)\n",
    "    if learning_type == 'map':\n",
    "        map_language_index = wta(list_of_posteriors) # For MAP learning, we pick the best language\n",
    "        map_language = list_of_all_languages[map_language_index]\n",
    "        return map_language\n",
    "    if learning_type == 'sample':\n",
    "        normalized_posteriors = normalize_logprobs(list_of_posteriors)\n",
    "        sampled_language_index = log_roulette_wheel(normalized_posteriors) # For sampling, we use the roulette wheel\n",
    "        sampled_language = list_of_all_languages[sampled_language_index]\n",
    "        return sampled_language"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Iterated learning\n",
    "\n",
    "`iterate`, is the top-level function which actually runs the simulation. It starts with a random language from the set of possible languages, and then each generation a learner learns from data produced by the previous agent. This function returns a list indicating whether the language is regular or not each generation. For convenience we encode the regular languages as `1` in this list and `0` otherwise. It also returns a second list, showing what each language was per generation in case you want more details. (Generally, we're just going to be using the first list for plotting graphs etc.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "def iterate(generations, bottleneck, log_bias, log_error_probability, learning_type):\n",
    "    language = random.choice(all_languages(variables, variants))\n",
    "    if regular(language):\n",
    "        accumulator = [1]\n",
    "    else:\n",
    "        accumulator = [0]\n",
    "    language_accumulator = [language]\n",
    "    for generation in range(generations):\n",
    "        data = []\n",
    "        for i in range(bottleneck):\n",
    "            data.append(produce(language, log_error_probability))\n",
    "        language = learn(data, log_bias, log_error_probability, learning_type)\n",
    "        if regular(language):\n",
    "            accumulator.append(1)\n",
    "        else:\n",
    "            accumulator.append(0)\n",
    "        language_accumulator.append(language)\n",
    "    return accumulator, language_accumulator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, for example, to do an iterated learning simulation for 100 generations, a bottleneck of 5, with a bias in favour of regularity of 0.6, an error probability of 0.05, with sampling as the learning type, you'd use:\n",
    "\n",
    "```\n",
    "iterate(1000, 5, log(0.6), log(0.05), 'sample')\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Questions\n",
    "\n",
    "**Note:** To answer some of these questions, you're going to have to run quite long simulation runs (up to 100000 generations to get super accurate results). In general, you probably want to keep the bottleneck values between 1 and 10.\n",
    "\n",
    "1. Using the default parameters suggested above, can you demonstrate convergence to the prior? You may want to start with a line graph of the results to get a sense of what's going on in the simulation. You could also plot a histogram to get the overall proportion of regular languages in the whole run, but notice that this is just the same as the average of your whole list so you don't really need a graph for this! You can get the average directly by using, for example, `print(mean(data[0]))` if `data` is the result of your simulation run (remember the `iterate` function returns whether the language at the each generation is regular or not as the first part of the data it returns, which is why we use `data[0]`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Let's start out using a bottleneck of 5 and run a bunch of runs and plot them all on the same graph*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'regularity')"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
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     },
     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
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   ],
   "source": [
    "for i in range(10):\n",
    "    data = iterate(1000, 5, log(0.6), log(0.05), 'sample')\n",
    "    plt.plot(data[0])\n",
    "\n",
    "plt.xlabel(\"generations\")\n",
    "plt.ylabel(\"regularity\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Hmmm... that's not very useful, is it? Let's do a histogram of one run instead!*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'regularity')"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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   "source": [
    "data = iterate(1000, 5, log(0.6), log(0.05), 'sample')\n",
    "plt.hist(data[0])\n",
    "plt.xlabel(\"regularity\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Ok, that seems a bit more informative. Notice that it does look like the prior bias is reflected here - it's 0.6 by default, so we'd expect approximately 60 of our 100 generations should be regular. But it's a bit of a silly graph. It's really only reflecting a single number (the proportion of 1's in the list). So let's just find out the mean instead.*"
   ]
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  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6773226773226774\n"
     ]
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   ],
   "source": [
    "print(mean(data[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*This doesn't look quite right. Maybe we need to run the simulation for longer?*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6022397760223978\n"
     ]
    }
   ],
   "source": [
    "data = iterate(10000, 5, log(0.6), log(0.05), 'sample')\n",
    "print(mean(data[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*OK, that's better! Now we can try some different values for the prior. Remember that the prior is given as a log.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2 0.1852814718528147\n",
      "0.4 0.43495650434956507\n",
      "0.6 0.5929407059294071\n",
      "0.8 0.785021497850215\n"
     ]
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   ],
   "source": [
    "biases = [0.2, 0.4, 0.6, 0.8]\n",
    "for bias in biases:\n",
    "    print(bias, mean(iterate(10000, 5, log(bias), log(0.05), 'sample')[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*It looks like the average of a long run is the same as the prior.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. How does changing the error rate and bottleneck size affect the results for the sample learner? Make sure you run the simulation long enough to get repeatable results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0005 0.5367463253674633\n",
      "0.005 0.5613438656134386\n",
      "0.05 0.6062393760623938\n",
      "0.5 0.5996400359964004\n"
     ]
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   ],
   "source": [
    "errors = [0.0005, 0.005, 0.05, 0.5]\n",
    "for error in errors:\n",
    "    print(error, mean(iterate(10000, 5, log(0.6), log(error), 'sample')[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Our convergence to the prior result appears to be unaffected by errors. That's a bit odd. It seems like it really doesn't matter what the speaker is saying! Let's try that with different bottleneck sizes too...*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 0.5994400559944005\n",
      "2 0.5953404659534046\n",
      "3 0.5966403359664033\n",
      "4 0.5867413258674132\n",
      "5 0.5822417758224178\n",
      "6 0.6092390760923908\n",
      "7 0.601039896010399\n",
      "8 0.5866413358664133\n",
      "9 0.538046195380462\n",
      "10 0.6543345665433457\n"
     ]
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   ],
   "source": [
    "bottlenecks = range(1,11)\n",
    "for bottleneck in bottlenecks:\n",
    "    print(bottleneck, mean(iterate(10000, bottleneck, log(0.6), log(0.05), 'sample')[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Just like with the error term, the size of the bottleneck seems to be irrelevant for the convergence result! Are they doing anything at all? Let's look at indivdual runs instead...*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
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     "execution_count": 83,
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       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pa449b445b9\">\n",
       "   <rect height=\"217.44\" width=\"334.8\" x=\"43.78125\" y=\"7.2\"/>\n",
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       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(iterate(10000, 10, log(0.6), log(0.0005), 'sample')[0], label='low error, large bottleneck')\n",
    "plt.xlabel('generations')\n",
    "plt.ylabel('regularity')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f4aa48cca30>"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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   ],
   "source": [
    "plt.plot(iterate(10000, 10, log(0.6), log(0.05), 'sample')[0], label='medium error, large bottleneck')\n",
    "plt.xlabel('generations')\n",
    "plt.ylabel('regularity')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f4aa4923bb0>"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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wIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDIyNyA+PgpzdHJlYW0KeJw1TzuyAyEM6zmFLpAZjG1gz7OZVC/3b59ksg0S/kjy9ERHJl7myAis2fG2FhmIGfgWU/GvPe3DhOo9uIcI5eJCmGEknDXruJun48W/XeUz1sG7Db5ilhcEtjCT9ZXFmct2wVgaJ3FOshtj10RsY13r6RTWEUwoAyGd7TAlyBwVKX2yo4w5Ok7kiediqsUuv+9hfcGmMaLCHFcFT9BkUJY97yagHRf039WN30k0i14CMpFgYZ0k5s5ZTvjVa0fHUYsiMSekGeQyEdKcrmIKoQnFOjsKKhUFl+pzyt0+/2hdW00KZW5kc3RyZWFtCmVuZG9iagoyMSAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDI0NSA+PgpzdHJlYW0KeJxFULuNQzEM6z0FFwhg/Sx7nndIldu/PUpGcIUhWj+SWhKYiMBLDLGUb+JHRkE9C78XheIzxM8XhUHOhKRAnPUZEJl4htpGbuh2cM68wzOMOQIXxVpwptOZ9lzY5JwHJxDObZTxjEK6SVQVcVSfcUzxqrLPjdeBpbVss9OR7CGNhEtJJSaXflMq/7QpWyro2kUTsEjkgZNNNOEsP0OSYsyglFH3MLWO9HGykUd10MnZnDktmdnup+1MfA9YJplR5Smd5zI+J6nzXE597rMd0eSipVX7nP3ekZbyIrXbodXpVyVRmY3Vp5C4PP+Mn/H+A46gWT4KZW5kc3RyZWFtCmVuZG9iagoyMiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM5MiA+PgpzdHJlYW0KeJw9UktuBTEI288puECl8E1ynqne7t1/W5vMVKoKLwO2MZSXDKklP+qSiDNMfvVyXeJR8r1samfmIe4uNqb4WHJfuobYctGaYrFPHMkvyLRUWKFW3aND8YUoEw8ALeCBBeG+HP/xF6jB17CFcsN7ZAJgStRuQMZD0RlIWUERYfuRFeikUK9s4e8oIFfUrIWhdGKIDZYAKb6rDYmYqNmgh4SVkqod0vGMpPBbwV2JYVBbW9sEeGbQENnekY0RM+3RGXFZEWs/PemjUTK1URkPTWd88d0yUvPRFeik0sjdykNnz0InYCTmSZjncCPhnttBCzH0ca+WT2z3mClWkfAFO8oBA7393pKNz3vgLIxc2+xMJ/DRaaccE62+HmL9gz9sS5tcxyuHRRSovCgIftdBE3F8WMX3ZKNEd7QB1iMT1WglEAwSws7tMPJ4xnnZ3hW05vREaKNEHtSOET0ossXlnBWwp/yszbEcng8me2+0j5TMzKiEFdR2eqi2z2Md1Hee+/r8AS4AoRkKZW5kc3RyZWFtCmVuZG9iagoyMyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDkwID4+CnN0cmVhbQp4nE2NQRLAIAgD77wiT1BE0P90etL/X6vUDr3ATgKJFkWC9DVqSzDuuDIVa1ApmJSXwFUwXAva7qLK/jJJTJ2G03u3A4Oy8XGD0kn79nF6AKv9egbdD9IcIlgKZW5kc3RyZWFtCmVuZG9iagoyNCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDMzOCA+PgpzdHJlYW0KeJxFUktyxTAI2+cUXCAz5mfj87xOV+n9t5VwOt089AwICTI9ZUim3DaWZITkHPKlV2SI1ZCfRo5ExBDfKaHArvK5vJbEXMhuiUrxoR0/l6U3Ms2u0Kq3R6c2i0Y1KyPnIEOEelbozO5R22TD63Yh6TpTFodwLP9DBbKUdcoplARtQd/YI+hvFjwR3Aaz5nKzuUxu9b/uWwue1zpbsW0HQAmWc95gBgDEwwnaAMTc2t4WKSgfVbqKScKt8lwnO1C20Kp0vDeAGQcYOWDDkq0O12hvAMM+D/SiRsX2FaCoLCD+ztlmwd4xyUiwJ+YGTj1xOsWRcEk4xgJAiq3iFLrxHdjiLxeuiJrwCXU6ZU28wp7a4sdCkwjvUnEC8CIbbl0dRbVsT+cJtD8qkjNipB7E0QmR1JLOERSXBvXQGvu4iRmvjcTmnr7dP8I5n+v7Fxa4g+AKZW5kc3RyZWFtCmVuZG9iagoyNSAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDY4ID4+CnN0cmVhbQp4nDMyt1AwULA0ARKGFiYK5mYGCimGXEC+qYm5Qi4XSAzEygGzDIC0JZyCiFtCNEGUglgQpWYmZhBJOAMilwYAybQV5QplbmRzdHJlYW0KZW5kb2JqCjI2IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggODEgPj4Kc3RyZWFtCnicPcy7FYAwCAXQPlO8EUJ8gOzjsdL9W8FEG7h81QMdIRnUDW4dh7SsS3eTfep6tYmkyIDSU2pcGk6MqGl9qX1q4Lsb5kvViT/Nz+cDh8cZawplbmRzdHJlYW0KZW5kb2JqCjI3IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggNDUgPj4Kc3RyZWFtCnicMzK3UDBQsDQBEoYWJgrmZgYKKYZclhBWLhdMLAfMAtGWcAoingYAn30MtQplbmRzdHJlYW0KZW5kb2JqCjI4IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjU1ID4+CnN0cmVhbQp4nEWRS5IDIAhE956CI4D85DyZmlVy/+00mEw2dpeo/YRKI6YSLOcUeTD9yPLNZLbptRyrnY0CiiIUzOQq9FiB1Z0p4sy1RLX1sTJy3Okdg+IN566cVLK4UcY6qjoVOKbnyvqq7vy4LMq+I4cyBWzWOQ42cOW2YYwTo81Wd4f7RJCnk6mj4naQbPiDk8a+ytUVuE42++olGAeCfqEJTPJNoHWGQOPmKXpyCfbxcbvzQLC3vAmkbAjkyBCMDkG7Tq5/cev83v86w53n2gxXjnfxO0xru+MvMcmKuYBF7hTU8z0XresMHe/JmWNy031D51ywy91Bps/8H+v3D1CKZogKZW5kc3RyZWFtCmVuZG9iagoyOSAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE2MSA+PgpzdHJlYW0KeJxFkEsSwyAMQ/ecQkfwRwZ8nnS6Su+/rSFNs4CnsUAGdycEqbUFE9EFL21Lugs+WwnOxnjoNm41EuQEdYBWpONolFJ9ucVplXTxaDZzKwutEx1mDnqUoxmgEDoV3u2i5HKm7s75R3D1X/VHse6czcTAZOUOhGb1Ke58mx1RXd1kf9JjbtZrfxX2qrC0rKXlhNvOXTOgBO6pHO39BalzOoQKZW5kc3RyZWFtCmVuZG9iagozMCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDIxNCA+PgpzdHJlYW0KeJw9ULsRQzEI6z0FC+TOfO03z8uly/5tJJykQjZCEpSaTMmUhzrKkqwpTx0+S2KHvIflbmQ2JSpFL5OwJffQCvF9ieYU993VlrNDNJdoOX4LMyqqGx3TSzaacCoTuqDcwzP6DW10A1aHHrFbINCkYNe2IHLHDxgMwZkTiyIMSk0G/61y91Lc7z0cb6KIlHTwrvnl9MvPLbxOPY5Eur35imtxpjoKRHBGavKKdGHFsshDpNUENT0Da7UArt56+TdoR3QZgOwTieM0pRxD/9a4x+sDh4pS9AplbmRzdHJlYW0KZW5kb2JqCjMxIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggODAgPj4Kc3RyZWFtCnicRYy7DcAwCER7pmAEfiZmnyiVs38bIErccE+6e7g6EjJT3mGGhwSeDCyGU/EGmaNgNbhGUo2d7KOwbl91geZ6U6v19wcqT3Z2cT3Nyxn0CmVuZHN0cmVhbQplbmRvYmoKMzIgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA0OSA+PgpzdHJlYW0KeJwzNrRQMFAwNDAHkkaGQJaRiUKKIRdIAMTM5YIJ5oBZBkAaojgHriaHKw0AxugNJgplbmRzdHJlYW0KZW5kb2JqCjMzIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTU3ID4+CnN0cmVhbQp4nEWQuRFDMQhEc1VBCRKwCOqxx9F3/6kX+Uq0bwAth68lU6ofJyKm3Ndo9DB5Dp9NJVYs2Ca2kxpyGxZBSjGYeE4xq6O3oZmH1Ou4qKq4dWaV02nLysV/82hXM5M9wjXqJ/BN6PifPLSp6FugrwuUfUC1OJ1JUDF9r2KBo5x2fyKcGOA+GUeZKSNxYm4K7PcZAGa+V7jG4wXdATd5CmVuZHN0cmVhbQplbmRvYmoKMzQgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAzMzIgPj4Kc3RyZWFtCnicLVI5jiQxDMv9Cn5gAOvy8Z4eTNT7/3RJVQUFqmzLPORyw0QlfiyQ21Fr4tdGZqDC8K+rzIXvSNvIOohryEVcyZbCZ0Qs5DHEPMSC79v4GR75rMzJswfGL9n3GVbsqQnLQsaLM7TDKo7DKsixYOsiqnt4U6TDqSTY44v/PsVzF4IWviNowC/556sjeL6kRdo9Ztu0Ww+WaUeVFJaD7WnOy+RL6yxXx+P5INneFTtCaleAojB3xnkujjJtZURrYWeDpMbF9ubYj6UEXejGZaQ4AvmZKsIDSprMbKIg/sjpIacyEKau6Uont1EVd+rJXLO5vJ1JMlv3RYrNFM7rwpn1d5gyq807eZYTpU5F+Bl7tgQNnePq2Wu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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(iterate(10000, 7, log(0.6), log(0.005), 'sample')[0], label='low error, medium bottleneck')\n",
    "plt.xlabel('generations')\n",
    "plt.ylabel('regularity')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f4aa46209d0>"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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    "plt.plot(iterate(10000, 7, log(0.6), log(0.05), 'sample')[0], label='medium error, medium bottleneck')\n",
    "plt.xlabel('generations')\n",
    "plt.ylabel('regularity')\n",
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   "source": [
    "*What these results show is that although the iterated learning chain spends the same proption of time in the set of regular languages across all these different parameter settings and this is determined only by the prior bias (which is what we mean by \"convergence to the prior\"), the other parameters change how quickly the language changes from one state to another.*"
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
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   "source": [
    "3. Switch the learning type to MAP learning, and rerun the simulation. Can you show the amplification of prior bias that is shown in the paper?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2 0.1151884811518848\n",
      "0.4 0.13508649135086492\n",
      "0.6 0.8845115488451155\n",
      "0.8 0.8884111588841116\n"
     ]
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   "source": [
    "biases = [0.2, 0.4, 0.6, 0.8]\n",
    "for bias in biases:\n",
    "    print(bias, mean(iterate(10000, 5, log(bias), log(0.05), 'map')[0]))"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*This looks like amplification of the prior bias. For biases in favour of regular languages (i.e. over 0.5), we get **more** regular languages than we would expect. For biases favouring the non-regular languages, we get **fewer** regular languages than we would expect.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4. How is bias amplification affected by the bottleneck size? Can you run a range of simulations for different bottleneck sizes, find the means, and plot these nicely in a graph?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*I'm going to try this with longer runs, because the numbers above look a bit messy to me, so maybe we need a longer run to get a more accurate reading of the proportions.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "results = []\n",
    "bottlenecks = range(1,11)\n",
    "for bottleneck in bottlenecks:\n",
    "    results.append(mean(iterate(100000, bottleneck, log(0.6), log(0.05), 'map')[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Notice that I've got a separate cell for running the simulation and plotting the results. That's because it's taking several minutes to run the simulations now, and I don't want to have to redo them every time I want to tweak my graph!*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f4aa49366d0>"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "application/pdf": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(range(1, 11), results, label=\"bias=0.6\")\n",
    "plt.xlabel(\"bottleneck size\")\n",
    "plt.ylabel(\"proportion regular\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*We're clearly seeing bias amplifcation (all these numbers are higher than 0.6), and it's affected by the bottleneck (tighter bottlenecks lead to more amplification). Let's try different priors (this really does take a while to run).*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "biases = [0.2, 0.4, 0.6, 0.8]\n",
    "all_results = []\n",
    "for bias in biases:\n",
    "    results = []\n",
    "    for bottleneck in range(1, 11):\n",
    "        results.append(mean(iterate(100000, bottleneck, log(bias), log(0.05), 'map')[0]))\n",
    "    all_results.append(results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f4aa4531be0>"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    },
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      ],
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in range(len(all_results)):\n",
    "    plt.plot(range(1, 11), all_results[i], \n",
    "             label=\"bias=\" + str(biases[i])) # trick to convert number to a string\n",
    "    \n",
    "plt.xlabel(\"bottleneck size\")\n",
    "plt.ylabel(\"proportion regular\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Well, look at that! This is almost the opposite of the result for samplers. Now, **the strength of the bias doesn't matter** - just which side of 0.5 it is, and as we saw about the size of the bottleneck **does** matter... Just for completeness, let's add the results for samplers just to make it super clear! I'm also going to take a bit more care with colours so the graph is a bit more readable.*\n",
    "\n",
    "*N.B. This simulation is very slow, I assume because there's a lot more maths involved in doing the roulette wheel compared to just picking the best. To stop myself getting bored waiting, I put a print in the loop so I had something to watch...*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2 1\n",
      "0.2 2\n",
      "0.2 3\n",
      "0.2 4\n",
      "0.2 5\n",
      "0.2 6\n",
      "0.2 7\n",
      "0.2 8\n",
      "0.2 9\n",
      "0.2 10\n",
      "0.4 1\n",
      "0.4 2\n",
      "0.4 3\n",
      "0.4 4\n",
      "0.4 5\n",
      "0.4 6\n",
      "0.4 7\n",
      "0.4 8\n",
      "0.4 9\n",
      "0.4 10\n",
      "0.6 1\n",
      "0.6 2\n",
      "0.6 3\n",
      "0.6 4\n",
      "0.6 5\n",
      "0.6 6\n",
      "0.6 7\n",
      "0.6 8\n",
      "0.6 9\n",
      "0.6 10\n",
      "0.8 1\n",
      "0.8 2\n",
      "0.8 3\n",
      "0.8 4\n",
      "0.8 5\n",
      "0.8 6\n",
      "0.8 7\n",
      "0.8 8\n",
      "0.8 9\n",
      "0.8 10\n"
     ]
    }
   ],
   "source": [
    "biases = [0.2, 0.4, 0.6, 0.8]\n",
    "all_results2 = []\n",
    "for b in biases:\n",
    "    results = []\n",
    "    bias = b\n",
    "    for bottleneck in range(1, 11):\n",
    "        print(b, bottleneck)\n",
    "        results.append(mean(iterate(100000, bottleneck, log(bias), log(0.05), 'sample')[0]))\n",
    "    all_results2.append(results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f4aa41af580>"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "colours = ['red', 'orange', 'purple', 'blue']\n",
    "\n",
    "for i in range(len(all_results)):\n",
    "    plt.plot(range(1, 11), all_results[i], color = colours[i],\n",
    "             label=\"map, bias=\" + str(biases[i]))\n",
    "    \n",
    "for i in range(len(all_results2)):\n",
    "    plt.plot(range(1, 11), all_results2[i], color = colours[i], linestyle='dashed',\n",
    "             label=\"sampler, bias=\" + str(biases[i]))\n",
    "    \n",
    "plt.xlabel(\"bottleneck size\")\n",
    "plt.ylabel(\"proportion regular\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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