Dummy

Baysig quick tour: Probability distributions

This is the second document in a quick tour of the Baysig programming language, following on from Baysig quick tour: Fundamentals.

Baysig has one elementary probability distribution built-in (unit) and many more defined in its library. We can plot the histogram approximately corresponding to the probability density functions for each distribution.

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distPlot unit

Further probability distributions can be built from this elementary one. Some are already defined in Baysig's standard library, such as the standard normal distribution unormal:

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distPlot (normal 0 1)

a uniform distribution with lower and upper bounds uniform 5.0 10.0

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7.83247163235966 5.849121783592068 9.03764552297947 7.489071284068998 9.703449644079303 6.879630194017578 9.083448501556052 5.97439347556196 9.498459001653469 7.597030057245174 6.674698372761416 5.190196099036885 5.43025898058672 5.90909982586957 5.265753268093279 7.283527339408995 9.592143380919506 5.599945016577827 6.658321213130448 8.15764770450947 9.946881789934562 5.515122877088171 6.944939141847412 9.288503738187371 5.073621543141459 7.319734882870627 5.062803333872651 6.776580241689594 5.037090359264307 8.06057027845191 5.307118002163972 6.014538624094459 7.309771086394267 6.987616345749148 7.28763318208224 8.341836272352802 5.473036083968598 7.2073862171283425 7.93748696519469 8.243172497778007 7.882731611114631 8.322461583493851 7.074353402032134 9.749327339173604 7.304214039422622 8.924314700747914 8.572439745434663 6.079291884340181 6.253358936229105 5.018260657093083 7.105067394931118 6.954637110921237 5.109197294830623 7.877799503709609 6.372352972713381 5.744923917982942 9.928486791888364 7.288158613324202 5.2545469126791104 5.50133128662159 9.389203681933974 9.4473437253496 7.912566974505914 6.802674173797172 9.489979453964619 6.912947682876361 5.206852729946108 8.591485957985624 9.852977130760667 8.124271037085975 7.540595891778496 9.276189518691085 9.926288955594945 9.464656289076586 7.193871487908501 5.819775192257896 8.93560413479237 7.400471708364895 8.564048825894494 9.357610556613096 8.692704243281089 8.578265458014389 8.333638578963846 5.582338591474652 9.85964975665781 8.710903240998952 8.188988731286718 6.4768802715644425 7.451576917071728 9.947159955032905 7.611783703931557 8.715524782593251 5.297523822006193 7.519789741495575 8.154451053858374 6.464097476905234 6.503551847185835 6.877675735596027 8.838399145481734 8.691812785969073 8.81374690188415 5.361146633597357 6.431942922396572 6.040448926165805 6.600102921156237 9.048705004166976 7.277366838482527 8.624740570929074 9.68534034436693 6.155478499400308 8.880650019012544 8.424464002816114 7.406432482508578 8.08017875605267 8.831294626889585 8.602336403821207 5.64650584143738 9.859833021149932 9.514938399548395 8.556034883613705 7.585168022798646 5.684643614963778 5.804625635688339 5.218946287692279 5.230979392125965 6.976946755772677 8.197102599823172 6.239982219701629 8.94426329386054 9.554043123394727 6.534623862601496 6.8968469630686675 9.013847498924902 8.76600743126264 5.946500040811137 6.23251064942044 5.003673986066988 6.390108983356066 9.531689355972151 6.5743939477425055 6.654462637764976 8.80613323971772 9.769779435538169 9.950597136037096 7.127245102805841 5.537851565875544 9.06944326801911 8.535458416104191 8.090663032627354 5.13115452489144 5.968314846038892 5.459691641260798 6.827781923773648 6.375659419311131 7.475887595786687 7.426235513395219 7.855867112021862 6.616104509824911 6.07119688323494 5.4118371173767 9.318685441817783 7.645683914340493 5.590955342769379 8.932681573539199 9.141462674397385 6.161490268111576 9.800231336802298 5.566691080007398 8.413886539268747 6.7424778912082495 7.1886437135256465 8.631994423603638 6.335772198237426 5.489451439466666 5.887551837574586 5.189155737214991 9.08753101925541 7.728544540640094 9.105991095794828 6.331105983427518 8.376233583170976 6.574823029007488 5.26125482871692 7.115264739169568 7.620311981909982 5.783336659842948 7.085587339728509 6.912231194300631 5.234589960946373 7.868443577985382 8.776747110247786 8.319997315602377 9.266219745452101 6.770556085223927 6.047599365945812 7.278472778219374 5.759771777260996 6.560517239049197 8.090674951228005 5.844158409359902 6.693823680210352 7.269360415467378 7.705110375002286 5.354453467380827 8.874376818741293 5.041157740643517 5.177437978605143 7.288001683142037 9.757954657487094 7.173850999021402 8.651998974069022 6.09200283444766 8.427472815017094 6.697036431100461 8.788901176021644 5.0166031997689515 5.418485305369479 5.5200788554549405 6.069634838744447 9.358954391060154 7.470174879410811 8.265057077693069 9.4955281760766 6.495777850853227 8.727778179740481 9.429747639091161 5.121774126338254 7.0074223300925045 8.525120666838513 5.919930999148621 6.924264856363708 5.67439280466046 8.5917049782797 5.007983938559886 8.092145237161812 7.025300738640432 9.3064545593938 9.625988358078988 7.556860729250326 8.69508729751573 6.246811282179973 8.089084694648413 5.081522575365734 5.918287132149164 7.399423680594056 9.746895128569308 6.772676347003475 8.803792682896319 9.135003919744632 6.722172351256519 8.084726536899042 5.584146630853448 7.784296017555588 6.653242427076503 9.963667140101006 8.840973623758408 8.280022517610346 9.370814859720603 9.622443551109741 9.493132338275466 7.514943178397358 6.501828935346992 9.734153227120208 9.474160040140655 5.594443676665994 7.223202096583136 7.508406569833962 5.974995444057275 8.135198377569951 8.271562237821417 8.338934742691679 8.387529942762928 9.815938259664907 5.155873199728288 5.4971803446526595 9.720843409821587 8.275474445725669 8.213559091839372 8.672244741737899 8.484626895162423 6.498373233829247 7.495489327119545 5.325340151396892 9.45199701340217 8.118653066483944 8.23180971393171 7.820481531745866 9.144991306621812 5.449845602374953 6.833440614126958 9.39371300730708 5.297195457456912 7.681514762267823 8.067633886371654 7.600063197920114 6.56295380009407 7.348371906519954 8.963894878738536 7.049411654483856 6.046670987514469 5.563740604688353 6.108581435659104 5.198775995569019 5.2020110551272545 9.647349635988324 5.471312704140161 8.4580910551497 8.184133123882145 9.577818033187143 8.732638673096112 8.30060046359445 6.645635803380431 9.165409413204856 7.818631071787941 9.954219489580538 9.677621648804664 8.56627281605412 9.149717906059166 5.691030494796818 5.101750751515326 6.055261645478955 7.8787268926107155 9.921606811676549 6.4621716182358035 7.571128400175629 6.1113747398162666 8.172473365713952 5.23291889932773 6.968887750284953 6.530009784393481 6.490532509398305 7.089205816641731 7.847419713588296 5.1335343881821105 5.233660094727218 9.645999671110417 5.300606975867554 7.273122504649946 6.906352894591715 8.965074701459397 7.601633762848012 9.774357543740038 5.4313589214420315 8.190455127784624 7.889708865641083 7.6628377143439925 6.235755823141838 9.054193801492032 6.716042616571309 8.03688959056439 7.289244215304322 9.41715322188501 9.27987311367874 8.523645129409536 7.246030694163878 9.830514575548126 9.032651015797782 8.999095912789716 9.076948154037108 8.788206743492793 6.948530299577055 6.023476924985525 5.86106571272506 8.665622793758665 5.82286674199227 7.954603044305602 9.303395252046965 5.926217884040954 8.58182973661037 5.490260094052707 6.376932857413062 5.412420114831165 6.675092296599199 9.600055654207257 5.484528701166745 5.772385409980171 8.182955396637077 9.035773371354624 7.138400184255652 8.378560362239034 7.826183295768263 8.45795809721061 6.171916581497508 5.6943826891235965 6.914594185518705 8.347153983392555 6.217089956909672 8.552834213141546 8.8392617923299 6.664875283992122 7.410399454508536 5.957358220116552 5.34408595071735 7.581059751701789 8.341921965671219 5.183439154686275 9.957935347910489 5.404154586234166 5.892179364080583 6.497751215123047 6.540120676285616 6.59635982513343 8.53696928558335 5.412770648350187 6.418461137548662 6.981615813444745 5.090061770773041 9.587225475019595 9.187454847662583 7.577128516877333 5.389684614583311 5.7517477603982545 8.632641675330849 8.714792473175281 8.102419599448893 9.878617259645512 7.6369589177512 7.789351936147863 6.023265123605557 9.2580240788539 8.7214459557369 6.074431295523986 8.881033544633778 5.968094581265024 6.918404812905566 9.137694391919695 6.584765572702691 6.664997992726276 8.292553021640646 7.505850071903749 9.610094403524439 9.563516802052856 7.753847786866558 5.779408372654442 8.178847296670742 7.900038518221127 7.56853349799489 5.803744859482102 9.442257837961023 6.844084216711131 8.497089711832135 9.38682685591905 9.693548553551908 5.584583184817942 6.87506126283553 8.552803387528472 7.604863859621078 5.430458881992488 8.034622583779186 5.087662092430438 6.99338366673174 9.270139268895178 9.3302655625926
distPlot $ uniform 5.0 10.0

and the gamma distribution, here gamma 3 2

12.222923004026335 3.99741160563868 4.781418509629383 5.9739330640742985 12.928800784309583 12.413034718400072 2.883421821212466 9.626890625915054 5.571256343637307 3.432569251161804 5.999950283843715 17.534801053737326 1.4008802804144618 2.1928323141079304 7.055876493660759 1.5730010101707066 8.085589233942395 5.71919680219519 4.910402510008614 7.647910100707925 10.116629912484834 12.65000366283666 1.619799659925008 1.7089291008749994 2.1096585321313026 10.049823777947402 3.8202273191276435 21.649986800806563 9.06238234913166 8.790430450288419 4.8986384002223495 2.684975367457059 1.8866082761787881 6.825524507768507 4.813572541051021 2.0794589134122283 5.630181228396037 3.1574922357124966 1.8924092592519957 5.40102909068103 3.1098947076248384 4.765149339368962 1.1487050624502688 5.967073618801742 7.272339374286032 6.9582498132098545 10.682171486111827 19.038374107678166 16.35461734006282 5.7525803940812 3.780116487875081 3.9842161895592154 4.724551281540654 7.2870014887368635 2.9217318065395586 7.622267251142656 11.750950467518088 8.578422101883419 3.700716587225693 9.166904823252242 6.427896476421301 6.683685585397994 3.6886780264335135 4.835081204878749 6.2684370941857335 4.148210692001619 6.752666739159076 4.980032746545164 2.5667623374424764 13.592056126560042 4.0063967723788965 5.449550175507788 4.304685502502846 7.876904063719349 3.645169548214942 7.0896821492407724 11.69913121454792 8.705950745312055 6.507463458777192 2.12465366629048 7.41427731760057 7.78334751107019 5.212902271070261 7.337346102918131 5.809933663706528 1.8476291231231214 4.3467114868819134 18.51972640258048 8.223517115755644 3.0176110465723394 1.0378425802426698 5.169093891627495 9.909955409262471 3.8194459443745097 3.9549766456411963 1.761669388547014 8.415849327442677 12.67992357745787 5.078534006394582 4.770075005916365 3.629932875730084 9.853611914220892 0.7901460256358455 10.153309955372311 4.457843299512483 5.789586401976807 6.423648496674628 15.4399867496481 8.263357238041541 4.376510460611856 4.027564358471777 8.136363318485706 1.5537954987020859 9.176906466987235 2.2885972429707615 6.592042415070177 9.268223403817828 7.2570179045052665 16.740719978542504 12.289746370234548 2.5546791992396756 6.322233390089902 6.905544960852713 5.300924763295889 8.688177359755045 5.78258334317422 8.212947661122355 7.113481893600907 3.842301028387381 14.515529680918167 8.584388854029505 6.91691769030721 5.461308647356237 8.146365857018122 6.903322118125953 7.599296163093888 2.3768126352425494 2.916058599926286 5.992801791913104 4.5536176092654905 6.282950776983281 11.584073025588276 9.936407231305568 8.068762352827687 1.9668147994708935 7.1567384405737124 3.598503280361034 8.545311004429037 5.349575016897692 5.603613327855303 2.3031611304103348 3.4462617744043467 8.405748216734633 6.368335037942492 7.993613069640482 9.279084127046442 4.770988370828097 5.026830974051804 2.8722143196113965 4.913412330344155 5.271010901221214 12.47193130467317 9.163973721257934 1.5553969562494068 8.952046831896647 8.800259655844663 10.533512491598541 5.470986465901622 8.794595159262709 1.592330302190213 5.147001286094456 6.486689147773626 2.6870504806541886 6.466906977623929 12.251522515264057 6.186553305775207 11.036142095205374 1.1556836216613506 7.226958341958033 3.921136225429686 8.067965688281912 3.780726829669236 13.387669476334946 7.972991163054022 8.583082515570752 3.9700531453541505 2.9114520064843217 10.223777419400863 7.960529826783425 6.548158733273959 5.76423635569553 5.022338792586894 2.74812879875438 12.838587989685232 3.2260070188893044 2.2185819388631605 6.090970872057668 6.672315903722634 5.574074816617821 3.458829741731175 7.537669736954681 2.717022838746688 5.340372145572911 3.9024374054117836 10.279225335098783 5.910213994460092 1.2048324503542112 6.854855254930602 5.588674175547145 4.385246554780766 5.098016427340923 4.239725076562527 3.1280063143745984 7.54688882908182 2.174617104729089 1.7196006397622026 4.089883841425291 11.950234322425636 2.6330366109990946 2.600013195488077 3.164440424503724 10.06882822831724 1.601167599605165 6.1600501477873415 7.099444496387112 4.507266033711307 14.66344369619706 1.9114186460083638 8.124787989014276 4.910792512850364 7.844961919541241 2.608512833911629 7.350406829239319 3.169854736873451 4.762923192146901 9.571578364174943 2.385366821234837 4.077596889633844 4.958048670773368 4.037873701262539 3.9218516471678493 4.341984524694496 3.743369439162509 8.356767182499727 6.253744902330781 6.941343602847271 3.187283384351366 4.104196802335979 0.5222145550968615 6.870961650731602 5.833206385309198 5.043729643769501 11.446196839657647 7.363918187985826 0.8094941397886812 13.329007009703554 5.4930223059144385 4.305593979581082 10.431065349979779 7.588600002810898 6.780680958179848 7.864207856142792 8.237968427982809 23.329610883862827 7.396698744318433 3.2886243225051794 2.0708403801825686 1.8268225598589711 2.429851608386273 6.7601628143619035 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13.482168546429378 9.578602841197664 24.50914366759753 3.4784274912950606 1.8396084218239173 1.7805836116821327 8.969810947545554 5.788404229933669 3.172294465593205 2.9908152179210887 6.100824901722232 1.1397808017314721 6.1302354783683315 6.369826794925263 6.743924791452747 2.555267563052274 3.8289555385953973 0.8457206431922382 3.9528800613043633 9.436056687141779 4.525742840883502 7.1965569529583435 3.4152479524398665 7.2027561735835635 6.662971162359478 4.189128079887789 5.3864020131375305 7.628901558701546 4.748001094498873 5.555272429505746 4.231151700260163 3.79223151133736 4.07200573877245 5.494948892481423 5.549101015404462 3.9772166373850917 1.1612883176175381 12.065126555051295 7.728033914897513 5.542230198028797 8.81070361408926 3.7930890583619363 7.817567891422238 7.688572039800406 9.06810395683814 3.8347558831616873 2.50871382183896 7.01475827724723 6.425940859051902 1.8262929842459141 10.696075537273732 6.29113283332056 6.23157434336334 3.3206740895693327 7.009613793442221 1.7511400158475816 4.001001945108008 11.45772725119102 7.3982388606372895 2.720876785042645 2.705268505245807 7.2130702927400225 5.227215590305468 2.557618768778511 1.2823498002857914 4.126455401559604 2.37833971107562 2.9909967263721224 4.868836139024449 2.1399332846984764 6.532732281172223 3.9392757049430243 6.698645353568804 2.5248656004153984 2.2171653363898454 6.643874328173202 4.155861549017633 2.860855036935423 2.827014008277284 6.191875661781584 3.384394860157937 9.988836108501115 12.735913474711701 7.657846775238483 6.987582042601397 2.5468340014460344 2.8138795988403746 5.454843249023128 4.498736733142542 1.1620984577558837 6.24778153445458 4.040922514107475 4.784567454481012 2.6754548703291463 10.938179110006068 10.035643431903795 3.913765617497342 9.79816070505987 7.014916150333741 4.78896203622616 10.990868465342546 11.871447689724274 7.089345864132349 4.38140264126991 8.304597580426924 8.06497457828027 8.068064305751246 2.404349839073385 2.9781478663155383 3.987732777494847 11.633872090227976 5.26378311874454 4.199497392762397 2.495604903492379 5.708263483117829 9.807625958622246 6.062304755209169 3.163068534584602 5.4798996316232405 7.850168585661843 13.302510625543327 5.2497228498526844 0.769308068445712 15.38136636930802 7.288957245714039 5.548560623559276 1.601110579484987 1.9750678465215805 4.574400026912975 6.171968580111671 8.841527151927895 6.818924405312637 9.820939829289525 9.356284870121645 8.255017810408253 8.223803917538238 3.8621135282906938 8.530219949291451 16.570485754834763 3.8650881691261016 9.413449885797712 1.845187199960706 10.819191840617169 2.692081045433313 4.295434973796057 7.267185431125804 4.358870836167034 3.4513933206971967 6.228837410312609 9.302973220456005 13.102763441602479 2.8531987755500783 3.8225191917358683 2.2131829301351162 3.042955927801407 8.633532920728301 6.450849904637853 3.269312602305903 4.070030915719242 7.55721435375304 6.214467995835539 15.031325827739453 6.104381827599188 4.836958707815005 10.165228166104866 5.0204469795839675 4.357723217938681 18.107861168557545 5.604572191734418 7.096190605883761 5.734327036106553 4.390587966579782 8.975289419121886 5.199225080303633 7.198877308339925 4.21583916764548 4.637765442051414 8.27532350190647 2.7003424939813585 3.0468831802167817 6.91650711686478 7.821702295601842 6.082260206290628 2.3054198043059477 6.186031891476853 3.159567536982734 8.627329889824422 3.712312151341604 6.777144642299833 7.068673485060309 9.788427742260417 5.4211311995692615 11.271239400985134 3.631818269439409 8.759502593794105 9.580171046833325 11.825071528976064 13.060826928778749 5.227798807225717 6.827071462783724 4.612313025443701 3.067934570097526 3.565284176465722 11.02383119905798 2.407186917539729 5.71018369879142 7.089122394572514 4.3311048850463845 1.047743471031371 6.323085220914575 3.7707447738404607 4.533286367028086 7.163860476571545 10.697897247771111 4.290318998853797 5.718533325181422 4.410709308489218 3.6404626227279806 2.8677448349438883 22.472251793049765 8.003435952285024 2.292188649091041 9.099290607176346 2.354796145758689 2.654125982958943 7.244454178193732 3.926762588534343 8.399385758141086 3.7375790851864696 8.150638887306544 5.045752793207654 11.162694193692696 7.681611131433787 2.9570115901708443 3.200358680811555 2.594604635308907 6.281434341253508 6.265563309006007 5.906384362522335 4.13723060547003 3.2086780477918566 1.4903675450062344 4.987664618310603 5.862733638288916 7.202073605808483 2.3784684430080647 5.814648101037786 8.30900774615713 10.104904260554136 4.170174342662338 7.078566634819168 13.772969629480096 3.2053898811330783 4.2488103086349 6.668031923016342 7.320843208378244 6.06247200286406 4.709835982859238 3.148405967876741 4.06530157470237 9.786142279777044 5.215460035153447 2.7147995735491453 3.627312957943224 15.315999361309617 16.56459593804775 2.4101151187722047 11.322034646582496 2.142985731371393 4.311205727065922 2.989097109867322 5.317792510905286 3.8945383036775945 8.086466526792936 5.39951976561739 0.7341844108728274 13.132864368619922 5.160130182410107 1.6567588335804366 4.587237500901723 4.030497192729761 13.677251435995682 5.44720884281484 6.756688395012844 6.361274808906954 7.325726372522256 3.310167252221062 3.5867290232169537 7.488877921865916 6.235616695685857 17.032342372506182 8.618834074045317 2.8944150592793783 8.9857714612802 8.54743528359974 9.680654498537697 8.566095433715894 4.82218876353979 9.450568769199364 5.068067830253814 7.062484765782156 4.560292267944075 1.134691865765538 6.3075917929275835 10.915402709096703 7.843404151275718 2.2697123632995373 20.921827068998034 5.397365072046327 6.078257292180656 4.287283979659527 7.4547832840023505 5.1150726712136985 5.292845788623856 2.348223135138337 4.720186377629068 9.596608557418367 2.9915720865418263 1.9967716383104617 1.5877206902075731 4.0245536826803665 2.8784016054048776 4.947292334424297 3.347720574395519 10.369374978228398 2.0705155665112494 5.075249978741153 11.1481138228947 4.267689041657825 1.8455955160544875 10.70376678262855 5.440630766944077 4.986163628504128 10.423176375889126 5.222274649713886 4.3658138469308785 4.9938933907591565 5.980489299448373 7.674535163110855 13.568583072156056 2.34686756475426 14.065398087280755 1.2159820929416696 2.8404509966851226 5.181638325569272 5.906786119408069 8.571022946550567 1.93938829490907 15.2689558861324 7.3177603821922705 3.5928706168663567 5.776295275973813 9.16612171355628 5.056272918318825 2.470170871020435 9.878049400490875 8.15334082043157 5.120614180131278 10.57771648779272 5.461327396323583 3.8510920581589088 4.239863979707411 2.2759992313583943 20.603303164401684 2.903961341024875 1.7016352226473068 4.507600042610536 12.17791227392302 6.372978485851133 1.8782745266945293 3.5855250597839907 9.10367089142639 3.6052506764584575 8.660320448676561 10.425023995966637 7.5231912462867 3.3597018906975027 5.050900428325056 1.486007985030796 5.112041246257748 1.8315225940667217 5.887112955396465 4.3949320691502844 4.064708140847957 7.613009446803156 4.1339493868788315 6.571551650665467 3.66602846996711 2.24529785278541 7.0790119569613426 13.680311528988206 2.9517855267466633 6.357326214725074 6.993648680211629 1.9918431538432335 2.6843276970988996 5.568255581073251 9.04853669799142 12.22033774507998 8.558756696440355 11.344095490435784 2.7377334296032663 2.220741795080138 4.695849014284407 1.5615409254780188 8.130339305509935 2.917096907075016 6.789634414951089 11.37505324143698 6.553211202154877 3.6505118421016567 3.0657553688780643 10.666752072887713 0.5539245227453624 6.383123538232258 4.054516754854098 9.945730067153438 3.774938350796103 2.203123500607143 0.8767172215536975 1.9784007195507138 3.426449793513947 7.611878961616052 6.374941427978056 0.8955337360419577 7.402924148867613 1.5321272998231898 3.7132730944648173 9.921593122051254 2.7954961982687987 3.330507840138735 5.418345276462898 4.460572322141222 4.812751973312109 12.769031967078053 6.5446621408097885 2.4203737078263 5.150987102235933 3.5000261391109353 4.681068084826274 1.8955217491519 10.493215354441102 12.54745496178867 4.872953927098047 5.354383317093238 5.250474190471758 10.11793932568606 7.374797985202186 4.299373584447129 5.571678007371173 11.230355106506863 4.4676977022869515 4.6866717882228786 6.904228464947948 8.355764470107177 2.646678679985725 4.973571793989828 6.693604712676991 12.221135752375881 2.6834792362075968 5.301845297863882 9.002746096720326 3.612970049853562 10.340647323046193 5.766183174655855 2.535317449577105 4.952083324126729 11.790205372146204 3.3521991240909874 8.947521911027458 3.372196293599938 3.855488792261007 22.796606618608497 4.561834280526331 1.4338143582780645 3.4830521813400037 4.558561067453938 4.8664340216216 13.335750398772413 4.716936248870711 10.92977313912132 4.513788888326982 6.796279840089302 6.988136782511629 4.790384298790409 9.189408310479275 2.8946904131470124 3.7737145748039946 4.752609213917 4.222346978664936 3.55072112680908 3.8048924466520164 5.4683070261846085 8.312486267714643 3.104372454832783 1.5889001541267453 7.039168584179482 5.566563652602818 4.0484707404687255 3.7185528505226113 3.0437960553382095 5.153072298431021 10.70403259549131 9.436958590625043 1.832733108540283 10.661353355439976 5.177520554220967 3.2370072501465863 8.765194225955115 2.5727785993219 5.375446281237061 6.175237228713474 1.7393911104912478 12.47168379588844 4.664961983011079 12.704822006524395 4.261476575807231 4.564243776693738 1.787106364487046 4.029246389992721 3.966463211230562 3.5466024781438623 5.185882976492673 5.166462497698954 3.9680957497087204 2.1428556729381016 2.93418516164715 2.7967674377638656 12.89463241995805 4.886054044332368 8.997261955685842 7.789034690569403 8.878489268149465 4.292492615409023 3.4517350538193057 5.621142674060524 2.717085750080103 9.88522740396247 1.8498684323221672 3.6845271386939524 3.0953731531560162 5.275689086050191 1.8622546718293036 2.5904569744020534 4.234767388542265 8.950162904027035 11.509068310744683 3.2646412799885245 4.725347635362867 4.427950704868165 5.926179215753046 10.102166396082218 4.742897919647852 7.19599251158022 2.5552948550061125 8.513208806491825 18.671806013728485 6.689813971526348 4.496780329551268 5.1500952375435745 4.121200329659102 6.070441678117477 12.348363933043103 7.246772954715379 5.407746244479897 12.88046896257151 7.138869672797051 3.8678412702708203 3.0092904969246956 3.8738963402069606 7.290873688149838 6.00105577700293 10.031174465766838 4.091155558645191 10.674233530573796 6.929393852631222 2.209399531602471 4.1005728580241785 2.9626838544009466 10.288237891448738 2.0770831396879226 7.03160920496827 11.334544182553431 2.884219074671032 5.89805204316231 9.721494971863105 5.7974605436278 11.638722521274861 13.986658667865608 4.217896175735966 6.191232610645113 15.729031448061711 4.876667994944284 4.588721081716464 7.372941572240493 8.47311480675194 3.983009646951758 3.3412117852174505 9.409421587942502 4.006401285442133 4.966875994389149 6.781062377682195 4.8926095507535425 4.737820940716963 4.562767971614599 9.190778957281747 3.533872692332649 6.879701364514096 4.476057912756059 7.352794856654417 3.397214526332569 2.0860163707093156 9.480807281728799 3.6536891445759876 6.179024227671605 6.87662022155045 3.786894910438332 6.0165160084664295 7.064389835540236 1.7137491985198037 2.688431124500429 3.492636008272941 9.302774085802481 3.2364839206546936 5.317118506402446 3.6024355348397052 3.840385389323461 12.675936701233802 11.754592063843049 4.528309525815035 7.540543800907747 4.91409016665834 3.6859107213453695 8.015218936680045 7.199283306023371 6.122117262679891 3.0466249113985553 7.235449954523867 3.275113600172656 2.0487487898941334 1.3514091800093235 12.538550873507521 6.239434277257105 3.6121733848293207 1.1094423080926954 1.6444219106536524 6.128534012981493 2.956393823522795 5.249334788599927 2.164798494777673 2.812248244376742 4.683979180308345 8.905866746901314 8.764799110819851 10.478059782003932 13.237122970754896 5.280990962680488 2.44434114066475 10.592674172207103 4.377289306349938 1.547576916450181 7.8374660305499635 4.225247917575828 5.739817137958611 11.20493256076185 4.643736144391277 10.227873773887257 8.086218752324926 9.323069450209598 6.276851737851958 9.340610890465989 4.236508592591354 1.4937497488021907 3.5793897459970627 3.1544885492756376 4.161474215450832 8.58314374672114 9.94779964386416 3.8758459543452943 1.5569451765677642 4.11239295810831 4.434189647860495 2.6659509677331155 5.0174343259132765 2.725741077136801 6.121012862859864 5.100679268657928 4.851514310873615 8.434825216607223 5.160873377320407 3.969525693184899 3.5715841016294387 6.802131895005532 6.688739581842441 5.076848250208715 4.497595709809737 4.744616676871801 8.276849618333571 2.244218085221305 5.5366349462231685 6.456793798668383 7.830159545093669 3.3101509770267796 7.332209447383624 1.368607462170503 5.91943484602106 8.223721344999012 6.115993910745702 1.1205219327036269 7.615920511705383 9.62347878284136 7.265008337669331 1.637410129192155 6.085160504901808 10.61101077503182 2.4925813521341986 4.4553432622609535 4.892768938955888 4.845450051369389 12.42066712672812 2.6889304826541407 16.669714763318723 2.603402669048081 11.629666724700147 11.655791176486957 4.6669952909036105 7.239753407301212 2.5198739373187062 3.53077984498281 12.206561795384832 2.5513773293325714 7.2815305293033505 4.251670431041788 5.85370902001959 0.8934943475608006 12.000878469507244 5.020856535968651 1.9549686822329497 3.2588989351257056 5.696846549033219 18.768084854979662 7.614080979798954 3.193736119384983 4.040824229990306 0.9001774118147872 9.919349862002848 6.8625984863542016 2.1890789656853142 5.1068895042231475 2.7788411133481863 9.864052733532594 7.41275855047255 2.456246894654561 3.322757844016265 3.8922632455358372 6.534812357890884 4.650103694195703 11.211164024195272 6.28575706681811 3.870140580478173 3.6972018628146555 7.506572977330334 2.8909590943087555 8.27614953005904 1.0880500924254384 3.47849620300524 1.4651090032428515 4.376220512915874 6.986618309712865 7.883647699530746 2.3784814245246304 3.720941898954115 6.2238840216669615 6.094532358834201 6.9109511269027415 4.676809725614736 4.818382936597278 7.9438940073340065 5.044159327092268 2.4222107044432133 7.965772902974722 10.561399947174205 3.259865031689511 1.529471283492219 4.505979832177174 7.360832660590966 8.42473830909917 3.7504618295926755 5.59315832079291 2.351753301700726 6.914474485107872 2.047665785965224 1.129660591555853 3.3003783332896894 3.584423276475573 7.249432160199032 4.432700315682816 15.66328817471007 8.742356715735413 7.340967443647708 5.0728151191340345 4.719641308073115 2.348544017175423 7.423517144540114 5.925275266114436 6.972773712137254
distPlot $ gamma 3 2

The real power of Baysig lies in the possibility of composing probability distributions. This is done by drawing a sample from one distribution, which can then be used as a parameter in another probability distribution. For instance, we could draw the mean of a normal distribution from a uniform distribution:

uniMean = prob
  mean ~ uniform 1 5
  normal mean 5
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distPlot uniMean

Now that uniMean has been defined, it too can be used to define further probability distributions.

uniMeanExp n = prob
   x ~ uniMean
   return (x^n)

This example illustrates a few new things: probability distributions can take arguments, just like functions. In fact, distributions are functions and are subject to the same flexibility: they can be stored in variables and passed as arguments to other functions or distributions. It may look as if these definitions introduce a special syntax which deviates from the functions, patterns and data types paradigm we introduced in the first section. And how to reconcile probabilistic programming with purely functional programming, in which a function applied to the same inputs must always give the same outputs? The last section in this tour demonstrates how probabilistic non-determinism can be elegantly wrapped up in purely functional programs and give a clear meaning to the notation introduced above.

uniMeanExp also uses the return function, which turns a deterministic expression into a probability distribution. It is most useful when expressing some function or combination of previously sampled values.

Can we see individual values drawn from a probability distribution? If we ask for the distribution to be displayed (more on how to do this in the section on reproducible research), Baysig will display the mean and standard deviation of the generated values.

uniMeanExp 2.0 ⇒ 14 ± 16

But we might also ask for a single value sampled from the generator to use in further calculation. In a sense, this is what happens inside a do-block. In x ~ D, x behaves like a particular value drawn from the distribution D. But the value of x can never "escape" from the distribution. This is an important point, because distributions in a sense represent uncertainty which must never be collapsed into a particular value. So when we use the operator sample to draw values from distributions, it is strictly speaking cheating. To prevent misuse of sample, it is not implemented as a function and can only be used at the top level (i.e. not inside a definition such as a function or a probability distribution). We also do not use equality to denote this assignment:

u <* sample (uniMeanExp 2.0)
u  ⇒ 7.1936

Similar syntax (<*) is used for other operations that are not purely functional, for instance loading data from disk.

One sample doesn't really tell much. The Baysig library has a number of utility functions for manipulating probability distributions. One such function is repeat which creates a new probability distribution over a list containing a integer number of draws from an existing distribution:

us <* sample (repeat 5 (uniMeanExp 2.0))
us ⇒ [5.0183, 5.2700, 19.5887, 3.1935, 4.5574]

The quick tour of Baysig continues in the document Baysig quick tour: Bayesian statistical inference